Question

How do you solve the following linear system: # -2x+5y=4, 3x+8y=1 #?

Answer 1

`(x,y)=color(blue)(""(-27/31,14/31))`

Explanation:

Given
[1]`color(white)("XXX")-2x+5y=4`
[2]`color(white)("XXX")3x+8y=1`

To eliminate the `x` components we can convert the given equations
into ones with identical coefficients for `x`
by multiplying [1] by `3` and [2] by `2`
[3]`color(white)("XXX")-6x+15y=12`
[4]`color(white)("XXX")6x+16y=2`

Adding [3] and [4]
[5]`color(white)("XXX")31y=14color(white)("XX")rarrcolor(white)("XX")y=14/31`

We cold plug this value back into one of the original equations and solve for `x`
or (the method I find simpler in this case) repeat the above process to eliminate `y` from the original given equations.
[6]`color(white)("XXX")-16x+40y=32`
[7]`color(white)("XXX")15x+40y=5`

Subtracting [7] from [6]
[8]`color(white)("XXX")-31x = 27color(white)("XX")rarrcolor(white)("XX")x=-27/31`