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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#

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