Question

How do I solve this system of equations?

`9a + 7b = -30`
`8b + 5c = 11`
`-3a + 10c = 73`

I've never seen a system of equations when more than one variable is missing in each equation.

Answer 1

`(a, b, c) = (-1, -3, 7)`

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

`-9a+30c = 219`

Adding this to the first equation, we get

`7b + 30c = 189" "`(*)

Multiplying the second equation by `6`, we get

`48b + 30c = 66`

Subtracting this from (*), we get

`-41b = 123`

`=>b = -123/41 = -3`

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

`-24+5c = 11`

`=> c = 7`

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

`9a - 21 = -30`

`=> a = -1`

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

`-3(-1) + 10(7) = 3+70 = 73`

Thus, we get the solution `{(a = -1), (b = -3), (c = 7):}`