Consider the system of linear equations in two variables x and y shown below: \begin{align… — GRE General Test

Consider the system of linear equations in two variables

x

and

y

shown below:

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

(x, y)

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

\frac{y}{x}

The value of

\frac{y}{x}

\frac{1}{2}

Solving the system by substitution gives

x = 4

and

y = 2

. Dividing

y

x

gives

\frac{2}{4} = \frac{1}{2}

, which is the target ratio.

Isolate

x

in the second equation.

x = 4y - 4

Expressing

x

in terms of

y

allows for substitution into the first equation.

Substitute

x = 4y - 4

into the first equation.

3(4y - 4) + 2y = 16 \implies 12y - 12 + 2y = 16 \implies 14y = 28 \implies y = 2

Solving the single-variable equation determines the value of

y

Substitute

y = 2

back into the isolated expression for

x

x = 4(2) - 4 = 4

Determines the corresponding value of

x

Compute the required ratio

\frac{y}{x}

\frac{y}{x} = \frac{2}{4} = \frac{1}{2}

Evaluates the final expression requested by the prompt.

Solving systems of linear equations using substitution or elimination to evaluate a combined expression.

Question