The system of linear equations shown below contains constants a and b. \begin{aligned} \fr… — SAT

The system of linear equations shown below contains constants

a

and

b

\begin{aligned} \frac{1}{2}(ax - 4y) &= x - \frac{1}{2}y + 6 \\ 2x - y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of

a + b

Answer: 12

Answer

The correct answer is 12.

To find the value of

a + b

that yields infinitely many solutions, we rewrite both equations in the standard form

Ax + By = C

. Simplifying the first equation gives

(a - 2)x - 3y = 12

. Multiplying the second equation

2x - y = b

3

gives

6x - 3y = 3b

. For the system to have infinitely many solutions, the two equations must be equivalent, meaning

a - 2 = 6

(which gives

a = 8

) and

3b = 12

(which gives

b = 4

). The sum of these values is

8 + 4 = 12

Step-by-Step Solution

Distribute the fraction

\frac{1}{2}

on the left side of the first equation.

\frac{1}{2}ax - 2y = x - \frac{1}{2}y + 6

To expand the expression and prepare it for simplification.

Group the

x

and

y

terms on the left side of the equation and the constants on the right side.

\left(\frac{1}{2}a - 1\right)x - \frac{3}{2}y = 6

To write the equation in standard linear form.

Multiply the entire equation by

2

to eliminate the fractional coefficients.

(a - 2)x - 3y = 12

To simplify comparison with the second equation by working with integer coefficients.

Multiply the second equation,

2x - y = b

, by

3

to align the

y

-coefficients with the first equation.

6x - 3y = 3b

Two linear equations have infinitely many solutions if they represent the same line, which requires matching coefficients and constants.

Equate the corresponding

x

-coefficients and constant terms from

(a - 2)x - 3y = 12

and

6x - 3y = 3b

a - 2 = 6 \implies a = 8

and

3b = 12 \implies b = 4

To solve for the values of the constants

a

and

b

Calculate the sum of

a

and

b

8 + 4 = 12

To find the final requested value of

a + b

Key Concept

Determining parameters for infinitely many solutions in a system of linear equations

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Step-by-Step Solution

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