Question

How do you solve the system of equations `8x + y = 31` and `7x + 5y = 23`?

Answer 1

`x = 4, y = -1` or `(4,-1)`

Explanation:

Using the elimination method, we can multiply the first equation by -5 to cancel out the y value and solve for x:

`[8x + y = 31] *`5 `=> -40x - 5y = -155`

By doing this, we have transformed `y` to `5y` which we can use to add to our other equation:

`-40x - 5y = -155`
`7x + 5y = 23`

When we add these two equations, `-5y` and `5y` cancel out, leaving only `x` to solve for. We are then left with the new equation (for `x`):

`-33x = -132`
`x=4`

Now that we have solved for x, we can plug this value into any of our other equations to find y. I would recommend the shorter equation (for your own ease):
`8(4) + y = 31`
`y = 31-32`
`y=-1`

By this point, we have establisehd that `x=4` and `y=-1`. All there is left to do is check:

`8(4) + (-1) = 31`
`32 - 1 = 31`
`31 = 31` `sqrt`

`7(4) +5(-1)=23`
`28-5=23`
`23=23` `sqrt`

Since our values both check out, our solution for this systems of equation is `x=4` and `y=-1` or `(4,-1)`.