Algebra Alıştırma Soruları — SAT — Sayfa 4

If $3(x - 2) - 4(2 - x) = \frac{1}{2}(6x - 4) + 1$ , what is the value of $2x - 3$ ?

-\frac{19}{2}

\frac{7}{2}

\frac{9}{2}

\frac{13}{4}

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{2}$

Cevap

The value of the expression is

\frac{7}{2}

.

The correct answer is

\frac{7}{2}

. Distributing the constants on both sides of the equation

3(x - 2) - 4(2 - x) = \frac{1}{2}(6x - 4) + 1

yields

3x - 6 - 8 + 4x = 3x - 2 + 1

. Combining like terms on both sides gives

7x - 14 = 3x - 1

. Subtracting

3x

from both sides and adding

14

to both sides results in

4x = 13

, which simplifies to

x = \frac{13}{4}

. Substituting this value into

2x - 3

gives

2\left(\frac{13}{4}\right) - 3 = \frac{13}{2} - 3 = \frac{7}{2}

.

Adım Adım Çözüm

1

Distribute the constants on both sides of the equation.

3x - 6 - 8 + 4x = 3x - 2 + 1

To simplify the linear terms and constants before isolating the variable.

2

Combine like terms on both sides of the equation.

7x - 14 = 3x - 1

To reduce the equation to a simpler form with one variable term and one constant term on each side.

3

Isolate the variable term by subtracting

3x

from both sides and adding

14

to both sides.

4x = 13

To group all variable terms on one side and constant terms on the other.

4

Solve for

x

by dividing both sides by

4

.

x = \frac{13}{4}

To isolate the variable

x

.

5

Substitute the value of

x

into the expression

2x - 3

.

2\left(\frac{13}{4}\right) - 3 = \frac{13}{2} - 3 = \frac{7}{2}

To calculate the final value requested by the question.

Anahtar Kavram

Solving linear equations in one variable by distributing terms, combining like terms, isolating the variable, and evaluating algebraic expressions.

Soru 62Soru →

If $\frac{3}{5}p - 4 = 8$ , what is the value of $p$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

To solve the equation

\frac{3}{5}p - 4 = 8

, we perform inverse operations to isolate

p

. First, add 4 to both sides of the equation to get

\frac{3}{5}p = 12

. Next, multiply both sides by the reciprocal of the coefficient of

p

, which is

\frac{5}{3}

. This gives

p = 12 \times \frac{5}{3} = 20

. Substituting 20 back into the original equation confirms it is the correct solution.

Adım Adım Çözüm

1

Add 4 to both sides of the equation.

\frac{3}{5}p = 12

To isolate the term containing the variable

p

.

2

Multiply both sides of the equation by

\frac{5}{3}

.

p = 20

To solve for

p

by multiplying by the reciprocal of its coefficient.

Anahtar Kavram

Solving one-variable linear equations using inverse operations.

Soru 63Soru →

In the equation below, $k$ is a constant.

\frac{1}{3}(2kx - 9) - \frac{5}{6}(x + 4) = \frac{11}{2}

If the equation has no solution, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 1.25

Cevap

1.25 (or 5/4)

The correct answer is 1.25 (or 5/4). A linear equation in one variable of the form

Ax + B = C

has no solution if the variable terms on both sides of the equation are equal (meaning

A = 0

) and the constant terms are unequal (

B \neq C

). Expanding the left side of the given equation yields

\frac{2}{3}kx - 3 - \frac{5}{6}x - \frac{10}{3} = \frac{11}{2}

. Combining the constant terms gives

\left(\frac{2}{3}k - \frac{5}{6}\right)x - \frac{19}{3} = \frac{11}{2}

. Setting the coefficient of

x

to

0

yields

\frac{2}{3}k - \frac{5}{6} = 0

. Solving for

k

gives

k = \frac{5}{6} \times \frac{3}{2} = \frac{5}{4}

, which is equivalent to 1.25. Since the remaining constant terms are unequal (

-\frac{19}{3} \neq \frac{11}{2}

), the equation has no solution when

k = 1.25

.

Adım Adım Çözüm

1

Expand the expression on the left side of the equation

\frac{2}{3}kx - 3 - \frac{5}{6}x - \frac{10}{3} = \frac{11}{2}

Apply the distributive property to remove the parentheses.

2

Group the

x

terms and combine the constants on the left side

\left(\frac{2}{3}k - \frac{5}{6}\right)x - \frac{19}{3} = \frac{11}{2}

Simplify the equation by combining like terms:

-3 - \frac{10}{3} = -\frac{9}{3} - \frac{10}{3} = -\frac{19}{3}

.

3

Set the coefficient of the

x

term equal to

0

\frac{2}{3}k - \frac{5}{6} = 0

For a linear equation to have no solution, the variable terms on both sides of the equation must cancel out (meaning the coefficient of the variable must be

0

), while the remaining constant terms must not be equal (

-\frac{19}{3} \neq \frac{11}{2}

).

4

Solve the resulting equation for

k

k = 1.25

Add

\frac{5}{6}

to both sides to get

\frac{2}{3}k = \frac{5}{6}

, then multiply by the reciprocal of

\frac{2}{3}

, which gives

k = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1.25

.

Anahtar Kavram

Conditions for a linear equation in one variable to have no solution

Soru 64Soru →

A rectangle has a length of $2x + 5$ centimeters and a width of $x - 2$ centimeters. If the perimeter of the rectangle is $42$ centimeters, what is the value of $x$ ?

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Cevap: 6

Cevap

6

The correct answer is 6. The perimeter of a rectangle is calculated using the formula

P = 2l + 2w

. Substituting the given expressions for length and width yields the linear equation

2(2x + 5) + 2(x - 2) = 42

. Distributing the 2 gives

4x + 10 + 2x - 4 = 42

. Combining like terms on the left side simplifies this to

6x + 6 = 42

. Subtracting 6 from both sides yields

6x = 36

. Finally, dividing both sides by 6 gives the value

x = 6

.

Adım Adım Çözüm

1

Write the perimeter equation in terms of

x

.

2(2x + 5) + 2(x - 2) = 42

The perimeter of a rectangle is the sum of all its sides, represented by the formula

P = 2l + 2w

.

2

Distribute the 2 into the parentheses.

4x + 10 + 2x - 4 = 42

Applying the distributive property yields

2(2x + 5) = 4x + 10

and

2(x - 2) = 2x - 4

.

3

Combine like terms on the left side.

6x + 6 = 42

Grouping the variable terms gives

4x + 2x = 6x

, and grouping the constant terms gives

10 - 4 = 6

.

4

Subtract 6 from both sides of the equation.

6x = 36

This isolates the variable term on the left side of the equation.

5

Divide both sides of the equation by 6.

x = 6

Dividing by the coefficient of

x

solves for the variable.

Anahtar Kavram

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

Soru 65Soru →

A line in the $xy$ -plane has the equation $3y - 2x = 12$ . What is the slope of this line?

\frac{3}{2}

4

\frac{2}{3}

-\frac{2}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{3}$

Cevap

The slope of the line is

\frac{2}{3}

.

To find the slope, the equation

3y - 2x = 12

is rewritten in slope-intercept form (

y = mx + b

). Adding

2x

to both sides gives

3y = 2x + 12

. Dividing by

3

yields

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

. The slope is the coefficient of

x

, which is

\frac{2}{3}

.

Adım Adım Çözüm

1

Isolate the term containing

y

by adding

2x

to both sides of the equation.

3y = 2x + 12

To express the equation in slope-intercept form (

y = mx + b

).

2

Divide all terms in the equation by

3

to solve for

y

.

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

To isolate

y

and identify the slope,

m

, which is the coefficient of

x

.

Anahtar Kavram

Finding the slope of a line from its linear equation by rewriting it in slope-intercept form.

Soru 66Soru →

A solar energy company offers two payment plans for installing solar panels. Under Plan A, the customer pays a one-time installation fee of $1,200 and a monthly maintenance fee of$ 25. Under Plan B, there is no installation fee, but the customer pays a monthly maintenance fee of $45 for the first 12 months, and$ 35 per month for each month thereafter. If a customer chooses Plan A, after how many months of service will the total cost of Plan A be exactly equal to the total cost of Plan B?

24

66

108

132

Cevabı ve açıklamayı göster

Cevap: 108

Cevap

The correct answer is 108 months.

The correct answer is 108. Setting the total cost of Plan A,

1200 + 25m

, equal to the total cost of Plan B,

45(12) + 35(m-12)

, yields the linear equation

1200 + 25m = 120 + 35m

. Isolating

m

by subtracting

25m

and

120

from both sides gives

10m = 1080

, which simplifies to

m = 108

.

Adım Adım Çözüm

1

Set up the total cost expression for Plan A.

\text{Cost}_A = 1200 + 25m

Plan A has a fixed setup fee of

1,200 plus a recurring cost of

25 for each of the

m

months.

2

Set up the total cost expression for Plan B, assuming the number of months

m > 12

.

\text{Cost}_B = 45(12) + 35(m - 12) = 540 + 35m - 420 = 120 + 35m

Plan B charges

45 per month for the first 12 months and

35 per month for the remaining

m - 12

months.

3

Equate the two expressions and solve for

m

.

1200 + 25m = 120 + 35m \Rightarrow 1080 = 10m \Rightarrow m = 108

Setting the two costs equal allows us to isolate the variable

m

by subtracting

25m

and

120

from both sides.

Anahtar Kavram

Formulating and solving multi-step linear equations in one variable from real-world word problems.

Tahmini Süre:2m 30s

Soru 67Soru →

If $5 - 3(2x - 4) = 2(x + 3) - 5$ , what is the value of $4x - 3$ ?

5

-7

2

6

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is the value of the expression when the linear equation is solved correctly. Distributing the terms on both sides of the equation yields

5 - 6x + 12 = 2x + 6 - 5

. Simplifying each side gives

17 - 6x = 2x + 1

. Adding

6x

to both sides and subtracting

1

from both sides results in

16 = 8x

, which means

x = 2

. Substituting

x = 2

into the expression

4x - 3

gives

4(2) - 3 = 5

.

Adım Adım Çözüm

1

Distribute the constants on both sides of the equation.

5 - 6x + 12 = 2x + 6 - 5

To eliminate the parentheses so that like terms can be combined.

2

Combine like terms on both sides of the equation.

17 - 6x = 2x + 1

To simplify the linear expression on each side.

3

Isolate the variable

x

by adding

6x

to both sides and subtracting

1

from both sides.

16 = 8x

, which simplifies to

x = 2

To find the value of

x

that satisfies the equation.

4

Substitute the value of

x

into the target expression

4x - 3

.

4(2) - 3 = 5

To find the final value requested by the question.

Anahtar Kavram

Solving linear equations in one variable by distributing constants, combining like terms, and evaluating expressions.

Soru 68Soru →

In the equation below, $c$ is a constant.

\frac{3c(2x - 1) - 2(x + 4c)}{5} = 2x - 3

If the equation has no solution for $x$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

To find the value of

c

that results in no solution, we first clear the fraction by multiplying both sides of the equation by 5, giving

3c(2x - 1) - 2(x + 4c) = 10x - 15

. Next, we expand the terms to get

6cx - 3c - 2x - 8c = 10x - 15

, and group the

x

terms and constant terms:

(6c - 2)x - 11c = 10x - 15

. For a linear equation to have no solution, the coefficients of

x

on both sides must be equal while the constant terms must be different. Setting the coefficients equal gives

6c - 2 = 10

, which solves to

c = 2

. Checking the constant terms when

c = 2

, we get

-11(2) = -22

on the left and

-15

on the right. Since

-22 \neq -15

, the equation has no solution, confirming

c = 2

is correct.

Adım Adım Çözüm

1

Multiply both sides of the equation by 5 to eliminate the denominator.

3c(2x - 1) - 2(x + 4c) = 10x - 15

Clearing the denominator simplifies the equation into standard polynomial terms.

2

Distribute the terms on the left side of the equation.

6cx - 3c - 2x - 8c = 10x - 15

Applying the distributive property expands the expression so terms can be grouped.

3

Group the

x

terms and constant terms on the left side.

(6c - 2)x - 11c = 10x - 15

Putting the equation in the standard form

Ax + B = Cx + D

allows us to easily set up the conditions for no solution.

4

Set the coefficients of

x

on both sides equal to each other.

6c - 2 = 10

, which simplifies to

6c = 12

, and thus

c = 2

.

For the equation to have no solution, the variable terms on both sides must cancel each other out.

5

Verify that the constant terms are not equal when

c = 2

.

The left-side constant is

-11(2) = -22

, and the right-side constant is

-15

. Since

-22 \neq -15

, the equation has no solution.

If the constant terms were equal, the equation would have infinitely many solutions instead of no solution.

Anahtar Kavram

Identifying conditions for a linear equation in one variable to have no solution.

Alternatif Yöntem

Instead of clearing the fraction first, write the left side of the equation as

(\frac{6c - 2}{5})x - \frac{11c}{5}

. For there to be no solution, the coefficient of

x

on the left side,

\frac{6c - 2}{5}

, must equal the coefficient of

x

on the right side, which is 2. Solving

\frac{6c - 2}{5} = 2

gives

6c - 2 = 10 \implies c = 2

.

Tahmini Süre:2m 30s

Soru 69Soru →

If $2(x - 6) = 10$ , what is the value of $x + 4$ ?

12

3

15

11

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

The correct answer is 15. Isolating the variable in the equation

2(x - 6) = 10

can be done by first dividing both sides by 2, which gives

x - 6 = 5

. Adding 6 to both sides yields

x = 11

. Substituting this value into the expression

x + 4

results in

11 + 4 = 15

. Alternatively, distributing the 2 gives

2x - 12 = 10

, and adding 12 to both sides gives

2x = 22

, which leads to

x = 11

and thus

x + 4 = 15

.

Adım Adım Çözüm

1

Distribute the 2 to the terms inside the parentheses on the left side of the equation.

2x - 12 = 10

To remove the parentheses and prepare to isolate the variable term.

2

Add 12 to both sides of the equation.

2x = 22

To isolate the variable term

2x

on the left side of the equation.

3

Divide both sides of the equation by 2.

x = 11

To find the value of

x

.

4

Substitute the value of

x

into the expression

x + 4

.

11 + 4 = 15

To find the final value requested by the question.

Anahtar Kavram

Solving linear equations in one variable by applying the distributive property, isolating the variable, and evaluating an expression.

Alternatif Yöntem

Instead of distributing first, divide both sides of the equation

2(x - 6) = 10

by 2 directly to get

x - 6 = 5

. Then, add 10 to both sides of the equation

x - 6 = 5

to get the value of

x + 4

directly:

(x - 6) + 10 = 5 + 10 \Rightarrow x + 4 = 15

.

Tahmini Süre:45s

Soru 70Soru →

A linear function $f$ has a slope of $\frac{2}{3}$ . The function $g$ is defined by $g(x) = f(x + 4) - 3$ . If the graph of $y = g(x)$ in the $xy$ -plane has an $x$ -intercept at $(5, 0)$ , what is the $x$ -intercept of the graph of $y = f(x)$ ?

Cevabı ve açıklamayı göster

Cevap: 4.5

Cevap

4.5

The correct answer is

4.5

(or

\frac{9}{2}

). By using the function transformation relation, we find

g(5) = f(9) - 3 = 0

, which yields

f(9) = 3

. With a slope of

\frac{2}{3}

, the equation of the line is

f(x) = \frac{2}{3}x - 3

. Setting

f(x) = 0

yields the

x

-intercept at

x = 4.5

.

Adım Adım Çözüm

1

Use the definition of function

g

and its given

x

-intercept to find a point on the graph of

f

.

f(9) = 3

Since the graph of

y = g(x)

has an

x

-intercept at

(5, 0)

, we know

g(5) = 0

. Substituting

x = 5

into the definition

g(x) = f(x + 4) - 3

gives

g(5) = f(5 + 4) - 3 = f(9) - 3

. Since

g(5) = 0

, it follows that

f(9) - 3 = 0

, or

f(9) = 3

.

2

Determine the equation of the linear function

f

using its slope and the point identified in Step 1.

f(x) = \frac{2}{3}x - 3

The function

f

is linear with a slope of

\frac{2}{3}

and passes through the point

(9, 3)

. Using the point-slope formula, we get

f(x) - 3 = \frac{2}{3}(x - 9)

, which simplifies to

f(x) = \frac{2}{3}x - 3

.

3

Find the

x

-intercept of the graph of

y = f(x)

by setting

f(x) = 0

.

x = 4.5

(or

\frac{9}{2}

)

To find the

x

-intercept, set

f(x) = 0

. This gives

\frac{2}{3}x - 3 = 0

. Adding

3

to both sides and multiplying by

\frac{3}{2}

yields

x = \frac{9}{2}

, which is equal to

4.5

.

Anahtar Kavram

Linear Functions and Graphs

Alternatif Yöntem

Alternatively, we can write the equation of

f

in slope-intercept form as

f(x) = \frac{2}{3}x + b

. Then the definition of

g(x)

becomes

g(x) = \frac{2}{3}(x + 4) + b - 3 = \frac{2}{3}x + \frac{8}{3} + b - 3 = \frac{2}{3}x + b - \frac{1}{3}

. Since the graph of

g

has an

x

-intercept at

(5, 0)

, we substitute

x = 5

and

g(5) = 0

to get

\frac{2}{3}(5) + b - \frac{1}{3} = 0

, which simplifies to

3 + b = 0

, so

b = -3

. This gives

f(x) = \frac{2}{3}x - 3

. Finally, the

x

-intercept is found by solving

\frac{2}{3}x - 3 = 0

, resulting in

x = 4.5

.

Tahmini Süre:2m 0s

Soru 71Soru →

In the $xy$ -plane, line $L$ passes through the points $(0, 4)$ and $(6, 0)$ . A second line, $M$ , is perpendicular to line $L$ and intersects line $L$ at a point on the line $y = x$ . If line $M$ is represented by the equation $y = px + q$ , where $p$ and $q$ are constants, what is the value of $p + q$ ?

\frac{6}{5}

\frac{10}{3}

\frac{3}{10}

-\frac{9}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{10}$

Cevap

The correct answer is the value stating three-tenths

To find the sum of the slope

p

and

y

-intercept

q

of line

M

, we first determine the equation of line

L

. Line

L

passes through

(0, 4)

and

(6, 0)

, so its slope is

\frac{0 - 4}{6 - 0} = -\frac{2}{3}

. The equation of line

L

is therefore

y = -\frac{2}{3}x + 4

. Setting

y = x

gives

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

, which yields

x = 2.4

. The intersection point is

(2.4, 2.4)

. Because line

M

is perpendicular to line

L

, its slope is the negative reciprocal of

-\frac{2}{3}

, which is

p = 1.5

. Using point-slope form with the intersection point

(2.4, 2.4)

, the equation of line

M

is

y - 2.4 = 1.5(x - 2.4)

, which simplifies to

y = 1.5x - 1.2

. Thus,

q = -1.2

. The sum

p + q

is

1.5 - 1.2 = 0.3

, or

\frac{3}{10}

.

Adım Adım Çözüm

1

Find the equation of line

L

using the given points

(0, 4)

and

(6, 0)

.

The slope of line

L

is

m_L = \frac{0 - 4}{6 - 0} = -\frac{2}{3}

. Since the

y

-intercept is

(0, 4)

, the equation of line

L

is

y = -\frac{2}{3}x + 4

.

To find where line

L

intersects another line, we first need to establish its linear equation.

2

Find the intersection point of line

L

and the line

y = x

.

Substitute

y = x

into the equation for line

L

:

x = -\frac{2}{3}x + 4 \implies \frac{5}{3}x = 4 \implies x = 2.4

. Thus, the intersection point is

(2.4, 2.4)

.

The problem states that line

M

intersects line

L

at a point on the line

y = x

.

3

Determine the equation of line

M

using the intersection point and the perpendicular relationship.

Since line

M

is perpendicular to line

L

, its slope

p

is the negative reciprocal of

-\frac{2}{3}

, which is

p = \frac{3}{2} = 1.5

. Using point-slope form at

(2.4, 2.4)

:

y - 2.4 = 1.5(x - 2.4) \implies y = 1.5x - 1.2

. This gives

q = -1.2

.

Perpendicular lines in the coordinate plane have slopes that multiply to negative one. We then use the point-slope formula to define the equation of line

M

.

4

Calculate the sum of

p

and

q

.

p + q = 1.5 + (-1.2) = 0.3 = \frac{3}{10}

.

The question asks for the value of the expression

p + q

.

Anahtar Kavram

Perpendicular lines and linear graph intersections

Tahmini Süre:3m 0s

Soru 72Soru →

In the $xy$ -plane, the graphs of the linear equations $y = 3x - 5$ and $y = -x + 7$ intersect at the point $(x, y)$ . What is the value of $x$ ?

1

3

4

6

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

To find the

x

-value of the intersection point of the two graphs, set the two equations equal to each other because both represent the same

y

-value at that point:

3x - 5 = -x + 7

. Adding

x

to both sides of the equation gives

4x - 5 = 7

. Adding

5

to both sides gives

4x = 12

. Dividing both sides by

4

yields

x = 3

.

Adım Adım Çözüm

1

Set the two expressions for

y

equal to each other to find the

x

-coordinate of the intersection point.

3x - 5 = -x + 7

Since both equations are solved for

y

, their right-hand sides must be equal at the point of intersection.

2

Add

x

to both sides of the equation to collect the variable terms on one side.

4x - 5 = 7

Moving the variable terms together allows us to isolate

x

.

3

Add

5

to both sides of the equation to isolate the term with

x

.

4x = 12

Grouping the constant terms on the opposite side prepares the equation for final division.

4

Divide both sides by

4

to find the value of

x

.

x = 3

This isolates

x

and gives the final solution.

Anahtar Kavram

Solving a system of linear equations by setting the equations equal to find the point of intersection in the coordinate plane.

Soru 73Soru →

In the equation below,

a

and

b

are constants.

a(3x - 5) - b(2x + 2) = 4x - 12

If the equation has infinitely many solutions for

x

, what is the value of

a + b

?

3

7

8

13

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

To find the value of

a + b

that results in infinitely many solutions, we first distribute the constants

a

and

b

to rewrite the equation as

(3a - 2b)x - (5a + 2b) = 4x - 12

. For a linear equation to have infinitely many solutions, the coefficient of

x

on both sides must be equal, and the constant terms on both sides must be equal. This gives the system of equations

3a - 2b = 4

and

5a + 2b = 12

. Adding these two equations yields

8a = 16

, which gives

a = 2

. Substituting

a = 2

into

3a - 2b = 4

yields

6 - 2b = 4

, which gives

b = 1

. Therefore, the value of

a + b

is

2 + 1 = 3

.

Adım Adım Çözüm

1

Distribute the constants

a

and

b

on the left side of the equation and group the terms.

(3a - 2b)x - (5a + 2b) = 4x - 12

To write the linear equation in the standard form

Ax + B = Cx + D

so that coefficients can be compared.

2

Set up a system of equations by equating the coefficient of

x

and the constant term on both sides of the equation.

3a - 2b = 4

and

-(5a + 2b) = -12

(which simplifies to

5a + 2b = 12

)

For a linear equation in one variable to have infinitely many solutions, the coefficient of

x

on both sides must be equal, and the constant terms on both sides must be equal.

3

Solve the system of equations by adding them to eliminate

b

.

8a = 16 \implies a = 2

Adding

3a - 2b = 4

and

5a + 2b = 12

eliminates

b

, allowing us to solve directly for

a

.

4

Substitute

a = 2

back into

3a - 2b = 4

to solve for

b

.

3(2) - 2b = 4 \implies 6 - 2b = 4 \implies -2b = -2 \implies b = 1

Substituting the value of

a

allows us to determine

b

.

5

Calculate the value of

a + b

.

a + b = 2 + 1 = 3

To find the sum of the two constants as requested.

Anahtar Kavram

Linear Equations in One Variable (Infinitely Many Solutions)

Soru 74Soru →

If $\frac{2}{3}(x - 4) = 6$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

To solve the linear equation

\frac{2}{3}(x - 4) = 6

, multiply both sides of the equation by

\frac{3}{2}

to isolate

x - 4

. This yields

x - 4 = 9

. Adding

4

to both sides of the equation isolates

x

and gives the value

x = 13

.

Adım Adım Çözüm

1

Multiply both sides of the equation

\frac{2}{3}(x - 4) = 6

by the reciprocal of the fraction, which is

\frac{3}{2}

.

x - 4 = 9

Multiplying a fraction by its reciprocal simplifies it to 1, leaving the term in parentheses isolated on the left side.

2

Add

4

to both sides of the equation

x - 4 = 9

.

x = 13

Adding

4

isolates the variable

x

on the left side.

Anahtar Kavram

Solving a one-variable linear equation using inverse operations.

Soru 75Soru →

In the $xy$ -plane, line $l$ passes through the point $(3, 4)$ and has a slope of $m$ , where $m > 0$ . Line $k$ is perpendicular to line $l$ and passes through the point $(2, 2)$ . If the sum of the $y$ -intercepts of line $l$ and line $k$ is $1$ , what is the value of $m$ ?

\frac{1}{3}

\frac{2}{3}

1

2

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The correct value of

m

is

2

.

The correct answer is the value

2

. By finding the equations of both lines in slope-intercept form, we express their

y

-intercepts in terms of

m

: the

y

-intercept of line

l

is

4 - 3m

and the

y

-intercept of line

k

is

\frac{2}{m} + 2

. Setting their sum to

1

gives the equation

3m^2 - 5m - 2 = 0

. Factoring this quadratic yields the solutions

m = -\frac{1}{3}

and

m = 2

. Since

m

must be positive,

m = 2

is the only valid solution.

Adım Adım Çözüm

1

Find the equation and

y

-intercept of line

l

.

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 4)

and slope

m

, the equation of line

l

is

y - 4 = m(x - 3)

, which simplifies to

y = mx + 4 - 3m

. Thus, the

y

-intercept of line

l

is

4 - 3m

.

Expressing the

y

-intercept of line

l

in terms of

m

allows us to use it in the sum equation.

2

Find the equation and

y

-intercept of line

k

.

Since line

k

is perpendicular to line

l

, its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Using the point-slope form with the point

(2, 2)

, the equation of line

k

is

y - 2 = -\frac{1}{m}(x - 2)

, which simplifies to

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

. Thus, the

y

-intercept of line

k

is

\frac{2}{m} + 2

.

Expressing the

y

-intercept of line

k

in terms of

m

allows us to use it in the sum equation.

3

Set up the equation for the sum of the

y

-intercepts and solve for

m

.

The sum of the

y

-intercepts is

(4 - 3m) + (\frac{2}{m} + 2) = 1

. Simplifying this equation gives

6 - 3m + \frac{2}{m} = 1 \implies 5 - 3m + \frac{2}{m} = 0

. Multiplying by

m

yields

3m^2 - 5m - 2 = 0

. Factoring the quadratic gives

(3m + 1)(m - 2) = 0

, which has solutions

m = -\frac{1}{3}

and

m = 2

. Since

m > 0

, we have

m = 2

.

To determine the unique positive slope that satisfies the given conditions.

Anahtar Kavram

Writing equations of perpendicular lines and finding their intercepts using slopes and points.

Soru 76Soru →

A food truck sells tacos and burritos. On Tuesday, the food truck sold a total of 120 tacos and burritos. The number of tacos sold, $t$ , was 20 more than the number of burritos sold, $b$ . This situation is represented by the system of equations below:

t + b = 120

t = b + 20

How many burritos did the food truck sell on Tuesday?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The food truck sold 50 burritos on Tuesday.

To find the number of burritos sold,

b

, we substitute

t = b + 20

into the first equation

t + b = 120

, giving

(b + 20) + b = 120

. Combining like terms yields

2b + 20 = 120

. Subtracting 20 from both sides results in

2b = 100

. Finally, dividing both sides by 2 gives

b = 50

. Thus, the food truck sold 50 burritos on Tuesday.

Adım Adım Çözüm

1

Substitute the expression for

t

from the second equation into the first equation

(b + 20) + b = 120

To eliminate the variable

t

and set up an equation with only one variable,

b

.

2

Combine like terms on the left side of the equation

2b + 20 = 120

To simplify the equation for solving.

3

Subtract 20 from both sides of the equation

2b = 100

To isolate the variable term on one side of the equation.

4

Divide both sides of the equation by 2

b = 50

To solve for the variable

b

representing the number of burritos.

Anahtar Kavram

Solving systems of linear equations using the substitution method

Soru 77Soru →

During a chemistry experiment, the temperature of a liquid sample is decreased at a constant rate. At the start of the experiment, the temperature of the liquid is $80^\circ\text{C}$ . After $4$ minutes, the temperature of the liquid is $68^\circ\text{C}$ . If $T(t)$ represents the temperature of the liquid, in degrees Celsius, $t$ minutes after the experiment starts, which of the following equations defines $T$ ?

T(t) = -3t + 80

T(t) = 3t + 80

T(t) = -\frac{1}{3}t + 80

T(t) = 80t - 3

Cevabı ve açıklamayı göster

Cevap: $T(t) = -3t + 80$

Cevap

The equation

T(t) = -3t + 80

correctly defines the temperature function.

The correct equation shows a starting value of

80

and a rate of change of

-3

, which corresponds to the initial temperature of

80^\circ\text{C}

and a decrease of

3^\circ\text{C}

per minute.

Adım Adım Çözüm

1

Identify the

y

-intercept (initial value) of the linear function.

The initial temperature at

t = 0

is

80^\circ\text{C}

, so the

y

-intercept

b

is

80

.

The initial value of a linear relationship in context corresponds to the

y

-intercept.

2

Calculate the slope (constant rate of change) using the coordinate points

(0, 80)

and

(4, 68)

.

The slope

m

is

\frac{68 - 80}{4 - 0} = \frac{-12}{4} = -3

.

The slope is defined as the change in the dependent variable (temperature) divided by the change in the independent variable (time).

3

Substitute the slope

m

and

y

-intercept

b

into the slope-intercept form

T(t) = mt + b

.

T(t) = -3t + 80

Substituting the specific slope and intercept values into the general form defines the particular function.

Anahtar Kavram

Determining a linear equation from a real-world scenario by identifying the initial value and constant rate of change.

Soru 78Soru →

If $\frac{3}{4}(8x - 12) - \frac{1}{2}(2x - 6) = 18$ , what is the value of $5x$ ?

24

30

12

27

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

By distributing the fractions to the terms inside the parentheses, we get

6x - 9 - x + 3 = 18

. Combining like terms on the left side yields

5x - 6 = 18

. Adding 6 to both sides gives the value of

5x

as 24.

Adım Adım Çözüm

1

Distribute the coefficients outside the parentheses to the terms inside.

\frac{3}{4}(8x) - \frac{3}{4}(12) - \frac{1}{2}(2x) + \frac{1}{2}(6) = 18

, which simplifies to

6x - 9 - x + 3 = 18

.

To eliminate the parentheses and prepare to combine like terms.

2

Combine like terms on the left side of the equation.

(6x - x) + (-9 + 3) = 18

, which simplifies to

5x - 6 = 18

.

To simplify the linear expression on the left side.

3

Isolate the term

5x

by adding 6 to both sides of the equation.

5x = 18 + 6

, which simplifies to

5x = 24

.

To find the value of

5x

directly as requested by the question.

Anahtar Kavram

Solving linear equations in one variable using the distributive property and combining like terms.

Soru 79Soru →

In the equation $3(4x + b) - 2(x - 5) = 2(5x + 8)$ , $b$ is a constant. If the equation has infinitely many solutions for $x$ , what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

b

is 2.

Distributing the constants in the equation

3(4x + b) - 2(x - 5) = 2(5x + 8)

yields

12x + 3b - 2x + 10 = 10x + 16

. Combining like terms on the left side simplifies the equation to

10x + 3b + 10 = 10x + 16

. For a linear equation in one variable to have infinitely many solutions, both sides of the equation must be identical. Since the coefficients of

x

are equal (

10 = 10

), the constant terms must also be equal:

3b + 10 = 16

. Solving for

b

yields

3b = 6

, which simplifies to

b = 2

.

Adım Adım Çözüm

1

Distribute the constants through the parentheses on both sides of the equation

3(4x + b) - 2(x - 5) = 2(5x + 8)

.

12x + 3b - 2x + 10 = 10x + 16

To eliminate parentheses and allow grouping of like terms.

2

Combine like terms on the left side of the equation.

10x + 3b + 10 = 10x + 16

To simplify the left-hand expression into the standard linear form.

3

Equate the constant terms on both sides of the equation.

3b + 10 = 16

A linear equation in one variable has infinitely many solutions when both sides are identical. Since the coefficients of the variable

x

are both 10, the constant terms must be equal.

4

Solve for

b

by isolating it.

b = 2

Subtracting 10 from both sides gives

3b = 6

, and dividing by 3 yields the final value.

Anahtar Kavram

Determining conditions for a linear equation in one variable to have infinitely many solutions (identity).

Soru 80Soru →

In the $xy$ -plane, line $l$ passes through the points $(-2, -3)$ and $(2, 5)$ . Line $k$ is parallel to line $l$ and has a $y$ -intercept of $(0, -1)$ . If the point $(a, 9)$ lies on line $k$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

a

is

5

.

To find the value of

a

, we first determine the slope of line

l

using the two given points,

(-2, -3)

and

(2, 5)

. The slope

m

is given by

\frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2

. Since line

k

is parallel to line

l

, it has the same slope of

2

. The

y

-intercept of line

k

is

(0, -1)

, so the equation of line

k

is

y = 2x - 1

. To find the value of

a

, we substitute the point

(a, 9)

into this equation:

9 = 2a - 1

. Solving for

a

gives

10 = 2a

, which simplifies to

a = 5

.

Adım Adım Çözüm

1

Calculate the slope of line

l

using the two given points.

The slope of line

l

is

2

.

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the points

(-2, -3)

and

(2, 5)

, we get

m = \frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2

.

2

Find the equation of line

k

.

The equation of line

k

is

y = 2x - 1

.

Line

k

is parallel to line

l

, so it has the same slope,

m = 2

. Its

y

-intercept is

(0, -1)

, which gives the equation

y = 2x - 1

in slope-intercept form.

3

Solve for

a

by substituting the point

(a, 9)

into the equation of line

k

.

The value of

a

is

5

.

Substituting

x = a

and

y = 9

into

y = 2x - 1

gives

9 = 2a - 1

. Solving for

a

yields

10 = 2a

, or

a = 5

.

Anahtar Kavram

Linear Functions and Graphs

Algebra

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