Question

Use the substitution method to solve the linear system. $$\begin{aligned} &7 g+h=-2\\\ &g-2 h=9 \end{aligned}$$

Short answer

The solution to the system of equations is \(g = 1\), \(h = -13/3\).

Step-by-step solution

Express one variable in terms of the other

From the second equation \(g-2 h=9\), express \(g\) in terms of \(h\). That means, isolate \(g\), hence \(g = 9 + 2h\)

Substitute the expression into the other equation

Substitute \(g = 9 + 2h\) from step 1 into the first equation which is \(7g + h = -2\). This gives: \(7(9 + 2h) + h = -2\), simplify this to give \(63 + 14h + h = -2\) which results in \(15h = -65\)

Solve for the first variable

Solve for \(h\) in step 2, you get \(h = -65/15 = -13/3\).

Solve for the second variable

Substitute \(h = -13/3\) into the equation found in step 1, which is \(g = 9 + 2h\), to get \(g = 9 + 2(-13/3) = 9 - 26/3 = 1\).