Question

Use substitution to solve the linear system. $$\begin{aligned} &g-5 h=20\\\ &4 g+3 h=34 \end{aligned}$$

Short answer

The solution to the linear system is \(g=10\) and \(h=-2\).

Step-by-step solution

Rearrange the first equation in terms of g

First we have the equation \(g-5h=20\). We want to express \(g\) in terms of \(h\), we can do that by adding \(5h\) to both sides of the equation. We get \(g=20+5h\).

Substitute the expression of g into the second equation

Next, we replace \(g\) in the second equation \(4g+3h=34\) by its expression in terms of \(h\) obtained from the first equation. So, we substitute \(g=20+5h\) in the second equation and get \(4(20+5h)+3h=34\) which simplifies to \(80+20h+3h=34\).

Solve for h

Now, simplify the equation to get the value of \(h\). The equation \(80+20h+3h=34\) simplifies to \(23h=-46\). We isolate \(h\) by dividing by 23 on both sides, that gives us \(h=-2\).

Substitute \(h=-2\) into the first equation to determine g

Substitute \(h=-2\) into the first equation \(g = 20+5h\) to obtain \(g\). Doing so gives \(g = 20+5(-2)\), which simplifies to \(g=20-10 =10\).