# How do I solve this system of equations?

## $9 a + 7 b = - 30$ $8 b + 5 c = 11$ $- 3 a + 10 c = 73$ I've never seen a system of equations when more than one variable is missing in each equation.

Oct 16, 2016

$\left(a , b , c\right) = \left(- 1 , - 3 , 7\right)$

#### Explanation:

We can solve a system of equations with missing variables the same way we would one in which each equation contained all variables. It is as if the variables are there and have a coefficient of $0$. For this example, let's use elimination.

Multiplying the third equation by $3$, we get

$- 9 a + 30 c = 219$

Adding this to the first equation, we get

$7 b + 30 c = 189 \text{ }$(*)

Multiplying the second equation by $6$, we get

$48 b + 30 c = 66$

Subtracting this from (*), we get

$- 41 b = 123$

$\implies b = - \frac{123}{41} = - 3$

Substituting $b = - 3$ into the second equation, we get

$- 24 + 5 c = 11$

$\implies c = 7$

Substituting $b = - 3$ into the first equation, we get

$9 a - 21 = - 30$

$\implies a = - 1$

Finally, we check our newfound values $a = - 1$ and $c = 7$ in the third equation to make sure our solution works:

$- 3 \left(- 1\right) + 10 \left(7\right) = 3 + 70 = 73$

Thus, we get the solution $\left\{\begin{matrix}a = - 1 \\ b = - 3 \\ c = 7\end{matrix}\right.$