Question

How do you solve this set of linear equations: `-4x - y = 8; 7x - y = - 3`?

Answer 1

`color(maroon)(x = -1, y = 4`

Explanation:

`-4x - y = 8, " Eqn (1)`

`7x - y = -3, " Eqn (2)`

Subtracting Eqn (1) from (2),

`7x - y - (-4x - y) = -3 - 8`

`7x + 4y - cancely + cancely = -11`

`11x = -11, " or " x = -11 / 11 = -1`

Substituting the value of x in Eqn (1),

`-4 * -1 - y = 8`

`-y = 8 - 4 = 4 " or " y = 4`

Answer 2

The solution is `(-1,-4)`.

Explanation:

Solve the system:

`"Equation 1:"` `-4x-y=8`

`"Equation 1:"` `7x-y=-3`

The solution is the point that both lines have in common, which is the point of intersection. I'm going to solve the system using substitution.

Solve Equation 1 for `y`.

`-y=4x+8`

Multiply both sides by `-1`. This will reverse the signs.

`y=-4x-8`

Substitute `-4x-8` for `y` in Equation 2 and solve for `x`.

`7x-(-4x-8)=-3`

Expand.

`7x+4x+8=-3`

Simplify.

`11x+8=-3`

Subtract `8` from both sides.

`11x=-3-8`

Simplify.

#11x=-11

Divide both sides by `11`.

`x=-11/11`

`x=-1`

Substitute `-1` for `x` in Equation 1. Solve for `y`.

`-4(-1)-y=8`

Simplify.

`4-y=8`

Subtract `4` from both sides.

`-y=8-4`

Simplify.

`-y=4`

Multiply both sides by `-1`. This will reverse the signs.

`y=-4`

Solution

The point of intersection is `(-1,-4)`.

graph{(-4x-y-8)(7x-y+3)=0 [-10.97, 11.53, -8.325, 2.925]}