Question

Explain the steps you would use to solve the system of equations using linear combinations. Then solve the system. $$\begin{aligned} &x-3 y=0\\\ &x+10 y=13 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 3\) and \(y = 1\).

Step-by-step solution

Align the Equations

To solve this system using linear combinations, it's helpful to align the equations by variable. They are already in this form: \[\begin{aligned} &x-3y=0\ &x+10y=13 \end{aligned}\]The variable \(x\) is aligned in both equations, and the terms involving \(y\) are also aligned.

Combine the Equations to Eliminate One Variable

In order to eliminate one variable, subtract the second equation from the first \[\begin{aligned} &(x - 3y) - (x + 10y) = 0 - 13\ &\Rightarrow -13y = -13\end{aligned}\]

Solve for the Remaining Variable

Next, solve for the remaining variable \(y\) by dividing each side of the equation by -13: \[\begin{aligned} &\frac{-13y}{-13} = \frac{-13}{-13}\ &\Rightarrow y = 1\end{aligned}\]

Substitute and Solve for the Other Variable

The value \(y = 1\) can now be substituted into one of the original equations to solve for \(x\). Using the first equation \(x - 3y = 0\): \[\begin{aligned} &x - 3(1) = 0\ &\Rightarrow x = 3\end{aligned}\]