\begin{aligned} 3x - 2y &= 14 \\ 5x + 6y &= 42 \end{aligned} If (x, y) is the solution to … — SAT

\begin{aligned} 3x - 2y &= 14 \\ 5x + 6y &= 42 \end{aligned}

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

The value of

x - y

4

To find the value of

x - y

, we first solve the system of linear equations. Multiplying the first equation,

3x - 2y = 14

, by

3

gives

9x - 6y = 42

. Adding this equation to the second equation,

5x + 6y = 42

, eliminates

y

and yields

14x = 84

, which simplifies to

x = 6

. Substituting

x = 6

back into the first equation gives

3(6) - 2y = 14

, or

18 - 2y = 14

, which simplifies to

2y = 4

, so

y = 2

. Therefore, the value of

x - y

6 - 2 = 4

Multiply the first equation by

3

to align the coefficients of the

y

terms for elimination.

9x - 6y = 42

This makes the coefficients of

y

opposite in sign and equal in magnitude to the second equation.

Add the modified first equation to the second equation to eliminate

y

and solve for

x

(9x - 6y) + (5x + 6y) = 42 + 42 \implies 14x = 84 \implies x = 6

Adding the equations eliminates

y

, resulting in a single-variable linear equation.

Substitute

x = 6

back into the first equation to solve for

y

3(6) - 2y = 14 \implies 18 - 2y = 14 \implies -2y = -4 \implies y = 2

Substituting the known value of

x

yields the value of the other coordinate.

Evaluate the expression

x - y

using the values

x = 6

and

y = 2

x - y = 6 - 2 = 4

This computes the requested quantity.

Solving systems of linear equations using the elimination method and evaluating a linear combination of the variables.

Question