Consider the system of equations below: \begin{aligned} \frac{1}{3}x + \frac{1}{4}y &= 5 \\ x - y &= 8 \end{aligned} If (x, y) is the solution to the system, what is the value of y?

Consider the system of equations below: \begin{aligned} \frac{1}{3}x + \frac{1}{4}y &= 5 \… — SAT

Consider the system of equations below:

\begin{aligned} \frac{1}{3}x + \frac{1}{4}y &= 5 \\ x - y &= 8 \end{aligned}

(x, y)

is the solution to the system, what is the value of

y

Answer: 4

The value of

y

4

Substituting

x = y + 8

from the second equation into the first equation yields

\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

. Multiplying the entire equation by

12

to clear denominators gives

4(y + 8) + 3y = 60

, which simplifies to

7y + 32 = 60

. Solving for

y

gives

7y = 28

, or

y = 4

Express

x

in terms of

y

from the second equation.

x = y + 8

This allows for substitution into the first equation to solve for

y

directly.

Substitute

x = y + 8

into the first equation.

_\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

This reduces the system to a single-variable linear equation in terms of

y

Multiply the entire equation by the least common multiple of the denominators, which is

12

4(y + 8) + 3y = 60

This clears the fractional coefficients and simplifies the arithmetic.

Distribute the

4

and combine like terms.

7y + 32 = 60

Simplifies the equation to prepare for isolating the variable

y

Isolate the variable term by subtracting

32

from both sides, then dividing by

7

y = 4

This gives the final value of

y

that satisfies the system.

Solving systems of linear equations using substitution and clearing fractional coefficients