For the system of equations shown below, (x, y) is the unique solution: \begin{aligned} \f… — SAT

For the system of equations shown below,

(x, y)

is the unique solution:

\begin{aligned} \frac{3}{4}(2x - y) &= \frac{2}{3}(x + 2y) \\ \frac{1}{2}(x - y) + \frac{5}{6}(2x + y) &= 11.5 \end{aligned}

What is the value of

x + y

Answer: 7

The value of

x + y

is 7.

To find the value of

x + y

, we first simplify the first equation by multiplying both sides by the least common multiple of

3

and

4

, which is

12

. This yields

9(2x - y) = 8(x + 2y)

, which simplifies to

18x - 9y = 8x + 16y

, and further simplifies to

10x = 25y

, or

x = 2.5y

. Next, we substitute

x = 2.5y

into the second equation:

\frac{1}{2}(2.5y - y) + \frac{5}{6}(2(2.5y) + y) = 11.5

. Simplifying the terms gives

0.75y + 5y = 11.5

, or

5.75y = 11.5

, which yields

y = 2

. Substituting

y = 2

back into

x = 2.5y

gives

x = 5

. Thus, the value of

x + y

5 + 2 = 7

Clear the fractions in the first equation by multiplying by the least common multiple of the denominators.

10x = 25y

, which simplifies to

x = 2.5y

To express one variable in terms of the other for substitution.

Substitute the expression for

x

into the second equation and solve for

y

y = 2

To find the numerical value of one of the variables.

Substitute the value of

y

back into the simplified first equation to find

x

x = 5

To find the numerical value of the remaining variable.

Add the values of

x

and

y

to find

x + y

To answer the specific question asked.

Solving systems of linear equations using algebraic simplification and substitution

Question