Question

Use the substitution method to solve the linear system. $$\begin{aligned} &x+y=12\\\ &x+\frac{3}{2} y=\frac{3}{2} \end{aligned}$$

Short answer

The solution to the system is \(x=33, y=-21\).

Step-by-step solution

Rearrange the first equation

First, rearrange the first equation to express \(x\) in terms of \(y\). So, equation \(x+y=12\) will become \(x=12-y\).

Substitution into the second equation

Plug \(x=12-y\) from the first equation into the second one. So now, instead of \(x+\frac{3}{2} y=\frac{3}{2}\), you will have \((12-y)+\frac{3}{2} y=\frac{3}{2}\).

Solve for \(y\)

Solving the equation in Step 2 will give \(y\). So, \((12-y)+\frac{3}{2} y=\frac{3}{2}\) simplifies to \(\frac{1}{2}y=-\frac{21}{2}\), which gives \(y=-21\).

Find \(x\)

Substitute \(y=-21\) back into the equation \(x=12-y\) from Step 1 to solve for \(x\). This gives \(x=12-(-21)\), which simplifies to \(x=33\).