Tüm alıştırma soruları

Soru 2061Soru →

In the $xy$ -plane, line $L_1$ passes through the points $(1, 1)$ and $(3, 5)$ . Line $L_2$ is defined by the equation $\frac{x}{4} + \frac{y}{7} = 2$ . If $(x, y)$ is the point of intersection of lines $L_1$ and $L_2$ , what is the value of $x + y$ ?

11

22

4

9.5

15

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

x + y

is 11.

The correct solution first determines the equation of the first line,

y = 2x - 1

, from its given points. Substituting this into the second line's equation

\frac{x}{4} + \frac{y}{7} = 2

and clearing fractions yields

x = 4

and

y = 7

. Adding these coordinates gives

4 + 7 = 11

.

Adım Adım Çözüm

1

Determine the slope and equation of line

L_1

.

The slope

m = \frac{5 - 1}{3 - 1} = 2

. Using point-slope form with

(1, 1)

,

y - 1 = 2(x - 1)

, which simplifies to

y = 2x - 1

.

Two points uniquely define a line, allowing us to express

y

in terms of

x

.

2

Substitute the expression for

y

into the equation for line

L_2

.

\frac{x}{4} + \frac{2x - 1}{7} = 2

.

At the intersection point, both equations share the exact same

(x, y)

values.

3

Clear the denominators by multiplying the equation by the least common multiple, 28.

7x + 4(2x - 1) = 56 \implies 7x + 8x - 4 = 56 \implies 15x = 60 \implies x = 4

.

Eliminating fractions simplifies solving for

x

.

4

Calculate the value of

y

and find

x + y

.

y = 2(4) - 1 = 7

, so

x + y = 4 + 7 = 11

.

The question specifically asks for the sum of the intersection coordinates.

Anahtar Kavram

Systems of Linear Equations and Line Intersections

Tahmini Süre:1m 30s

Soru 2062Soru →

Three consecutive integers $a$ , $b$ , and $c$ satisfy $a < b < c$ . If $a + b + c = -9$ and $a \cdot b \cdot c < 0$ , what is the value of $(-1)^a + (-1)^b + (-1)^c$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of the expression is 1.

The sum of three consecutive integers

a + b + c = 3b = -9

determines

b = -3

, making

a = -4

and

c = -2

. The product

(-4)(-3)(-2) = -24

is negative, confirming the given condition. Applying exponent sign rules, raising

-1

to an even integer power yields

1

, while raising

-1

to an odd integer power yields

-1

. Thus,

(-1)^{-4} = 1

,

(-1)^{-3} = -1

, and

(-1)^{-2} = 1

. Summing these three terms gives

1 + (-1) + 1 = 1

.

Adım Adım Çözüm

1

Find the values of integers

a

,

b

, and

c

.

a = -4

,

b = -3

,

c = -2

Three consecutive integers centered at

b

sum to

3b = -9

, so

b = -3

.

2

Check the sign condition of the product

a \cdot b \cdot c

.

(-4)(-3)(-2) = -24 < 0

The product of three negative numbers is negative.

3

Evaluate

(-1)^n

for each integer power.

(-1)^{-4} = 1

,

(-1)^{-3} = -1

,

(-1)^{-2} = 1

Negative one raised to an even integer power is 1; raised to an odd integer power is -1.

4

Sum the three evaluated terms.

1 + (-1) + 1 = 1

Addition of the resulting values.

Anahtar Kavram

Even-odd exponent rules for negative bases and sign rules for product of signed integers.

Soru 2063Soru →

The distribution of scores on a graduate admissions examination is normally distributed with a mean of $540$ and a standard deviation of $35$ . An applicant scored $610$ on this examination. If $800$ applicants scored higher than this applicant, which of the following is closest to the total number of applicants who took the examination?

8,000

16,000

32,000

50,000

64,000

Cevabı ve açıklamayı göster

Cevap: $32,000$

Cevap

32,000

The z-score for a score of

610

is calculated as

z = (610 - 540) / 35 = 2.0

. By the 68-95-99.7 empirical rule for normal distributions,

95\%

of all scores lie within

2

standard deviations of the mean. Because the distribution is symmetric, the remaining

5\%

is split equally between the two tails, meaning

2.5\%

of scores lie above

z = 2.0

. Given that

800

applicants scored higher than

610

, we set

0.025 N = 800

, which gives total applicants

N = 32,000

.

Adım Adım Çözüm

1

Calculate the z-score for a test score of

610

.

z = \frac{610 - 540}{35} = \frac{70}{35} = 2.0

The z-score measures how many standard deviations the score is above the mean.

2

Determine the proportion of scores that lie above

z = 2.0

using the 68-95-99.7 empirical rule.

Proportion

= \frac{100\% - 95\%}{2} = 2.5\% = 0.025

According to the empirical rule,

95\%

of values lie within

2

standard deviations of the mean (between

z = -2.0

and

z = 2.0

). Due to symmetry, half of the remaining

5\%

, or

2.5\%

, lies strictly above

z = 2.0

.

3

Set up an equation relating the number of higher-scoring applicants (

800

) to the total number of applicants (

N

).

0.025 \times N = 800 \implies N = \frac{800}{0.025} = 32,000

Since

2.5\%

of all applicants scored higher than

610

, dividing

800

by

0.025

yields the total population size.

Anahtar Kavram

Empirical Rule (68-95-99.7 Rule) and Standard Deviation Tail Areas

Tahmini Süre:1m 30s

Soru 2064Soru →

A specialty paint manufacturing facility creates a custom dye by mixing three liquid concentrates: Red, Yellow, and Blue. In the formulation, the ratio of the volume of Red concentrate to Yellow concentrate is $2 : 3$ , and the ratio of the volume of Yellow concentrate to Blue concentrate is $4 : 5$ . If a single storage vat contains $140\text{ liters}$ of this fully mixed custom dye, how many liters of Yellow concentrate are in the vat?

32\text{ liters}

48\text{ liters}

60\text{ liters}

70\text{ liters}

84\text{ liters}

Cevabı ve açıklamayı göster

Cevap: $48\text{ liters}$

Cevap

48\text{ liters}

of Yellow concentrate

To find the volume of Yellow concentrate, first combine the given ratios (Red : Yellow = 2 : 3 and Yellow : Blue = 4 : 5) into a unified ratio by expressing Yellow with a common term. Multiplying the first ratio by 4 gives Red : Yellow = 8 : 12, and multiplying the second ratio by 3 gives Yellow : Blue = 12 : 15. The combined ratio Red : Yellow : Blue is 8 : 12 : 15, yielding a total of 8 + 12 + 15 = 35 parts. Yellow accounts for 12 out of 35 parts. Multiplying 12/35 by the total volume of 140 liters yields (12/35) * 140 = 48 liters.

Adım Adım Çözüm

1

Unify the two separate ratios into a single three-part ratio.

Red : Yellow =

2 : 3 = 8 : 12

and Yellow : Blue =

4 : 5 = 12 : 15

. Thus, Red : Yellow : Blue =

8 : 12 : 15

.

Yellow is the common component in both ratios. Finding a common multiple for Yellow's ratio term (LCM of 3 and 4 is 12) allows us to express all three quantities in a unified ratio scale.

2

Calculate the total number of ratio parts and determine Yellow's fraction of the total mixture.

Total parts =

8 + 12 + 15 = 35

. Yellow's fraction of the total volume =

\frac{12}{35}

.

To convert a part-to-part ratio into a part-to-whole ratio, divide the target component's ratio parts by the sum of all ratio parts.

3

Multiply Yellow's fraction by the total volume of the mixture.

Yellow volume =

\frac{12}{35} \times 140 = 12 \times 4 = 48\text{ liters}

.

Multiplying the part-to-whole fraction by the total volume yields the exact volume of Yellow concentrate present.

Anahtar Kavram

Combining pairwise ratios into a unified three-part ratio to determine part-to-whole proportions.

Tahmini Süre:1m 30s

Soru 2065Soru →

Let $r$ and $s$ be integers such that $(-1)^{r+s} = -1$ and $r^2 s + r$ is an odd integer. Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

r - s

is an odd integer.

r + 2s

is an even integer.

r^2 + s^2

is an odd integer.

s(r + 1)

is an odd integer.

r^2 s + s

is an even integer.

Cevabı ve açıklamayı göster

Cevap: $r - s$ is an odd integer.; $r^2 + s^2$ is an odd integer.; $r^2 s + s$ is an even integer.

Cevap

The statements that must be true are '

r - s

is an odd integer', '

r^2 + s^2

is an odd integer', and '

r^2 s + s

is an even integer'.

From

(-1)^{r+s} = -1

, the sum

r+s

must be odd, meaning

r

and

s

have opposite parity. Factoring

r^2 s + r

yields

r(rs + 1) = \text{odd}

, which requires both

r

and

rs + 1

to be odd. Hence,

r

is odd, which forces

s

to be even. Testing the options shows that subtracting an even number from an odd number gives an odd number, adding the squares of an odd and an even number gives an odd number, and multiplying any integer by the even number

s

gives an even number.

Adım Adım Çözüm

1

Determine the parity of

r + s

from

(-1)^{r+s} = -1

r + s

is an odd integer

For

(-1)^k = -1

, the exponent

k

must be an odd integer. Therefore,

r + s

is odd, which implies that one variable is even and the other is odd.

2

Analyze the given expression

r^2 s + r

r

is odd and

s

is even

Factor

r^2 s + r

as

r(rs + 1)

. For the product of two integers to be odd, both factors must be odd. Thus,

r

must be odd. Since

r + s

is odd and

r

is odd,

s

must be even. (Verification: if

s

is even and

r

is odd,

rs + 1

is even + 1 = odd, so

r(rs + 1)

is odd

\times

odd = odd).

3

Evaluate each given statement using

r = \text{odd}

and

s = \text{even}

Statements '

r - s

is an odd integer', '

r^2 + s^2

is an odd integer', and '

r^2 s + s

is an even integer' are true.

1)

\text{odd} - \text{even} = \text{odd}

(True).
2)

\text{odd} + 2(\text{even}) = \text{odd} + \text{even} = \text{odd}

(False for even).
3)

(\text{odd})^2 + (\text{even})^2 = \text{odd} + \text{even} = \text{odd}

(True).
4)

(\text{even})(\text{odd} + 1) = \text{even} \times \text{even} = \text{even}

(False for odd).
5)

r^2 s + s = s(r^2 + 1) = \text{even} \times \text{even} = \text{even}

(True).

Anahtar Kavram

Parity rules for integer addition, multiplication, and exponents

Soru 2066Soru →

If

a

,

b

, and

c

are pairwise distinct real numbers, which of the following expressions is equivalent to

\frac{a^3(b - c) + b^3(c - a) + c^3(a - b)}{(a - b)(b - c)(c - a)}?

-(a + b + c)

a + b + c

a^2 + b^2 + c^2

ab + bc + ca

-(ab + bc + ca)

Cevabı ve açıklamayı göster

Cevap: $-(a + b + c)$

Cevap

-(a + b + c)

Factoring the numerator by using the Factor Theorem and cyclic symmetry reveals that

a^3(b - c) + b^3(c - a) + c^3(a - b) = -(a - b)(b - c)(c - a)(a + b + c)

. Dividing this by the denominator

(a - b)(b - c)(c - a)

cancels the pairwise difference terms, leaving

-(a + b + c)

.

Adım Adım Çözüm

1

Analyze the numerator for polynomial factors using cyclic symmetry

Let

P(a, b, c) = a^3(b - c) + b^3(c - a) + c^3(a - b)

. If

a = b

, then

P(b, b, c) = b^3(b - c) + b^3(c - b) + 0 = 0

. By the Factor Theorem,

(a - b)

is a factor. By cyclic symmetry,

(b - c)

and

(c - a)

are also factors.

Identifying linear factors reduces the polynomial simplification problem.

2

Determine the degree and form of the remaining factor

P(a, b, c)

is a homogeneous polynomial of degree 4, while

(a - b)(b - c)(c - a)

has degree 3. Therefore, the remaining factor must be a homogeneous symmetric polynomial of degree 1, which takes the form

k(a + b + c)

for some constant

k

.

Homogeneous degree properties dictate the algebraic structure of the quotient.

3

Find the constant

k

by substituting test values

Let

a = 0

,

b = 1

, and

c = 2

. Evaluating

P(0, 1, 2) = 0 + 1^3(2 - 0) + 2^3(0 - 1) = 2 - 8 = -6

. The factor product gives

(0 - 1)(1 - 2)(2 - 0) = (-1)(-1)(2) = 2

. Setting

2 \cdot k(0 + 1 + 2) = -6 \implies 6k = -6 \implies k = -1

.

Evaluating at convenient integer values determines the missing constant scalar.

4

Divide the factored numerator by the denominator

\frac{-(a - b)(b - c)(c - a)(a + b + c)}{(a - b)(b - c)(c - a)} = -(a + b + c)

.

Canceling common non-zero factors yields the simplified expression.

Anahtar Kavram

Factoring Cyclic Symmetric Polynomials

Soru 2067Soru →

The annual snowfall totals in a high-altitude meteorological district are normally distributed with a mean of $140$ inches and a standard deviation of $12$ inches. According to the 68–95–99.7 empirical rule for normal distributions, what percent of the years have an annual snowfall between $116$ inches and $152$ inches?

Cevabı ve açıklamayı göster

Cevap: 81.5

Cevap

81.5%

To find the percentage of data between

116

inches and

152

inches, first calculate the standard deviation distances (

z

-scores) from the mean of

140

inches.

116

inches is

24

inches below the mean, which corresponds to

z = -2

.

152

inches is

12

inches above the mean, which corresponds to

z = +1

. According to the 68–95–99.7 empirical rule,

95\%

of the data lies within

2

standard deviations of the mean, meaning

47.5\%

lies between

z = -2

and

z = 0

. Similarly,

68\%

of the data lies within

1

standard deviation of the mean, meaning

34\%

lies between

z = 0

and

z = +1

. Summing these two symmetric halves gives

47.5\% + 34\% = 81.5\%

.

Adım Adım Çözüm

1

Convert the boundary values (

116

inches and

152

inches) into standard z-scores.

z_{lower} = \frac{116 - 140}{12} = -2

and

z_{upper} = \frac{152 - 140}{12} = +1

Standardizing raw values into z-scores allows the application of standard normal distribution properties.

2

Apply the 68–95–99.7 empirical rule to split the area relative to the mean (

z = 0

).

Area from

z = -2

to

z = 0

is

47.5\%

; Area from

z = 0

to

z = +1

is

34\%

.

The normal curve is symmetrical around the mean. Thus,

95\%

between

-2\sigma

and

+2\sigma

yields

47.5\%

below the mean, and

68\%

between

-1\sigma

and

+1\sigma

yields

34\%

above the mean.

3

Sum the percentages of the two disjoint regions bounded by

z = -2

and

z = +1

.

47.5\% + 34\% = 81.5\%

Combining the area below the mean and the area above the mean gives the total percentage of values falling within the specified interval.

Anahtar Kavram

Normal distribution empirical rule (68–95–99.7 rule) with asymmetric standard deviation boundaries

Tahmini Süre:1m 30s

Soru 2068Soru →

A startup hired two freelance software developers, Developer A and Developer B, for a combined total of $45$ hours on a single project. Developer A charges $\$65$ per hour, and Developer B charges $\$80$ per hour. If the total amount paid to both developers was $\$3{}225$ , how many hours did Developer A work on the project?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

Developer A worked on the project for 25 hours.

Let

x

be the hours Developer A worked and

y

be the hours Developer B worked. From

x + y = 45

, we get

y = 45 - x

. Substituting into

65x + 80y = 3225

yields

65x + 80(45 - x) = 3225

. Expanding gives

65x + 3600 - 80x = 3225

, which simplifies to

-15x = -375

, so

x = 25

.

Adım Adım Çözüm

1

Set up a system of linear equations representing the total hours and total cost.

x + y = 45

and

65x + 80y = 3225

, where

x

is Developer A's hours and

y

is Developer B's hours.

Translating the verbal conditions into mathematical equations.

2

Express

y

in terms of

x

from the hours equation.

y = 45 - x

Prepares the linear system for substitution.

3

Substitute

y = 45 - x

into the total cost equation and solve for

x

.

65x + 80(45 - x) = 3225 \implies -15x = -375 \implies x = 25

Solves for the requested variable

x

directly.

Anahtar Kavram

Solving word problems using 2x2 systems of linear equations via substitution or elimination.

Soru 2069Soru →

A commercial bakery prepares a specialty grain blend using oats, wheat, and rye. Initially, the ratio of oats to wheat to rye by weight in the blend is $4 : 3 : 2$ . After $15\text{ kg}$ of oats and $15\text{ kg}$ of rye are added to the mixture while the amount of wheat remains unchanged, the ratio of oats to rye in the new blend becomes $3 : 2$ . Which of the following statements about the final grain blend must be true? Select all such statements.

Geçerli olan tümünü seçin

The weight of wheat in the final blend is

22.5\text{ kg}

.

The total weight of the final grain blend is

97.5\text{ kg}

.

The ratio of wheat to rye in the final grain blend is

3 : 4

.

Oats account for exactly

50\%

of the total weight of the final grain blend.

The total weight of the blend increased by

50\%

from its initial weight.

Cevabı ve açıklamayı göster

Cevap: The weight of wheat in the final blend is $22.5\text{ kg}$ .; The total weight of the final grain blend is $97.5\text{ kg}$ .; The ratio of wheat to rye in the final grain blend is $3 : 4$ .

Cevap

The weight of wheat in the final blend is

22.5\text{ kg}

, the total weight of the final grain blend is

97.5\text{ kg}

, and the ratio of wheat to rye in the final grain blend is

3 : 4

.

Solving the ratio proportion equation yields a multiplier constant of

x = 7.5

. Substituting

x = 7.5

yields final weights of

45\text{ kg}

for oats,

22.5\text{ kg}

for wheat, and

30\text{ kg}

for rye. The weight of wheat is indeed

22.5\text{ kg}

, the total final weight is

45 + 22.5 + 30 = 97.5\text{ kg}

, and the ratio of wheat to rye is

22.5 : 30 = 3 : 4

. Thus, these three statements are correct.

Adım Adım Çözüm

1

Define variables using the initial ratio.

Let the initial weight of oats be

4x

, wheat be

3x

, and rye be

2x

.

Expressing quantities in terms of a common ratio multiplier

x

ensures proportional relationships are preserved.

2

Set up an equation using the updated quantities and given new ratio.

\frac{4x + 15}{2x + 15} = \frac{3}{2}

15 kg was added to both oats and rye, establishing a new ratio of 3 to 2 between oats and rye.

3

Solve for the multiplier

x

.

2(4x + 15) = 3(2x + 15) \implies 8x + 30 = 6x + 45 \implies 2x = 15 \implies x = 7.5

.

Cross-multiplying yields the unique value for the ratio constant

x

.

4

Calculate the final weight of each component and the total weight.

Initial oats =

30\text{ kg}

, final oats =

45\text{ kg}

. Wheat (unchanged) =

22.5\text{ kg}

. Initial rye =

15\text{ kg}

, final rye =

30\text{ kg}

. Final total weight =

45 + 22.5 + 30 = 97.5\text{ kg}

.

Evaluating each component confirms all specific properties of the final mixture.

5

Verify each offered statement against calculated values.

Wheat weight is

22.5\text{ kg}

(True). Total weight is

97.5\text{ kg}

(True). Ratio of Wheat to Rye is

22.5 : 30 = 3 : 4

(True). Oats percentage is

\frac{45}{97.5} \approx 46.15\%

(False). Weight percent increase is

\frac{30}{67.5} \approx 44.44\%

(False).

Comparing calculated facts directly determines which statements must be true.

Anahtar Kavram

Ratio adjustment and algebraic formulation of multi-part mixture problems

Soru 2070Soru →

If $m$ and $n$ are negative integers such that $m < n$ , which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

2^m < 2^n

\left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n

m^2 < n^2

\sqrt{m^2} = m

2^{m+n} = 2^m + 2^n

Cevabı ve açıklamayı göster

Cevap: $2^m < 2^n$ ; \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n

Cevap

The statements

2^m < 2^n

and \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n must be true.

For the statement involving base

2

, since

2 > 1

, the exponential function is strictly increasing, so

m < n

guarantees

2^m < 2^n

. For the statement involving base

\frac{1}{2}

, since

0 < \frac{1}{2} < 1

, the function is strictly decreasing, meaning a smaller input

m

produces a larger output, so \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n.

Adım Adım Çözüm

1

Analyze the expression

2^m < 2^n

for base greater than 1

Since

b = 2 > 1

, raising

2

to a larger exponent yields a larger value. Because

m < n

,

2^m < 2^n

is always true.

Exponential functions with a base

b > 1

are strictly increasing.

2

Analyze the expression \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n for fractional base between 0 and 1

Since

b = \frac{1}{2}

is between

0

and

1

, raising

\frac{1}{2}

to a smaller exponent yields a larger value. Because

m < n

, \left(\frac{1}{2}\right)^m > \left(\frac{1}{2}\right)^n is always true.

Exponential functions with a base

0 < b < 1

are strictly decreasing.

3

Evaluate the remaining algebraic statements using counterexamples

For

m = -3

and

n = -2

:

m^2 = 9 > 4 = n^2

, so

m^2 < n^2

is false.

\sqrt{m^2} = \sqrt{9} = 3 \neq -3

, so

\sqrt{m^2} = m

is false.

2^{-5} = \frac{1}{32} \neq \frac{1}{8} + \frac{1}{4} = \frac{3}{8}

, so

2^{m+n} = 2^m + 2^n

is false.

A single counterexample disproves that a statement MUST be true.

Anahtar Kavram

Monotonicity of exponential functions and properties of square roots of negative bases

Soru 2071Soru →

For all real numbers $x$ and $y$ such that $|x| \neq |y|$ and $x^2 + y^2 \neq 0$ , consider the algebraic expression:

P(x, y) = \left(\frac{x^3 + y^3 - x^2 y - xy^2}{x^4 - y^4}\right) \cdot \left(\frac{x^3 + y^3}{x^2 - xy + y^2}\right)

Which of the following expressions are equivalent to $P(x, y)$ for all valid values of $x$ and $y$ ? Indicate all such expressions.

Geçerli olan tümünü seçin

\frac{x^4 - y^4}{(x^2 + y^2)^2}

1 - \frac{2y^2}{x^2 + y^2}

\frac{(x - y)^2}{x^2 + y^2}

\frac{x - y}{x + y}

\frac{x^2 + (-y)^2}{x^2 - y^2}

Cevabı ve açıklamayı göster

Cevap: $\frac{x^4 - y^4}{(x^2 + y^2)^2}$ ; $1 - \frac{2y^2}{x^2 + y^2}$

Cevap

The expressions equivalent to

P(x, y)

are

\frac{x^4 - y^4}{(x^2 + y^2)^2}

and

1 - \frac{2y^2}{x^2 + y^2}

.

The given expression simplifies to

\frac{x^2 - y^2}{x^2 + y^2}

. The option

\frac{x^4 - y^4}{(x^2 + y^2)^2}

simplifies directly to

\frac{x^2 - y^2}{x^2 + y^2}

after factoring the numerator. The option

1 - \frac{2y^2}{x^2 + y^2}

simplifies to

\frac{x^2 + y^2 - 2y^2}{x^2 + y^2} = \frac{x^2 - y^2}{x^2 + y^2}

when combined over a common denominator.

Adım Adım Çözüm

1

Factor the numerator of the first rational term by grouping terms.

x^3 + y^3 - x^2 y - xy^2 = x^2(x - y) - y^2(x - y) = (x^2 - y^2)(x - y) = (x - y)^2 (x + y)

Grouping allows rewriting four polynomial terms into product of linear/quadratic factors.

2

Factor the denominator of the first rational term as a difference of squares.

x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (x - y)(x + y)(x^2 + y^2)

Decomposing

x^4 - y^4

into simpler factors reveals common factors with the numerator.

3

Simplify the first rational term by canceling common factors

(x - y)(x + y)

.

\frac{(x - y)^2(x + y)}{(x - y)(x + y)(x^2 + y^2)} = \frac{x - y}{x^2 + y^2}

Canceling non-zero common factors simplifies the fraction.

4

Factor the numerator of the second rational term using the sum of cubes formula.

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

Applying the sum of cubes identity exposes the irreducible quadratic factor present in the denominator.

5

Simplify the second rational term and multiply the result by the simplified first term.

P(x, y) = \left(\frac{x - y}{x^2 + y^2}\right) \cdot (x + y) = \frac{(x - y)(x + y)}{x^2 + y^2} = \frac{x^2 - y^2}{x^2 + y^2}

Multiplying the simplified forms yields the simplest explicit representation of

P(x, y)

.

6

Verify equivalence of the options against

\frac{x^2 - y^2}{x^2 + y^2}

.

The option

\frac{x^4 - y^4}{(x^2 + y^2)^2} = \frac{(x^2 - y^2)(x^2 + y^2)}{(x^2 + y^2)^2} = \frac{x^2 - y^2}{x^2 + y^2}

, and the option

1 - \frac{2y^2}{x^2 + y^2} = \frac{x^2 + y^2 - 2y^2}{x^2 + y^2} = \frac{x^2 - y^2}{x^2 + y^2}

. Both match

P(x, y)

.

Transforming algebraic expressions under common denominators or factoring confirms equivalence.

Anahtar Kavram

Simplifying complex algebraic expressions using factoring by grouping, difference of squares, sum of cubes, and common denominator manipulation.

Soru 2072Soru →

Working alone at its constant rate, Printer $X$ can complete a printing job in $4\text{ hours}$ . Working alone at its constant rate, Printer $Y$ can complete the same printing job in $6\text{ hours}$ . Printer $X$ begins working on the job alone and works for $1\text{ hour}$ . At that point, Printer $Y$ joins Printer $X$ , and both printers work together at their respective constant rates until the job is completed. What is the total time, in hours, required to complete the entire job from start to finish?

1.8\text{ hours}

2.4\text{ hours}

2.8\text{ hours}

3.2\text{ hours}

3.5\text{ hours}

Cevabı ve açıklamayı göster

Cevap: $2.8\text{ hours}$

Cevap

2.8\text{ hours}

The correct answer is

2.8\text{ hours}

. Printer

X

works alone for

1\text{ hour}

at a rate of

\frac{1}{4}

job per hour, completing

\frac{1}{4}

of the total job. This leaves

\frac{3}{4}

of the job unfinished. When Printer

Y

joins, their combined rate is

\frac{1}{4} + \frac{1}{6} = \frac{5}{12}

job per hour. Dividing the remaining

\frac{3}{4}

of the job by

\frac{5}{12}

yields

\frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8\text{ hours}

for the joint work phase. Adding the initial

1\text{ hour}

of solo work yields a total of

1 + 1.8 = 2.8\text{ hours}

.

Adım Adım Çözüm

1

Calculate individual work rates and the portion of the job completed in the first hour

Printer

X

's rate is

\frac{1}{4}

job/hour and Printer

Y

's rate is

\frac{1}{6}

job/hour. In the first hour, Printer

X

completes

1 \times \frac{1}{4} = \frac{1}{4}

of the job.

Printer

X

works alone for the first hour before Printer

Y

joins.

2

Determine the remaining fraction of the job

Remaining job =

1 - \frac{1}{4} = \frac{3}{4}

.

The entire job is represented by

1

, so subtracting the completed portion yields the remaining portion.

3

Calculate the combined rate of both printers working together

Combined rate =

\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}

job/hour.

Rates add when workers or machines work simultaneously.

4

Calculate the time required for both printers to finish the remaining job and find total time

Time together =

\frac{3/4}{5/12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8\text{ hours}

. Total time =

1 + 1.8 = 2.8\text{ hours}

.

Time equals remaining work divided by combined rate, plus the

1\text{ hour}

already elapsed.

Anahtar Kavram

Combined Work Rates and Multi-Stage Work Problems

Alternatif Yöntem

Convert the job into arbitrary work units. Let the job equal

12\text{ units}

(the LCM of

4

and

6

). Printer

X

produces

12 / 4 = 3\text{ units/hour}

and Printer

Y

produces

12 / 6 = 2\text{ units/hour}

. In the first hour, Printer

X

produces

3\text{ units}

, leaving

12 - 3 = 9\text{ units}

. Working together, their combined rate is

3 + 2 = 5\text{ units/hour}

. The remaining

9\text{ units}

take

9 / 5 = 1.8\text{ hours}

. Total time is

1 + 1.8 = 2.8\text{ hours}

.

Tahmini Süre:1m 30s

Soru 2073Soru →

A bakery packages two types of gift baskets containing gourmet croissants and blueberry muffins. Basket X contains $4$ croissants and $3$ muffins, with a total production cost of $\$19.00$ . Basket Y contains $2$ croissants and $5$ muffins, with a total production cost of $\$16.50$ . Assuming the cost per croissant and the cost per muffin are constant across all baskets, what is the production cost, in dollars, of a single croissant?

Cevabı ve açıklamayı göster

Cevap: 3.25

Cevap

The production cost of a single croissant is 3.25 dollars.

Let

c

represent the cost of a croissant and

m

represent the cost of a muffin. The given situation translates to the system of equations

4c + 3m = 19.00

and

2c + 5m = 16.50

. Multiplying the second equation by

2

gives

4c + 10m = 33.00

. Subtracting

4c + 3m = 19.00

from

4c + 10m = 33.00

results in

7m = 14.00

, which gives

m = 2.00

. Substituting

m = 2.00

into

2c + 5(2.00) = 16.50

yields

2c + 10 = 16.50

, so

2c = 6.50

and

c = 3.25

. Thus, a single croissant costs $3.25 dollars.

Adım Adım Çözüm

1

Set up a system of linear equations

4c + 3m = 19.00

and

2c + 5m = 16.50

Translate the contents and costs of Basket X and Basket Y into algebraic equations where

c

is the price of a croissant and

m

is the price of a muffin.

2

Eliminate variable c

4c + 10m = 33.00

, then subtracting

4c + 3m = 19.00

yields

7m = 14.00

, so

m = 2.00

Multiplying the second equation by 2 aligns the coefficients of

c

, allowing elimination by subtraction.

3

Solve for variable c

2c + 5(2.00) = 16.50 \implies 2c = 6.50 \implies c = 3.25

Substitute the value found for

m

back into one of the original linear equations to calculate the cost of a croissant.

Anahtar Kavram

Solving Systems of Linear Equations via Elimination

Alternatif Yöntem

Express

c

in terms of

m

using the second equation:

c = 8.25 - 2.5m

. Substitute this expression into the first equation:

4(8.25 - 2.5m) + 3m = 19.00 \implies 33 - 10m + 3m = 19.00 \implies -7m = -14.00 \implies m = 2.00

. Finally, calculate

c = 8.25 - 2.5(2.00) = 3.25

.

Tahmini Süre:1m 30s

Soru 2074Soru →

The pie chart below shows the distribution of a regional health network's total annual operating budget of $\$50,000,000$ among four divisions in 2025: Inpatient Care ( $40\%$ ), Outpatient Care ( $25\%$ ), Surgical Services ( $20\%$ ), and Medical Research ( $15\%$ ). The accompanying table details the breakdown of expenditures within the Medical Research division into three categories: Personnel ( $50\%$ ), Equipment ( $30\%$ ), and Operational Overhead ( $20\%$ ). How much greater is the total dollar amount allocated to Surgical Services than the combined dollar amount allocated to Equipment and Operational Overhead within Medical Research?

$3,750,000

$6,250,000

$7,750,000

$8,750,000

$15,000,000

Cevabı ve açıklamayı göster

Cevap: $6,250,000

Cevap

The total dollar amount allocated to Surgical Services is $6,250,000 greater than the combined allocation for Equipment and Operational Overhead within Medical Research.

Surgical Services receives

20\%

of the total

\$50,000,000

budget (

\$10,000,000

). Medical Research receives

15\%

of

\$50,000,000

(

\$7,500,000

). Equipment (

30\%

) and Operational Overhead (

20\%

) within Medical Research sum to

50\%

of the Medical Research budget, which equals

0.50 \times \$7,500,000 = \$3,750,000

. The difference is

\$10,000,000 - \$3,750,000 = \$6,250,000

.

Adım Adım Çözüm

1

Calculate the total dollar amount allocated to Surgical Services.

Surgical Services budget =

20\% \times \$50,000,000 = \$10,000,000

.

Surgical Services constitutes 20 percent of the network's overall budget.

2

Calculate the total dollar amount allocated to the Medical Research division.

Medical Research budget =

15\% \times \$50,000,000 = \$7,500,000

.

Medical Research constitutes 15 percent of the overall budget.

3

Determine the percentage and dollar amount for Equipment and Operational Overhead combined within Medical Research.

Combined percentage =

30\% + 20\% = 50\%

. Dollar amount =

50\% \times \$7,500,000 = \$3,750,000

.

The sub-table shows Equipment is 30 percent and Overhead is 20 percent of the Medical Research division's total budget.

4

Calculate the difference between the Surgical Services budget and the combined Research Equipment/Overhead budget.

Difference =

\$10,000,000 - \$3,750,000 = \$6,250,000

.

Subtracting the sub-category expenditures from the main division budget gives the required dollar difference.

Anahtar Kavram

Multi-tier data interpretation involving pie chart sector calculations combined with sub-table nested percentage breakdowns.

Soru 2075Soru →

For all real numbers

x

and

y

such that

|x| \neq |y|

, which of the following expressions is equivalent to

\frac{x^3 + x^2y + x^2 - y^3 - xy^2 - y^2}{x^2 - y^2}?

x + y + 1

x - y + 1

x + y - 1

(x + y)^2 + 1

x^2 + xy + y^2 + 1

Cevabı ve açıklamayı göster

Cevap: $x + y + 1$

Cevap

x + y + 1

Factoring the numerator by grouping yields

(x - y)(x + y)(x + y + 1)

. Factoring the denominator gives

(x - y)(x + y)

. Canceling the non-zero common factor

(x - y)(x + y)

leaves the simplified expression

x + y + 1

.

Adım Adım Çözüm

1

Group terms in the numerator to identify common factor pairs.

N = (x^3 - y^3) + (x^2y - xy^2) + (x^2 - y^2)

Grouping cubic terms, quadratic cross-terms, and difference of squares separately allows factoring out fundamental algebraic patterns.

2

Apply standard algebraic formulas to each grouped term.

N = (x - y)(x^2 + xy + y^2) + xy(x - y) + (x - y)(x + y)

Using difference of cubes

x^3 - y^3 = (x-y)(x^2+xy+y^2)

and difference of squares

x^2 - y^2 = (x-y)(x+y)

reveals a common

(x-y)

factor across all terms.

3

Factor out

(x - y)

from the numerator and simplify the remaining polynomial.

N = (x - y)\left[(x^2 + xy + y^2) + xy + (x + y)\right] = (x - y)\left[x^2 + 2xy + y^2 + x + y\right]

Combining like terms inside the bracket simplifies the expression.

4

Recognize the perfect square trinomial inside the expression.

N = (x - y)\left[(x + y)^2 + (x + y)\right] = (x - y)(x + y)(x + y + 1)

Rewriting

x^2 + 2xy + y^2

as

(x + y)^2

allows factoring out

(x + y)

.

5

Divide the factored numerator by the denominator.

\frac{(x - y)(x + y)(x + y + 1)}{(x - y)(x + y)} = x + y + 1

Since

|x| \neq |y|

, both

(x - y)

and

(x + y)

are non-zero and can be canceled.

Anahtar Kavram

Factoring multivariable polynomials using grouping, difference of cubes, and difference of squares formulas.

Soru 2076Soru →

If $x$ is a real number such that $\frac{4^{x+2} - 4^x}{15} = 2^{x+3}$ , what is the value of $x$ ?

1

2

3

5

6

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

Factoring out

4^x

from the numerator gives

4^x(4^2 - 1) = 15 \cdot 4^x

. Dividing by 15 simplifies the left-hand side to

4^x

. Expressing

4^x

as

2^{2x}

allows setting

2^{2x} = 2^{x+3}

. Equating the exponents

2x = x + 3

yields

x = 3

.

Adım Adım Çözüm

1

Factor the numerator of the left-hand side

4^{x+2} - 4^x = 4^x(4^2 - 1) = 4^x(16 - 1) = 15 \cdot 4^x

Factoring out the common exponential term

4^x

simplifies the subtraction.

2

Simplify the fraction on the left-hand side

\frac{15 \cdot 4^x}{15} = 4^x

The factor of 15 in the numerator cancels with 15 in the denominator.

3

Rewrite

4^x

in terms of base 2

4^x = (2^2)^x = 2^{2x}

Both sides must have a common base to equate their exponents.

4

Set the exponential expressions equal and solve for

x

2^{2x} = 2^{x+3} \implies 2x = x + 3 \implies x = 3

Since the bases are equal and positive (base 2), their exponents must be equal.

Anahtar Kavram

Factoring common exponential terms and equating exponents with identical bases

Tahmini Süre:1m 30s

Soru 2077Soru →

A logistics facility utilizes two automated sorting lines, Line A and Line B, to process incoming shipments. Line A processes shipments at a constant rate that is 20 percent greater than the constant rate of Line B. When both lines operate simultaneously, they process a combined total of 3,300 shipments in 3 hours. Working alone at its constant rate, how many hours will it take Line B to process 2,250 shipments?

Cevabı ve açıklamayı göster

Cevap: 4.5

Cevap

4.5 hours

Let

r

represent the rate of Line B in shipments per hour. Because Line A processes 20% faster than Line B, Line A's rate is

1.20r

. Together, their combined processing rate is

r + 1.20r = 2.20r

shipments per hour. Operating for 3 hours, the total shipments processed is

3 \times 2.20r = 6.60r = 3,300

. Solving for

r

gives

r = 500

shipments per hour. To process 2,250 shipments alone, Line B requires

\frac{2,250}{500} = 4.5

hours.

Adım Adım Çözüm

1

Define the relationship between the individual processing rates.

If Line B processes at rate

r

shipments per hour, Line A processes at

1.20r

shipments per hour.

Line A's rate is 20 percent greater than Line B's rate.

2

Find the combined processing rate.

Combined rate =

r + 1.20r = 2.20r

shipments per hour.

When working together, individual rates add up.

3

Determine Line B's rate (

r

).

3 \times 2.20r = 3,300 \implies 6.60r = 3,300 \implies r = 500

shipments per hour.

Total work equals combined rate multiplied by time.

4

Calculate the time required for Line B to complete 2,250 shipments.

\text{Time} = \frac{2,250}{500} = 4.5

hours.

Time taken equals total work divided by the individual rate.

Anahtar Kavram

Combined Rates and Ratio Relationships

Soru 2078Soru →

Consider the system of linear equations in two variables

x

and

y

shown below, where

k

is a constant:

\begin{aligned} 2x + ky &= 10 \\ 3x - 6y &= 15 \end{aligned}

Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

If

k = -4

, the system has infinitely many solutions.

If

k = 0

, the system has a unique solution

(x, y) = (5, 0)

.

If

k = 4

, the system has no solution.

If

k = 2

, the solution to the system satisfies

x + y = 7

.

There is no real value of

k

for which the system has a solution with

y > 0

.

Cevabı ve açıklamayı göster

Cevap: If $k = -4$ , the system has infinitely many solutions.; If $k = 0$ , the system has a unique solution $(x, y) = (5, 0)$ .

Cevap

The correct statements are the ones stating that if

k = -4

, the system has infinitely many solutions, and if

k = 0

, the system has a unique solution

(x, y) = (5, 0)

.

Analyzing the simplified relation

(k + 4)y = 0

demonstrates that setting

k = -4

makes the equation identity

0y = 0

, yielding infinitely many solutions. For any other value of

k

, including

k = 0

,

y

must equal

0

, which gives

x = 5

, establishing a unique solution at

(5, 0)

.

Adım Adım Çözüm

1

Simplify the second equation to express

x

in terms of

y

.

3x - 6y = 15 \implies x - 2y = 5 \implies x = 2y + 5

.

Expressing

x

explicitly allows direct substitution into the first linear equation.

2

Substitute

x = 2y + 5

into the first equation

2x + ky = 10

.

2(2y + 5) + ky = 10 \implies 4y + 10 + ky = 10 \implies (k + 4)y = 0

.

This reduces the 2x2 system to a single linear equation in

y

parameterized by

k

.

3

Analyze the conditions for

y

based on the parameter

k

.

If

k = -4

, the equation becomes

0y = 0

, which is true for all real

y

(infinitely many solutions). If

k \neq -4

, then

y = 0

and

x = 5

(a unique solution).

Determines system consistency and solution multiplicity across all values of

k

.

Anahtar Kavram

Parametric Analysis of 2x2 Linear Systems

Soru 2079Soru →

For pairwise distinct real numbers $x$ , $y$ , and $z$ , consider the algebraic expression:

E(x, y, z) = \frac{(x^2 - y^2)^3 + (y^2 - z^2)^3 + (z^2 - x^2)^3}{(x - y)^3 + (y - z)^3 + (z - x)^3}

If $x = 5$ , $y = 3$ , and $z = 1$ , what is the numerical value of $E(5, 3, 1)$ ?

Cevabı ve açıklamayı göster

Cevap: 192

Cevap

192

Using the identity that

a^3 + b^3 + c^3 = 3abc

whenever

a + b + c = 0

, both the numerator and denominator can be factored directly. Factoring the difference of squares in the numerator yields

3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)

. Dividing this by the factored denominator

3(x - y)(y - z)(z - x)

simplifies the expression to

(x + y)(y + z)(z + x)

. Substituting

x = 5

,

y = 3

, and

z = 1

yields

(8)(4)(6) = 192

.

Adım Adım Çözüm

1

Use the conditional cubic identity

a + b + c = 0 \implies a^3 + b^3 + c^3 = 3abc

on the denominator.

Denominator becomes

3(x - y)(y - z)(z - x)

.

The sum of the three terms

(x - y) + (y - z) + (z - x)

equals 0.

2

Apply the same identity to the numerator.

Numerator becomes

3(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)

.

The sum of the squared difference terms

(x^2 - y^2) + (y^2 - z^2) + (z^2 - x^2)

also equals 0.

3

Factor each difference of squares in the numerator.

Numerator becomes

3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)

.

Using the difference of squares identity

u^2 - v^2 = (u - v)(u + v)

on each term.

4

Simplify the fraction by dividing the common factors in the numerator and denominator.

E(x, y, z) = (x + y)(y + z)(z + x)

.

The factors

3

,

(x - y)

,

(y - z)

, and

(z - x)

cancel out completely.

5

Evaluate the simplified product for

x = 5

,

y = 3

, and

z = 1

.

(5 + 3)(3 + 1)(1 + 5) = 8 \times 4 \times 6 = 192

.

Direct evaluation after algebraic simplification.

Anahtar Kavram

Simplifying rational expressions involving sum of cubes identity

a^3 + b^3 + c^3 = 3abc

when

a + b + c = 0

and difference of squares factoring.

Soru 2080Soru →

If $n$ is a real number such that $\frac{27^n + 27^n + 27^n}{3^{n+2}} = 243$ , what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

Rewriting

27^n + 27^n + 27^n

as

3 \cdot (3^3)^n = 3^{3n+1}

allows the left-hand side to simplify to

\frac{3^{3n+1}}{3^{n+2}} = 3^{2n-1}

. Equating this to

243 = 3^5

gives

2n - 1 = 5

, which solves to

n = 3

.

Adım Adım Çözüm

1

Express repeated addition in the numerator as multiplication.

27^n + 27^n + 27^n = 3 \cdot 27^n

Adding three identical quantities is equivalent to multiplying one quantity by 3.

2

Convert base 27 to base 3 and apply exponent multiplication.

3 \cdot (3^3)^n = 3^1 \cdot 3^{3n} = 3^{3n+1}

Since

27 = 3^3

, using the power rule

(a^b)^c = a^{bc}

and product rule

a^b \cdot a^c = a^{b+c}

converts the numerator to a single power of 3.

3

Simplify the fraction using the quotient rule of exponents.

\frac{3^{3n+1}}{3^{n+2}} = 3^{(3n+1) - (n+2)} = 3^{2n-1}

Dividing exponential terms with the same base requires subtracting the exponent in the denominator from the exponent in the numerator.

4

Rewrite 243 with base 3 and equate exponents across the equal sign.

3^{2n-1} = 3^5 \implies 2n - 1 = 5

Since

243 = 3^5

, two exponential expressions with the same base are equal if and only if their exponents are equal.

5

Solve the linear equation for

n

.

2n = 6 \implies n = 3

Adding 1 to both sides yields

2n = 6

, and dividing by 2 yields

n = 3

.

Anahtar Kavram

Combining repeated addition of exponential terms and converting expressions to a common base using exponent rules.

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap