In the equation below, k is a constant. 15x - 2(3x - k) = 9x + 10 If the equation has infi… — SAT

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

15x - 2(3x - k) = 9x + 10

If the equation has infinitely many solutions, what is the value of $k$ ?

A
-5
B
20
5Answer
D
8

Answer

The correct value of

k

5

, which makes the equation have infinitely many solutions.

The correct answer is

5

because distributing

-2

through the parentheses on the left side of the equation yields

15x - 6x + 2k = 9x + 10

. Combining the

x

terms gives

9x + 2k = 9x + 10

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

x

are already equal (

9 = 9

), we set the constants equal to each other (

2k = 10

), which simplifies to

k = 5

Step-by-Step Solution

Distribute the constant

-2

to both terms inside the parentheses on the left side of the equation.

15x - 6x + 2k = 9x + 10

Applying the distributive property correctly simplifies the terms inside the parentheses.

Combine the like terms of

x

on the left side of the equation.

9x + 2k = 9x + 10

Subtracting

6x

from

15x

simplifies the left side of the equation to have a single

x

term.

Set the constant terms on both sides equal to each other to find the condition for infinitely many solutions.

2k = 10

For a linear equation in one variable to have infinitely many solutions, the coefficients of the variable on both sides must be equal, and the constant terms on both sides must also be equal.

Solve for

k

by dividing both sides of the equation by

2

k = 5

Isolating the variable

k

yields the final value.

Key Concept

For a linear equation in one variable to have infinitely many solutions, the equation must simplify to an identity where the variable coefficients are equal and the constant terms are equal on both sides of the equation.

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic