Question

How do solve the following linear system?: `11x + 3y + 7 = 0 , -6x-2y=-8`?

Answer 1

`x=-19/2,y=65/2`

Explanation:

Multiplying the first equation by `2` and the second equation by `3` and adding both

`4x=-38`
so
`x=-19/2`
plugging this in the second equation
`-6(-19/2)-2y=-8`
`57-2y=-8`
`-2y=-65`

`y=65/2`

Answer 2

`x=-19/2` and `y=65/2`

Explanation:

Given
[1]`color(white)("XXX")11x+3y+7=0`
[2]`color(white)("XXX")-6x-2y=-8`

To make this easier to work with I will convert [1] into the same standard form as [2] by subtracting `7` from both sides
[3]`color(white)("XXX")11x+3y=-7`

We need to eliminate one of the variables (either `x` or `y`) by combining these two equations.
For example, we might decide to eliminate the term containing the variable `y`.

To do this we need to convert equations [2] and [3] into equivalent forms with identical magnitude coefficients for `y`.

The least common multiple for the (magnitude of)coefficients of `y` in [2] and [3] is `6`, so we will convert each of [2] and [3] into equivalent forms with a term `+-6y`.

Multiplying [2] by `3`
[4]`color(white)("XXX")-18x-6y=-24`
Multiplying [3] by `2`
[5]`color(white)("XXX")22x+6y=-14`

Now if we add [4] and [5], we can eliminate the `y` term:
[6]`color(white)("XXX")4x=-38`

Dividing both sides of [6] by `4` we get
[7]`color(white)("XXX")x=-19/2`

We can now substitute `(-19/2)` for `x` back in one of our original equations. For demonstration purposes I will use equation [2]
[8]`color(white)("XXX")-6 * (-19/2)-2y=-8`

Simplifying
[9]`color(white)("XXX")57-2y=-8`

[10]`color(white)("XXX")-2y=-65`

[11]`color(white)("XXX")y=65/2`

Giving the final solution
`color(white)("XXX")x=-19/2 " or " -9 1/2`
and
`color(white)("XXX")y=65/2" or " 32 1/2`

[Since we used [2] to develop the value for `y` we should check this answer using [1] to verify that our results are correct].