Question

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?

Answer 1

`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#