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.