Question

How do solve the following linear system?: `-3x -8y = -9 , 4 x+ 16y = 2`?

Answer 1

`color(red)((x,y)=(8,-15/8))`

Explanation:

Given
[1]`color(white)("XXX")-3x-8y=-9`
[2]`color(white)("XXX")4x+16y=2`

Multiplying [1] by `2`
[3]`color(white)("XXX")-6x-16y=-18`

Adding [2] and [3]
`color(white)("XXX+(-")4x+16y=color(white)("xxxx")2`
`color(white)("XXX")+ul(-6x-16y=-18)`
[4]`color(white)("XXX")-2xcolor(white)("xxxxx")=-16`

Dividing both sides of [4] by `(-2)`
[5]`color(white)("XXX")x=8`

Substituting `8` for `x` in [1]
[6]`color(white)("XXX")(-3)xx8-8y=-9`
`color(white)([6])` `color(white)("XXX")-24-8y=-9`
`color(white)([6])` `color(white)("XXX")-8y=15`
[7]`color(white)("XXX")y=-15/8`

So we have `(x,y)=(8,-15/8)`

We can verify this result by substituting these values for `x` and `y` in equation [2]

Answer 2

See a solution process below:

Explanation:

**Step 1) Solve both of the equations for `8y`:

Equation 1:

`-3x - 8y = -9`

`color(red)(3x) - 3x - 8y = color(red)(3x) - 9`

`0 - 8y = 3x - 9`

`-8y = 3x - 9`

`color(red)(-1) xx -8y = color(red)(-1)(3x - 9)`

`8y = (color(red)(-1) xx 3x) - (color(red)(-1) xx 9)`

`8y = -3x + 9`

Equation 2:

`4x + 16y = 2`

`(4x + 16y)/color(red)(2) = 2/color(red)(2)`

`(4x)/color(red)(2) + (16y)/color(red)(2) = 1`

`2x + 8y = 1`

`-color(red)(2x) + 2x + 8y = -color(red)(2x) + 1`

`0 + 8y = -2x + 1`

`8y = -2x + 1`

Step 2) Because the left side of each equation are equal we can now equate the right side of each equation and solve for `x`:

`-3x + 9 = -2x + 1`

`color(red)(3x) - 3x + 9 - color(red)(1) = color(red)(3x) - 2x + 1 - color(red)(1)`

`0 + 8 = (color(red)(3) - 2)x + 0`

`8 = 1x`

`8 = x`

`x = 8`

Step 3) Substitute `8` for `x` into the solution for either equation in Step 1 and solve for `y`:

`8y = -2x + 1` becomes:

`8y = (-2 xx 8) + 1`

`8y = -16 + 1`

`8y = -15`

`(8y)/color(red)(8) = -15/color(red)(8)`

`(color(red)(cancel(color(black)(8)))y)/cancel(color(red)(8)) = -15/8`

`y = -15/8`

The Solution Is: `x = 8` and `y = -15/8` or `(8, -15/8)`