Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 3x - y = -2 and -4x + 2y= 5?

Answer 1

Two lines with different slopes will always intersect at exactly one point (i.e. they will have exactly one solution).

Explanation:

`3x-y=-2` has a slope of `3`
`-4x+2y=5` has a slope of `2`

This system of lines has one solution.

Bonus 1: Slope Determination
Given a linear equation in standard form `Ax+By=C`
its slope is `-A/B`.

Bonus 2: Determining the Solution for the Given System
[1]`color(white)("XXX")3x-y=-2`
[2]`color(white)("XXX")-4x+2y=5`

Multiply [1] by `2`
[3]`color(white)("XXX")6x-2y=-4`

Add [2] and [3]
[4]`color(white)("XXX")2x=1`

Divide both sides by `2`
[5]`color(white)("XXX")x=1/2`

Substitute `1/2` for `x` in [2]
[6]`color(white)("XXX")-4(1/2)+2y=5`

Simplify
[7]`color(white)("XXX")2y=7`

Divide by `2`
[8]`color(white)("XXX")y=3 1/2`

`(x,y)=(1/2,3 1/2)`

Bonus 3: For Those Who Like To Be Picky
The rule stated in the "Answer" assumes the two lines lie in a common plane. Two lines (in 3 dimensional space, for example) may not intersect even with different slopes.