Question

How do you solve `y-6x=22` and `4y+17=-11x` using substitution?

Answer 1

`x = -3`
`y = 4`

Explanation:

Solving through Substitution

`y - 6x = 22`
`4y + 17 = -11x`

First, we want to solve for a variable that can be easily found. Then, we'll plug the variable in to solve for the variables. Let's start with finding what y is since it looks easy to solve for.

`y - 6x = 22`

Add 6x to both sides to negate `-6x`. This'll give us `y`'s value. You should now have:

`y = 6x + 22`

Substitute in this value for y in the second equation to solve for the variables:

`4y + 17 = -11x`
`4(6x + 22) + 17 = -11x`

Distribute.

`24x + 88 + 17 = -11x`

Combine like terms. `88 + 17` is `105`, so:

`24x + 105 = -11x`

Subtract `24x` from both sides to negate `24x` in order to solve for x. The reason why we aren't adding `11x` to both sides is because it'll make the equation unbalanced. `-11x - 24x` is basically the same as `11 + 24`because they're the same sign, so they combine to become `-35x`. So:

`105 = -35x`

Divide by -35 to solve for x.

`105/-35` = `x`

`x = -3`

Now, plug this variable back into this equation to solve for y:

`y = 6x + 22`
`y = 6(-3) + 22`

Distribute.

`y = -18 + 22`

Add.
`y = 4`

Confirm this by plugging it back into the two equations:

`y - 6x = 22`
`4 - 6(-3) = 22`

Distribute.

`4 + 18 = 22`
`22 = 22`

`4y + 17 = -11x`
#4(4) + 17 = -11(-3) #16 + 17 = 33# #33 = 33#

Confirmed: these are the correct variables.

Answer 2

See a solution process below:

Explanation:

Step 1) Solve the first equation for `y`:

`y - 6x + color(red)(6x) = 22 + color(red)(6x)`

`y - 0 = 22 + 6x`

`y = 22 + 6x`

Step 2) Substitute `(22 + 6x)` for `y` in the second equation and solve for `x`:

`4y + 17 = -11x` becomes:

`4(22 + 6x) + 17 = -11x`

`(4 * 22) + (4 * 6x) + 17 = -11x`

`88 + 24x + 17 = -11x`

`88 + 17 + 24x = -11x`

`105 + 24x = -11x`

`105 - color(red)(105) + 24x + color(blue)(11x) = -11x + color(blue)(11x) - color(red)(105)`

`0 + (24 + color(blue)(11))x = 0 - 105`

`35x = -105`

`(35x)/color(red)(35) = -105/color(red)(35)`

`(color(red)(cancel(color(black)(35)))x)/cancel(color(red)(35)) = -3`

`x = -3`

Step 3) Substitute `-3` for `x` in the solution to the first equation at the end of Step 1 and calculate `y`;

`y = 22 + 6x` becomes:

`y = 22 + (6 xx -3)`

`y = 22 - 18`

`y = 4`

The Solution Is:

`x = -3` and `y = 4`

Or

`(-3, 4)`