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?

1 Answer
Oct 23, 2015

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.