Question

Use substitution to solve the linear system. $$\begin{aligned} &2 x+y=4\\\ &-x+y=1 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 1\) and \(y = 2\).

Step-by-step solution

Express One Variable

First, let's express \(x\) from the second equation: \(-x+y=1\) becomes \(x=y-1\).

Substitute Expression Into Other Equation

Now, substitute \(x = y - 1\) from step 1 into the first equation \(2x+y=4\). This gives the equation \(2(y-1)+y=4\).

Solve Equation

Next, solve the equation from step 2. The equation \(2(y-1)+y=4\) simplifies to \(2y - 2 + y = 4\), then further simplifies to \(3y - 2 = 4\) and finally to \(3y = 6\). Solving for y, we get \(y = 2\).

Substitute y into Expression

Substitute the found value for \(y\) (2) into the expression from Step 1. This gives \(x = 2 - 1 = 1\). Thus, we found the value of \(x\).