In the equation below, a and b are constants. a(2x - 9) - 3(x - b) = 7x + 12 If the equati… — SAT

In the equation below, $a$ and $b$ are constants.

a(2x - 9) - 3(x - b) = 7x + 12

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

A
-19
B
10
C
11
19Answer

Answer

To find the value of

b

for which the equation has infinitely many solutions, we expand the left side to get

2ax - 9a - 3x + 3b = 7x + 12

. Grouping the terms by variable yields

(2a - 3)x + (3b - 9a) = 7x + 12

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

x

and the constant term on both sides must be equal. Setting the

x

-coefficients equal gives

2a - 3 = 7

, which simplifies to

2a = 10

and

a = 5

. Setting the constant terms equal gives

3b - 9a = 12

. Substituting

a = 5

into this equation yields

3b - 9(5) = 12

, which simplifies to

3b - 45 = 12

. Adding

45

to both sides gives

3b = 57

, which results in

b = 19

Step-by-Step Solution

Expand and simplify the left side of the equation.

(2a - 3)x + (3b - 9a) = 7x + 12

To compare coefficients with the right side of the equation, we need to group the terms on the left side.

Set the coefficients of

x

on both sides equal to each other to solve for

a

2a - 3 = 7 \implies 2a = 10 \implies a = 5

For an equation to have infinitely many solutions, the coefficients of the variable on both sides must be equal.

Set the constant terms on both sides equal to each other, substitute the value of

a

, and solve for

b

3b - 9a = 12 \implies 3b - 9(5) = 12 \implies 3b - 45 = 12 \implies 3b = 57 \implies b = 19

For an equation to have infinitely many solutions, the constant terms on both sides must also be equal.

Key Concept

Linear equations with infinitely many solutions require the coefficients of the variable to be equal and the constant terms to be equal on both sides of the equation.

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