Algebra Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) — Sayfa 5

Determine the value of $h$ that satisfies the equation below:

5 - 2(3h - 4) = 6h - 4(h + 4) - 3

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct value is 4.

The equation is solved by first distributing the coefficients across the parentheses, which yields

5 - 6h + 8 = 6h - 4h - 16 - 3

. Combining like terms on both sides simplifies the equation to

13 - 6h = 2h - 19

. Adding

6h

and

19

to both sides isolates the variable, resulting in

32 = 8h

. Dividing by

8

gives the final answer of

4

.

Adım Adım Çözüm

1

Apply the distributive property.

5 - 6h + 8 = 6h - 4h - 16 - 3

Multiply the terms inside the parentheses by their outside coefficients, carefully distributing the negative signs.

2

Combine like terms on the left side.

13 - 6h = 6h - 4h - 16 - 3

Combine the constant terms

5

and

8

.

3

Combine like terms on the right side.

13 - 6h = 2h - 19

Combine the variable terms (

6h - 4h = 2h

) and the constant terms (

-16 - 3 = -19

).

4

Isolate the variable

h

.

13 + 19 = 2h + 6h

Add

6h

to both sides and add

19

to both sides to group the variables and constants.

5

Solve for the final value.

32 = 8h \rightarrow h = 4

Simplify the addition and divide by

8

to find

h

.

Anahtar Kavram

Solving One-Variable Linear Equations

Tahmini Süre:1m 30s

Soru 82Soru →

Expand the given product of a binomial and a trinomial, combine like terms, and determine the values of the missing coefficients.

Aşağıdaki boşlukları doldurun

When the expression

(3x - 5)(2x^2 + 4x - 7)

is fully expanded and simplified, the resulting polynomial in standard form is

6x^3 +

x^2 +

x + 35

.

Cevabı ve açıklamayı göster

Cevap

The coefficient for the x^2 term is 2, and the coefficient for the x term is -41.

By distributing each term of the binomial

(3x - 5)

to every term of the trinomial

(2x^2 + 4x - 7)

, we get six terms. Grouping and combining the like terms yields an

x^2

coefficient of

12 - 10 = 2

and an

x

coefficient of

-21 - 20 = -41

.

Adım Adım Çözüm

1

Distribute the first term of the binomial,

3x

, to each term in the trinomial.

3x(2x^2) + 3x(4x) + 3x(-7) = 6x^3 + 12x^2 - 21x

To apply the distributive property for the first part of the polynomial multiplication.

2

Distribute the second term of the binomial,

-5

, to each term in the trinomial.

-5(2x^2) - 5(4x) - 5(-7) = -10x^2 - 20x + 35

To apply the distributive property for the second part, being careful to carry the negative sign.

3

Combine the results and group the like terms.

6x^3 + (12x^2 - 10x^2) + (-21x - 20x) + 35

To arrange the expression so that coefficients of terms with the same exponent can be simplified.

4

Simplify the grouped terms.

6x^3 + 2x^2 - 41x + 35

By adding the coefficients:

12 - 10 = 2

and

-21 - 20 = -41

.

Anahtar Kavram

Distributive Property with Polynomials

Alternatif Yöntem

You can use the 'box method' (a grid) by creating a 2x3 table with

3x

and

-5

on the rows, and

2x^2

,

4x

, and

-7

on the columns. Multiply the row and column headers to fill each cell, then add all the cells together along the diagonals to combine like terms.

Tahmini Süre:1m 30s

Soru 83Soru →

Evaluate the algebraic expression $\frac{3a^2b - 4bc}{a - c}$ given that $a = -2$ , $b = 3$ , and $c = -5$ . What is the final numerical value?

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The numerical value of the expression is 32.

After properly substituting the given values, evaluating the exponents, and following the order of operations, the numerator calculates to 96 and the denominator simplifies to 3. The final division of 96 by 3 yields 32.

Adım Adım Çözüm

1

Substitute the assigned values of

a

,

b

, and

c

into the given algebraic expression.

\frac{3(-2)^2(3) - 4(3)(-5)}{-2 - (-5)}

Direct variable substitution is required to compute the numerical value.

2

Evaluate the exponent applied to the negative base.

(-2)^2 = 4

According to the order of operations (PEMDAS), exponents must be resolved prior to multiplication.

3

Carry out the multiplication operations in the numerator.

3(4)(3) = 36

and

4(3)(-5) = -60

Multiplication is processed from left to right within the grouping of the numerator.

4

Complete the subtraction operation in the numerator.

36 - (-60) = 36 + 60 = 96

Subtracting a negative quantity is algebraically identical to adding its positive counterpart.

5

Evaluate the subtraction operation in the denominator.

-2 - (-5) = -2 + 5 = 3

The entire denominator acts as a single grouping and must be fully simplified before division.

6

Divide the evaluated numerator by the evaluated denominator.

\frac{96}{3} = 32

This final arithmetic step produces the precise evaluation of the fractional expression.

Anahtar Kavram

Evaluating complex rational algebraic expressions using substitution and the order of operations with signed integers.

Soru 84Soru →

Which of the following represents the fully simplified form of the expression $(3x - 4)^2 - x(2x + 5)$ ?

7x^2 - 19x + 16

7x^2 - 5x + 16

7x^2 - 29x - 16

7x^2 - 29x + 16

Cevabı ve açıklamayı göster

Cevap: $7x^2 - 29x + 16$

Cevap

The correct simplified expression is

7x^2 - 29x + 16

.

Expanding the squared binomial yields

9x^2 - 24x + 16

. Distributing the

-x

yields

-2x^2 - 5x

. Combining the like terms gives

(9x^2 - 2x^2) + (-24x - 5x) + 16

, which simplifies perfectly to

7x^2 - 29x + 16

.

Adım Adım Çözüm

1

Expand the squared binomial

(3x - 4)^2

.

(3x)^2 - 2(3x)(4) + 4^2 = 9x^2 - 24x + 16

Using the perfect square formula

(a - b)^2 = a^2 - 2ab + b^2

is required to properly multiply the binomial by itself.

2

Distribute the term

-x

across the binomial

(2x + 5)

.

-x(2x) - x(5) = -2x^2 - 5x

The distributive property must be applied, ensuring the negative sign is carried to both terms inside the parentheses.

3

Combine the results from the previous steps and combine like terms.

(9x^2 - 24x + 16) + (-2x^2 - 5x) = (9x^2 - 2x^2) + (-24x - 5x) + 16 = 7x^2 - 29x + 16

Like terms (those with the same variable raised to the same power) must be grouped and added together to find the fully simplified polynomial form.

Anahtar Kavram

Expanding perfect square binomials and distributing monomial terms correctly.

Soru 85Soru →

What is the exact numerical value of the variable $p$ that balances the given algebraic equation?

Aşağıdaki boşlukları doldurun

-2(3p - 5) + 4p = 3(p + 5) - 20

Value:

Cevabı ve açıklamayı göster

Cevap

3

The correct answer is found by first distributing the outer values into the parentheses, taking special care to distribute the negative sign accurately so that

-2(-5) = 10

. After expanding, the equation becomes

-6p + 10 + 4p = 3p + 15 - 20

. Combining the like terms on each side reduces the equation to

-2p + 10 = 3p - 5

. Adding

2p

to both sides and adding

5

to both sides isolates the terms, resulting in

15 = 5p

. Finally, dividing both sides by 5 yields a value of 3.

Adım Adım Çözüm

1

Distribute the constants into the parentheses on both sides of the equation.

-6p + 10 + 4p = 3p + 15 - 20

Expanding the grouped terms is necessary to access and combine all the variable and constant terms.

2

Combine the like terms on each side of the equation separately.

-2p + 10 = 3p - 5

Simplifying each side independently reduces the complexity before moving variables across the equals sign.

3

Isolate the variable by moving all

p

terms to one side and all constants to the opposite side.

10 + 5 = 3p + 2p

, which simplifies to

15 = 5p

Adding

2p

and adding

5

to both sides gathers the unknown variable exclusively to the right and constants to the left.

4

Divide by the coefficient of the variable to solve for

p

.

p = 3

Dividing both sides by 5 leaves the variable

p

completely isolated, providing the final solution.

Anahtar Kavram

Applying the distributive property and balancing operations to solve multi-step linear equations.

Soru 86Soru →

Consider the following algebraic equation:

5(2m - 1) - 3m = 2(m + 10)

What is the value of the variable $m$ ?

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Cevap: 5

Cevap

5

By correctly applying the distributive property on both sides of the equation, combining the resulting like terms, and applying algebraic operations to isolate the variable, the equation simplifies to

5m = 25

, which yields exactly

5

.

Adım Adım Çözüm

1

Distribute the constants outside the parentheses into the terms inside.

10m - 5 - 3m = 2m + 20

This removes the parentheses and allows us to combine like terms.

2

Combine the like variable terms on the left side (

10m

and

-3m

).

7m - 5 = 2m + 20

Simplifying each side independently makes the equation easier to solve.

3

Subtract

2m

from both sides and add

5

to both sides.

5m = 25

This groups all variable terms on the left and all constant terms on the right.

4

Divide both sides by

5

.

m = 5

This isolates the variable

m

to give the final solution.

Anahtar Kavram

Solving one-variable linear equations requiring distribution and combining like terms

Tahmini Süre:1m 0s

Soru 87Soru →

Solve the following linear equation for $k$ :

3 - 2(4k - 5) = -5(k + 1)

What is the exact value of $k$ ?

0

-\frac{2}{3}

6

\frac{8}{3}

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The correct value is

6

.

The equation is correctly solved by first using the distributive property on both sides to get

3 - 8k + 10 = -5k - 5

. Combining the constants on the left side yields

13 - 8k = -5k - 5

. Adding

8k

to both sides gives

13 = 3k - 5

. Finally, adding

5

to both sides results in

18 = 3k

, and dividing by

3

gives the solution

k = 6

.

Adım Adım Çözüm

1

Distribute the coefficients into the parentheses on both sides of the equation.

3 - 8k + 10 = -5k - 5

To eliminate the parentheses and expose all terms for simplification.

2

Combine the constant terms on the left side of the equation.

13 - 8k = -5k - 5

To simplify the left side.

3

Add

8k

to both sides of the equation.

13 = 3k - 5

To gather all terms with the variable

k

on the right side.

4

Add

5

to both sides of the equation.

18 = 3k

To isolate the variable term.

5

Divide both sides by

3

.

k = 6

To solve for

k

.

Anahtar Kavram

Solving one-variable linear equations by applying the distributive property and combining like terms.

Soru 88Soru →

Multiply the binomial $(3x - 4)$ by the trinomial $(5x^2 - 2x - 7)$ . After fully expanding the product and combining all like terms, what is the numerical coefficient of the $x$ term?

Cevabı ve açıklamayı göster

Cevap: -13

Cevap

The coefficient of the x term is -13.

The coefficient of the

x

term is found by locating all the

x

terms in the expanded expression (

-21x

and

8x

) and adding their coefficients together (

-21 + 8 = -13

).

Adım Adım Çözüm

1

Distribute the term

3x

to every term in the trinomial

(5x^2 - 2x - 7)

.

3x(5x^2) + 3x(-2x) + 3x(-7) = 15x^3 - 6x^2 - 21x

The distributive property requires multiplying each term of the first polynomial by each term of the second polynomial.

2

Distribute the term

-4

to every term in the trinomial

(5x^2 - 2x - 7)

.

-4(5x^2) - 4(-2x) - 4(-7) = -20x^2 + 8x + 28

This completes the distribution for the second term of the binomial, taking care to properly multiply the negative signs.

3

Write out the fully expanded polynomial before combining like terms.

15x^3 - 6x^2 - 21x - 20x^2 + 8x + 28

Listing all terms together makes it easier to identify which terms share the same variables and exponents.

4

Identify and combine the like terms that contain exactly the variable

x

to the first power.

-21x + 8x = -13x

The question specifically asks for the coefficient of the

x

term, so we combine

-21x

and

8x

.

Anahtar Kavram

Multiplying polynomials and combining like terms

Soru 89Soru →

An architect is designing a rectangular layout where the area is represented by the algebraic expression $4y^2 + 12y - 40$ . To determine the possible dimensions of the layout in terms of $y$ , the architect needs to rewrite this expression in its completely factored form. Which of the following represents the completely factored expression?

4(y + 5)(y - 2)

4(y - 5)(y + 2)

(4y - 8)(y + 5)

4(y + 5)(y + 2)

Cevabı ve açıklamayı göster

Cevap: $4(y + 5)(y - 2)$

Cevap

The completely factored expression is

4(y + 5)(y - 2)

.

The correct answer completely factors out the GCF of 4 from all terms and correctly factors the remaining trinomial

y^2 + 3y - 10

into

(y + 5)(y - 2)

.

Adım Adım Çözüm

1

Identify and extract the greatest common factor (GCF) from all terms in the expression

4y^2 + 12y - 40

.

4(y^2 + 3y - 10)

Factoring out the GCF simplifies the polynomial and makes the remaining trinomial easier to factor.

2

Find two integers that multiply to the constant term (-10) and add to the coefficient of the middle term (3).

The correct integers are 5 and -2.

These integers are required to construct the binomial factors of the simplified trinomial

y^2 + 3y - 10

.

3

Write the trinomial as a product of two binomials using the integers found, and combine them with the GCF.

4(y + 5)(y - 2)

This represents the original expression fully broken down into its prime polynomial factors.

Anahtar Kavram

Factoring Polynomials by Extracting the GCF and Grouping

Soru 90Soru →

Which of the following represents the completely factored form of the quadratic expression $3x^2 - 3x - 18$ ?

3(x - 3)(x + 2)

3(x + 3)(x - 2)

(3x - 9)(x + 2)

3(x - 6)(x + 1)

Cevabı ve açıklamayı göster

Cevap: $3(x - 3)(x + 2)$

Cevap

The completely factored form is

3(x - 3)(x + 2)

.

The correct answer properly extracts the GCF of 3 first, leaving the trinomial

x^2 - x - 6

. This trinomial factors into

(x - 3)(x + 2)

because

-3

and

2

multiply to

-6

and add up to

-1

.

Adım Adım Çözüm

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression.

3(x^2 - x - 6)

The coefficients

3, -3,

and

-18

are all divisible by

3

.

2

Find two numbers that multiply to the constant term (

-6

) and add to the coefficient of the middle term (

-1

).

The numbers are

-3

and

2

.

-3 \times 2 = -6

and

-3 + 2 = -1

.

3

Rewrite the quadratic trinomial as the product of two binomials using these numbers.

3(x - 3)(x + 2)

This represents the final, completely factored form of the original expression.

Anahtar Kavram

Factoring out a Greatest Common Factor (GCF) followed by factoring a quadratic trinomial.

Soru 91Soru →

Pair the given polynomial expressions on the left with their fully factored equivalents on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings are:

4x^2 - 25

with

(2x - 5)(2x + 5)

;

x^2 - 8x + 15

with

(x - 3)(x - 5)

;

3x^2 + 12x - 15

with

3(x + 5)(x - 1)

; and

2x^3 - 8x

with

2x(x - 2)(x + 2)

.

Each polynomial expression was broken down into its fundamental algebraic components by identifying patterns such as the difference of squares, standard trinomial factoring, and extracting the greatest common factor.

Adım Adım Çözüm

1

Analyze the first expression

4x^2 - 25

.

It factors to

(2x - 5)(2x + 5)

.

It follows the difference of squares pattern

a^2 - b^2 = (a - b)(a + b)

with

a = 2x

and

b = 5

.

2

Analyze the second expression

x^2 - 8x + 15

.

It factors to

(x - 3)(x - 5)

.

The constant term is positive and the middle coefficient is negative, so both factors must contain negative constants that multiply to

15

and add to

-8

.

3

Analyze the third expression

3x^2 + 12x - 15

.

It factors to

3(x + 5)(x - 1)

.

Factoring out the GCF of

3

gives

3(x^2 + 4x - 5)

. The resulting quadratic factors into binomials with constants

5

and

-1

.

4

Analyze the fourth expression

2x^3 - 8x

.

It factors to

2x(x - 2)(x + 2)

.

Factoring out the GCF of

2x

gives

2x(x^2 - 4)

. The expression

x^2 - 4

is a difference of squares that further factors into

(x - 2)(x + 2)

.

Anahtar Kavram

Factoring Polynomials

Tahmini Süre:2m 0s

Soru 92Soru →

If $(x, y)$ is the solution to the system of equations below, what is the value of the expression $x + 3y$ ?

3x - 2y = 14

x = y + 5

21

-9

1

41

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The value of the expression is 1.

The system can be efficiently solved using substitution because the second equation already isolates

x

. Substituting

x = y + 5

into the first equation yields

3(y + 5) - 2y = 14

. Distributing the

3

gives

3y + 15 - 2y = 14

, which simplifies to

y = -1

. Substituting this back into the second equation gives

x = -1 + 5 = 4

. Finally, the question asks for the value of

x + 3y

, which evaluates to

4 + 3(-1) = 1

.

Adım Adım Çözüm

1

Substitute the expression for

x

from the second equation into the first equation.

3(y + 5) - 2y = 14

This isolates a single variable, allowing us to solve for

y

.

2

Distribute the 3 and solve the resulting equation for

y

.

3y + 15 - 2y = 14 \Rightarrow y + 15 = 14 \Rightarrow y = -1

Expanding the terms allows combining like terms to find the value of

y

.

3

Substitute

y = -1

back into the second equation to determine

x

.

x = -1 + 5 \Rightarrow x = 4

Both values are needed for the final expression.

4

Evaluate the target expression

x + 3y

using the values found for

x

and

y

.

4 + 3(-1) = 4 - 3 = 1

The question specifically asks for the evaluated value of this expression, not just the ordered pair.

Anahtar Kavram

Solving a system of linear equations using the substitution method and evaluating an algebraic expression.

Soru 93Soru →

Given the system of linear equations below, what is the value of $y$ ?

5x + 2y = 12

3x - 4y = 2

2

11

1

-2

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The correct value for y is 1.

By multiplying the first equation by 2, we get 10x + 4y = 24. Adding this to the second equation (3x - 4y = 2) eliminates y, resulting in 13x = 26, which simplifies to x = 2. Substituting x = 2 back into the first equation gives 5(2) + 2y = 12, or 10 + 2y = 12. Subtracting 10 from both sides leaves 2y = 2, yielding the final correct answer of y = 1.

Adım Adım Çözüm

1

Multiply the first equation by 2 to align the coefficients of y for elimination.

10x + 4y = 24

Creating a +4y term in the first equation allows y to be eliminated when added to the -4y term in the second equation.

2

Add the modified first equation to the second equation.

(10x + 3x) + (4y - 4y) = 24 + 2 \Rightarrow 13x = 26

This eliminates the variable y, leaving a single-variable equation that can be easily solved for x.

3

Divide both sides by 13 to solve for x.

x = 2

Isolating x provides the first part of the coordinate solution.

4

Substitute x = 2 back into the first original equation to solve for y.

5(2) + 2y = 12 \Rightarrow 10 + 2y = 12 \Rightarrow 2y = 2 \Rightarrow y = 1

Since the question asks specifically for the value of y, this final substitution step is required.

Anahtar Kavram

Solving Systems of Linear Equations using the Elimination Method

Soru 94Soru →

A logistics team is packing a shipment containing standard crates and heavy crates. A load of $4$ standard crates and $3$ heavy crates weighs a total of $170$ pounds. Another load consisting of $2$ standard crates and $5$ heavy crates weighs a total of $190$ pounds. Based on this information, what is the weight, in pounds, of a single heavy crate?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

The correct answer is found by translating the word problem into the system of equations

4x + 3y = 170

and

2x + 5y = 190

. By multiplying the second equation by

-2

, the

x

terms are eliminated when the equations are added, yielding

-7y = -210

, which simplifies to

y = 30

.

Adım Adım Çözüm

1

Define variables for the unknown quantities.

Let

x

be the weight of a standard crate and

y

be the weight of a heavy crate.

To translate the word problem into a mathematical system of equations.

2

Set up a system of linear equations based on the given information.

Equation 1:

4x + 3y = 170

Equation 2:

2x + 5y = 190

To mathematically represent the total weights of the two different loads.

3

Use the elimination method to cancel out the

x

variable. Multiply Equation 2 by

-2

.

-2(2x + 5y) = -2(190) \Rightarrow -4x - 10y = -380

Creating a

-4x

term allows the

x

variables to be eliminated when added to the

4x

in Equation 1.

4

Add the new Equation 2 to Equation 1.

(4x + 3y) + (-4x - 10y) = 170 - 380 \Rightarrow -7y = -210

This leaves a single-variable equation that can be directly solved.

5

Solve for

y

.

y = \frac{-210}{-7} = 30

To isolate

y

, which represents the target value: the weight of the heavy crate.

Anahtar Kavram

Solving Systems of Linear Equations

Soru 95Soru →

The two linear equations below form a system.

2x + 3y = 12

2x - y = 4

Based on the solution to this system, what is the value of the expression $x + 2y$ ?

5

7

10

12

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The correct answer is 7.

The correct answer is found by first solving the system of equations. Subtracting the second equation from the first eliminates the x variable, giving 3y - (-y) = 12 - 4, which simplifies to 4y = 8 and y = 2. Substituting y = 2 into the second equation yields 2x - 2 = 4, meaning 2x = 6 and x = 3. Finally, evaluating the expression x + 2y with these values gives 3 + 2(2) = 3 + 4 = 7.

Adım Adım Çözüm

1

Eliminate the x variable by subtracting the second equation from the first.

(2x - 2x) + (3y - (-y)) = 12 - 4 \Rightarrow 4y = 8

Because the coefficient of x is identical in both equations, subtracting them cancels out the x terms entirely, leaving an equation with only y.

2

Solve for y.

y = 2

Dividing both sides of the equation by 4 isolates the y variable.

3

Substitute the value of y back into the second equation to solve for x.

2x - (2) = 4 \Rightarrow 2x = 6 \Rightarrow x = 3

Using the known value of y allows us to find the corresponding x value for the system's ordered pair.

4

Evaluate the target expression

x + 2y

using the found values.

3 + 2(2) = 3 + 4 = 7

This answers the specific question asked in the stem by applying the solution to the given expression.

Anahtar Kavram

Solving Systems of Linear Equations by Elimination

Soru 96Soru →

A motor pool contains a total of 24 vehicles, consisting exclusively of transport trucks and utility vehicles. The number of transport trucks is 6 less than twice the number of utility vehicles. How many utility vehicles are in the motor pool?

Cevabı ve açıklamayı göster

Cevap: 10; ten; 10 utility vehicles

Cevap

10

By defining

t

as transport trucks and

u

as utility vehicles, we can create the system

t + u = 24

and

t = 2u - 6

. Substituting the value of

t

into the first equation yields

(2u - 6) + u = 24

. Combining the terms gives

3u - 6 = 24

. Adding 6 to both sides results in

3u = 30

, and dividing by 3 shows there are

10

utility vehicles.

Adım Adım Çözüm

1

Define variables for the unknown quantities.

Let

t

be the number of transport trucks and

u

be the number of utility vehicles.

This allows us to translate the word problem into a system of algebraic equations.

2

Set up the system of equations based on the problem statement.

Equation 1:

t + u = 24

(total vehicles).
Equation 2:

t = 2u - 6

(transport trucks are 6 less than twice the utility vehicles).

These equations accurately represent the given conditions in mathematical terms.

3

Substitute the expression for

t

from Equation 2 into Equation 1.

(2u - 6) + u = 24

Substitution eliminates the variable

t

, leaving a single-variable equation that can be solved directly for

u

.

4

Combine like terms and solve for

u

.

3u - 6 = 24 \Rightarrow 3u = 30 \Rightarrow u = 10

Isolating

u

gives the exact number of utility vehicles requested by the problem.

Anahtar Kavram

Solving Systems of Linear Equations Using Substitution

Alternatif Yöntem

This can also be solved by elimination if we rearrange the second equation to

-t + 2u = 6

. Adding this to

t + u = 24

yields

3u = 30

, which gives

u = 10

.

Tahmini Süre:1m 0s

Soru 97Soru →

Calculate the solution $(x, y)$ for the system of linear equations shown below.

3x - 2y = 7

2x + 4y = 10

What is the value of the expression $x - y$ ?

3

6

2

8

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct value of the expression is 2.

The system can be efficiently solved using either substitution or elimination. By dividing the second equation by 2, we get

x + 2y = 5

, which means

x = 5 - 2y

. Substituting this into the first equation gives

3(5 - 2y) - 2y = 7

. Distributing the 3 results in

15 - 6y - 2y = 7

, which simplifies to

15 - 8y = 7

. Subtracting 15 from both sides gives

-8y = -8

, so

y = 1

. Substituting

y = 1

back into

x = 5 - 2y

yields

x = 5 - 2(1) = 3

. Finally, the requested expression is

x - y

, which evaluates to

3 - 1 = 2

.

Adım Adım Çözüm

1

Simplify the second equation by dividing all terms by 2.

x + 2y = 5

Dividing the entire equation by a common factor makes the coefficients smaller and allows us to easily isolate

x

for the substitution method.

2

Isolate

x

in the simplified equation.

x = 5 - 2y

Expressing

x

in terms of

y

prepares us to substitute this expression into the first equation.

3

Substitute the expression for

x

into the first equation and solve for

y

.

3(5 - 2y) - 2y = 7 \Rightarrow 15 - 6y - 2y = 7 \Rightarrow 15 - 8y = 7 \Rightarrow -8y = -8 \Rightarrow y = 1

This step eliminates

x

entirely, creating a single-variable linear equation that can be solved to find the exact value of

y

.

4

Substitute the value of

y

back into the isolated

x

equation to find

x

.

x = 5 - 2(1) \Rightarrow x = 5 - 2 \Rightarrow x = 3

Now that we have the numerical value for

y

, we can plug it back in to find the corresponding value for

x

.

5

Evaluate the final expression

x - y

using the calculated values.

3 - 1 = 2

The question asks specifically for the value of

x - y

, not just the ordered pair.

Anahtar Kavram

To solve a system of linear equations, you can use substitution by isolating a variable in one equation and substituting it into the other, or use elimination by combining the equations to cancel out a variable.

Soru 98Soru →

If $5m + 12 = 2m + 27$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The correct value of the variable is 5.

By using inverse operations to collect the variable terms on one side and the constant terms on the other, the equation simplifies to

3m = 15

, which directly yields a result of

5

.

Adım Adım Çözüm

1

Move all terms containing the variable to one side of the equation.

3m + 12 = 27

Subtracting

2m

from both sides eliminates the variable from the right side and groups it on the left.

2

Move all constant terms to the opposite side of the equation.

3m = 15

Subtracting

12

from both sides isolates the term containing the variable.

3

Solve for the variable.

m = 5

Dividing both sides by the coefficient

3

provides the final value of

m

.

Anahtar Kavram

Solving Linear Equations with Variables on Both Sides

Tahmini Süre:45s

Soru 99Soru →

Which of the following represents the simplified form of the expression $4b^5 \cdot 2b^{-2}$ ?

8b^{-10}

6b^3

8b^3

8b^7

Cevabı ve açıklamayı göster

Cevap: $8b^3$

Cevap

The simplified expression is

8b^3

.

The correct expression is found by multiplying the numerical coefficients (

4 \times 2 = 8

) and adding the exponents for the common base

b

(

5 + (-2) = 3

), resulting in

8b^3

.

Adım Adım Çözüm

1

Multiply the numerical coefficients.

4 \times 2 = 8

When multiplying algebraic terms, their numerical coefficients are multiplied together.

2

Apply the product rule of exponents to the variable terms.

b^5 \cdot b^{-2} = b^{5 + (-2)} = b^3

When multiplying exponential expressions with the same base, add their exponents.

3

Combine the coefficient and the variable term.

8b^3

This is the final simplified form of the expression.

Anahtar Kavram

When multiplying two exponential terms with the same base, multiply their coefficients and add their exponents.

Soru 100Soru →

Which simplified algebraic expression corresponds to each unsimplified expression? Match each item on the left with its correctly simplified equivalent on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings are:

4x + 3x - 5

matches with

7x - 5

;

2(3x + 4)

matches with

6x + 8

;

9x - 4x + 2

matches with

5x + 2

; and

3(x - 2) + 5

matches with

3x - 1

.

Each expression is correctly matched to its simplified equivalent by strictly applying the distributive property to eliminate any grouping symbols, followed by combining like terms (terms with identical variables) and grouping constants.

Adım Adım Çözüm

1

Simplify the first expression,

4x + 3x - 5

.

7x - 5

Combining the like variable terms (

4x

and

3x

) gives

7x

. The constant

-5

remains unchanged.

2

Simplify the second expression,

2(3x + 4)

.

6x + 8

Applying the distributive property multiplies each term inside the parentheses by

2

.

3

Simplify the third expression,

9x - 4x + 2

.

5x + 2

Combining the like variable terms (

9x

and

-4x

) gives

5x

. The constant

2

remains unchanged.

4

Simplify the fourth expression,

3(x - 2) + 5

.

3x - 1

Distributing the

3

yields

3x - 6

. Combining the constant terms (

-6

and

5

) results in

-1

.

Anahtar Kavram

Simplifying Algebraic Expressions by Combining Like Terms and Distributing

Algebra

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Alternatif Yöntem

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