In the system of equations below, a and b are constants. \begin{aligned} 3x - 4y &= 7 \\ a… — SAT

In the system of equations below, $a$ and $b$ are constants.

\begin{aligned} 3x - 4y &= 7 \\ ax - 12y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of $a + b$ ?

Answer: 30

Answer

For a system of two linear equations in two variables to have infinitely many solutions, the two equations must represent the same line. Comparing the

y

-coefficients shows that

-12

3

times

-4

. Multiplying the first equation by

3

gives the equivalent equation

9x - 12y = 21

. For this to be identical to the second equation,

ax - 12y = b

, the corresponding coefficients must be equal, meaning

a = 9

and

b = 21

. Therefore, the value of

a + b

9 + 21 = 30

Step-by-Step Solution

Identify the condition for a system of linear equations to have infinitely many solutions.

The two equations must represent the same line, making them equivalent equations.

Infinitely many solutions occur when the two equations in a system are scalar multiples of each other.

Align the coefficients of the

y

-terms in both equations.

Multiplying the first equation by

3

gives

9x - 12y = 21

The coefficient of

y

in the second equation is

-12

, which is

3

times the coefficient of

y

in the first equation (

-4

Equate the coefficients of the remaining terms to find the values of

a

and

b

a = 9

and

b = 21

For

9x - 12y = 21

and

ax - 12y = b

to be equivalent, their corresponding coefficients and constants must be equal.

Calculate the value of

a + b

a + b = 9 + 21 = 30

The problem asks for the value of the sum

a + b

Key Concept

Determining constants in a system of linear equations with infinitely many solutions

Estimated Time:1m 30s

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