Question

Use linear combinations to solve the system. $$\begin{aligned}&2 x+y=120\\\&x+2 y=120\end{aligned}$$

Short answer

The solution to the system of equations is \(x = 40\) and \(y = 40\).

Step-by-step solution

Set up the system of equations

From the given, the system of equations is \(\begin{aligned}2x+y = 120 \ x+2y = 120 \end{aligned}\)

Apply a linear combination to eliminate one variable

To eliminate 'y', multiply the first equation by 2 and the second by -1. This gives \(\begin{aligned}4x + 2y = 240 \ -x - 2y = -120 \end{aligned}\)

Add the multiplied equations together

Adding the two transformed equations together will eliminate 'y'. Doing this results in the equation \(3x = 120\)

Solve for x

Since we now have an equation with only 'x', we can solve for 'x'. Dividing both sides by 3, we get \(x = 40\)

Substitute x = 40 into the original equation

Substitute \(x = 40\) into the first equation of the system, we get \(2*40 + y = 120\). Simplifying this gives \(y = 120 - 80 = 40\)