Question

Explain the steps you would use to solve the system of equations using linear combinations. Then solve the system. $$\begin{aligned} &x+3 y=6\\\ &x-3 y=12 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 9\) and \(y = -1\).

Step-by-step solution

Identify the System

The given system of equations is: \(x + 3y = 6\) and \(x - 3y = 12\)

Add the Equations

Add the two equations together: \((x + 3y) + (x - 3y) = 6 + 12\). This eliminates the 'y' variable from the equation, as \(3y - 3y\) equals zero.

Solve for x

Solving for 'x', we have \(2x = 18\), which simplifies to \(x = 9\).

Substitute x value

Now substitute \(x = 9\) into the first equation - \(x + 3y = 6\) - which gives \(9 + 3y = 6\).

Solve for y

Solving this equation for 'y', we have \(3y = -3\), which simplifies to \(y = -1\)