Question

Use the linear system below. $$-x+y=5 \quad \text { Equation } 1$$ $$\frac{1}{2} x+y=8 \quad \text { Equation } 2$$ Solve for \(y\) in the equation that you chose.

Short answer

The solution for \(y\) is 7 and the solution for \(x\) is 2.

Step-by-step solution

Identify the variables

Looking at Equation 1, we see it is in the form of -x + y = 5, where \(x\) and \(y\) are the variables to solve for.

Solve for y

We are asked to solve for \(y\), so rearrange Equation 1, adding x to both sides to get an expression for y in terms of x: \(y = x + 5\).

Substitute into the other equation

Now take this expression for y and substitute it into Equation 2: \(\frac{1}{2}x + (x + 5) = 8\). This creates an equation in one variable, x, which can be solved.

Simplify and solve for x

Solve for \(x\) by first simplifying the equation: \(\frac{3}{2}x + 5 = 8\), then subtract 5 from each side: \(\frac{3}{2}x = 3\), and finally multiply each side by \(\frac{2}{3}\) to isolate x: \(x = 2\).

Substitute x = 2 into the equation for y

Now substitute \(x = 2\) into the expression for \(y\) derived in step 2 to get \(y = 2 + 5 = 7\).