Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &3 p-2=-q\\\ &-q+2 p=3 \end{aligned}$$

Short answer

The solution to the system of linear equations is \( p = 1 \) and \( q = -1 \).

Step-by-step solution

Rewrite the system maintaining the order of variables.

First, rewrite the system of equations maintaining the order of variables. To do so, rewrite \(-q\) as \(q\) on the left side. The system becomes: \[ \begin{aligned} &3p - q=2 \ &2p + q=3 \end{aligned} \]

Apply the linear combination.

Second, add the two equations together to eliminate 'q'. Performing \( (3p - q) + (2p + q) \) gives \( 5p = 5 \).

Solve for p.

Third, solve for 'p' by dividing both sides by 5. It gives \( p = 1 \).

Substitute p into the first original equation.

Fourth, replace 'p' with '\(1\)' into the first original equation \(3p - q = 2\), hence \(3(1) - q = 2\), which simplifies as \(3 - q = 2\).

Solve for q.

Finally, isolate 'q' on the left side by following these steps: Subtract '3' from both sides to get \(-q = 2 - 3\), then multiply each side by '-1' to get '\(q = -1\)'.