Question

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Substitute the value of \(x\) into your equation from Exercise 2 . What is the solution of the linear system?

Short answer

The solution to the linear system is \( x = 4 \) and \( y = 9 \).

Step-by-step solution

Rearrange Equation 1 for x

Starting from Equation 1, which is \( -x+y=5 \), we rearrange it to solve for x. This gives us \( x = y-5 \).

Substitute x in Equation 2

Next, we substitute the expression we found for x into Equation 2, this results in the new equation \( \frac{1}{2}(y-5)+y=8 \).

Solve the new equation

We can simplify and solve the equation for y. First we can distribute the \(\frac{1}{2}\) to get \( \frac{y}{2}-\frac{5}{2}+y=8 \). Adding like terms gives \( \frac{3}{2}y - \frac{5}{2}=8 \). Solving for y gives \( y=\frac{2}{3}(8+\frac{5}{2})=9 \).

Find the value of x

Now we go back to the expression we found for x in step 1, which was \( x = y-5 \). Substituting \( y = 9 \) gives \( x = 9-5 = 4 \).