Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &3 y=-5 x+15\\\ &-y=-3 x+9 \end{aligned}$$

Short answer

The solution for the system of equations is \( x = 3 \) and \( y = -2 \)

Step-by-step solution

Rewrite Equations

First, rewrite the equations so that all variable terms are on the left side and the constants are on the right side. The system becomes: \[ \begin{aligned} &5x + 3y = 15\ &3x + y = 9 \end{aligned} \]

Scale Equations

Next, scale the equations so the coefficients of y's in both equations are the same. Multiply the first equation by 1 and the second equation by 3 to achieve this. \[ \begin{aligned} &5x + 3y = 15\ &9x + 3y = 27 \end{aligned} \]

Subtract Equations

Subtract the second equation from the first to solve for x. The second equation should vanish leaving us with \( -4x = -12 \). Solving this equation gives \( x = 3 \).

Find y Value

Plug x = 3 into the first original equation to solve for \( y \). The first equation yields \( 5*3 + 3y = 15 \) which simplifies to \( y = -2 \).