Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given y = x + 2 and –4x + y = –1 ?

Answer 1

Convert both equations to slope-intercept form
if they are

# { ("identical", rarr, "many solutions"), ("same slope, different intercepts",rarr,"no solutions"), ("otherwise",rarr,"one solution"):}#

Explanation:

Identical equations have solutions for every coordinate on either line.

Lines with the same slope but not through the same y-intercept are parallel (but not identical).

Otherwise the lines represented by the equations cross at exactly one location

For the given example:
[1]`color(white)("XXX")y=x+2`
[2]`color(white)("XXX")-4x+y=-1`

converting into explicit slope-intercept form: `y=color(red)(m)x+color(blue)(b)`

[3]`color(white)("XXX")y=color(red)(1)x+color(blue)(2)`
[4]`color(white)("XXX")y=color(red)(4)x+color(blue)((-1))`
and
there is exactly one solution.

Although it isn't requested by this question, we could go on to determine this solution:

Multiply [3] by `4`
[5]`color(white)("XXX")4y=4x+8`

Subtract [4] from [5]
[6]#color(white)("XXX")-3y=-9

Divide both sides by `(-3)`
[7]`color(white)("XXX")y=3`

Substitute `(3` for `y` in [1]
[8]`color(white)("XXX")3=x+2`

`rarrcolor(white)("XXX")x=1`