Question

How do you solve the system of equations of `6x + 9y = - 18` and `4x - 9y = - 42`?

Answer 1

Simultaneous equations

`x = -6`

`y =2`

Explanation:

Let `6x+9y = -18` be equation 1
Let `4x-9y = -42` be equation 2

From 1,

`9y = -18-6x " "`(equation 3)

Substitute into 2,

`4x -(-18 -6x) = -42`

`4x+18+6x=-42`

`10x = -42-18`

`10x = -60`

`x = -6`

Substitute `x = -6` into equation 3

`9y = -18-6 * (-6)`

`9y = -18 +36`

`y = 18/9 = 2`

Answer 2

Set them up as "layered" algebra problems.

Explanation:

We align the variables under each other, and then use algebra to manipulate the expressions to eliminate one unknown and solve for the other, Then we take that solved variable and use it to find the second (or additional) one. Finally, take both solutions and put them in first solved equation again to validate the answer.

`6x + 9y = −18`
`4x − 9y = −42` We already have equal (and opposite) coefficients on the 'y' terms, so we can just 'add' the expressions.
`6x + 9y = −18`
`4x − 9y = −42`
`10x = -60` ; `x = -6` The first solution.

Taking the second equation:
`4(-6) − 9y = −42` ; `-24 -9y = -42` ; `-9y = -18` ; `y = 2` The second solution.

Put them both into the first equation to check:
`6(-6) + 9(2) = −18` ; `-36 + 18 = -18` ; `-18 = -18` CORRECT!