Consider the system of linear equations below. \begin{aligned} 3x - 4y &= 7 \\ 2x + 3y &= … — SAT

Consider the system of linear equations below.

\begin{aligned} 3x - 4y &= 7 \\ 2x + 3y &= 16 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + y$ ?

7Answer
B
6
C
14
D
-9

Answer

The correct answer is 7. By multiplying the first equation by 3 and the second by 4, we get

9x - 12y = 21

and

8x + 12y = 64

. Adding these equations eliminates the

y

terms and gives

17x = 85

, which simplifies to

x = 5

. Substituting

x = 5

into the second equation yields

2(5) + 3y = 16 \implies 10 + 3y = 16 \implies 3y = 6 \implies y = 2

. The sum of the coordinates is

x + y = 5 + 2 = 7

Step-by-Step Solution

Multiply the equations by constants to align the coefficients of one variable for elimination.

Multiplying the first equation by 3 gives

9x - 12y = 21

. Multiplying the second equation by 4 gives

8x + 12y = 64

This sets up the system so that the

y

terms have opposite coefficients (

-12

and

+12

), allowing them to be eliminated by addition.

Add the two modified equations to eliminate the

y

variable and solve for

x

(9x - 12y) + (8x + 12y) = 21 + 64 \implies 17x = 85 \implies x = 5

Adding the equations eliminates

y

and leaves a single linear equation in terms of

x

Substitute the value of

x

back into one of the original equations to solve for

y

Substituting

x = 5

into the second equation

2x + 3y = 16

gives

2(5) + 3y = 16 \implies 10 + 3y = 16 \implies 3y = 6 \implies y = 2

Plugging the known variable value back in allows us to solve for the remaining unknown variable.

Calculate the value of the requested expression

x + y

x + y = 5 + 2 = 7

The question specifically asks for the sum of

x

and

y

Key Concept

Solving systems of linear equations using the elimination method and evaluating linear combinations of the solutions.

Alternative Method

Alternatively, you can solve the first equation for

x

in terms of

y

3x = 7 + 4y \implies x = \frac{7+4y}{3}

. Substitute this expression into the second equation:

2\left(\frac{7+4y}{3}\right) + 3y = 16

. Multiply the entire equation by 3 to clear the fraction:

2(7+4y) + 9y = 48 \implies 14 + 8y + 9y = 48 \implies 17y = 34 \implies y = 2

. Substitute

y = 2

back to find

x

x = \frac{7+4(2)}{3} = 5

. Thus,

x + y = 5 + 2 = 7

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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