Question

Decide which variable to eliminate when using linear combinations to solve the system. Explain your thinking. $$\begin{array}{l}\frac{1}{3} x+6 y=6 \\\\-x+3 y=3\end{array}$$

Short answer

The variable chosen to eliminate was x. After elimination, the solution to the system is x = 0, y = 1.

Step-by-step solution

Multiply the Equations

To eliminate the x-variable, both equations should be multiplied such that the coefficients of x are opposites. Therefore, multiply the first equation by 3 and the second by 1. The equations become: \\(1) \: x + 18y = 18 \\(2) -x + 3y = 3

Addition of Modified Equations

Add equation (1) and equation (2), which results in the cancellation of the x-variable by adding matching terms: \\0 + 21y = 21. This simplifies to y = 1.

Solving for Remaining Variable

Now substitute y = 1 into one of the original equations. It's simpler to use the second equation, -x + 3y = 3, for this. With y = 1, the equation becomes -x + 3 = 3, which simplifies to x = 0.