How do you determine whether a linear system has one solution, many solutions, or no solution when given 4x - 7y = 10 and y = x - 7?

1 Answer
Oct 17, 2015

4x-7y=10 and y=x-7 have different slopes and therefore their lines must intersect in exactly one place.

Explanation:

Two straight lines will intersect in one location unless they are collinear or parallel; in either of these cases they will have the same slope.

4x-7y=10 can be written as y=4/7x-10/7
color(white)("XXXX")which is a linear equation in slope-intercept form with a slope of 4/7

y=x-7 is in slope intercept form with a slope of 1

The slopes are not equal
and therefore the lines intersect.

Bonus: Solution for the given equations
[1]color(white)("XXXX")4x-7y=10
[2]color(white)("XXXX")y=x-7

Substituting (x-7) for y from [2] in [1]
[3]color(white)("XXXX")4x-7(x-7)=10

[4]color(white)("XXXX")-3x=-39

[5]color(white)("XXXX")x=13

Substituting (-13) for x in [2]
[6]color(white)("XXXX")y = 13-7 = 6#