In the equation \frac{3}{4}(8x - 12) + kx = 10x - 9, k is a constant. If the equation has … — SAT

In the equation $\frac{3}{4}(8x - 12) + kx = 10x - 9$ , $k$ is a constant. If the equation has infinitely many solutions, what is the value of $k$ ?

Answer: 4

The value of

k

4

Distributing the fraction on the left side of the equation yields

\frac{3}{4}(8x) - \frac{3}{4}(12) = 6x - 9

. Substituting this back gives

6x - 9 + kx = 10x - 9

. Factoring out

x

on the left side gives

(6 + k)x - 9 = 10x - 9

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

x

on both sides must be identical, and the constants must be identical. Since the constants on both sides are already

-9

, we set the coefficients equal:

6 + k = 10

. Subtracting 6 from both sides gives the correct value

k = 4

Distribute the fraction

\frac{3}{4}

to the terms inside the parentheses.

6x - 9 + kx = 10x - 9

To simplify the expression and eliminate the parentheses.

Factor out

x

from the terms on the left side of the equation.

(6 + k)x - 9 = 10x - 9

To group the

x

terms together to easily compare coefficients.

Set the coefficient of

x

on the left side equal to the coefficient of

x

on the right side.

6 + k = 10

For the equation to have infinitely many solutions, the coefficients of the variable on both sides must be equal when the constant terms are equal.

Solve for

k

by subtracting 6 from both sides of the equation.

k = 4

To isolate the constant

k

A linear equation in one variable has infinitely many solutions when it can be simplified to an identity of the form

Ax + B = Ax + B

, meaning both the coefficients of

x

and the constant terms on both sides of the equation are equal.

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