Question

How do you write the equation of a line in slope intercept, point slope and standard form given Point: (-5, 4) Slope: m = 2?

Answer 1

Slope intercept: `y=2x+14`
Point-slope: `y=2x+14`
Standard form: `2x-y=-14`

Explanation:

Slope intercept equation is `y=mx+c`, where `m` is the slope/gradient, `c` is the y-intercept and `(x, y)` is the format of the points/coordinates given `(-5, 4)`.

The point-slope is given as `y-y_1=m(x-x_1)`.

The standard form of an equation is `Ax+By=C`.

Now, for slope intercept, substitute all the values you have been given and solve for `c`, the y-intercept:

`y=mx+c`
`4=2(-5)+c`
`4=-10+c`
`c=14`

Then substitute only `m` and `c` to get the equation of the slope-intercept:

`y=2x+14`

Now for the point-slope (which is just another method of finding the equation of the slope), you find that there's `y_1` and `x_1` in the formula, this corresponds to the `x` and `y` of the coordinates of the points you have been given, `(-5, 4)`. Where `(x_1, y_1)`.

So, substitute the values of `y_1`, `x`, and `x_1` and isolate `y` to find the equation of the point-slope:

`y-y_1=m(x-x_1)`
`y-(4)=2(x-(-5))`
`y-4=2(x+5)`
`y-4=2x+10`
`y=2x+10+4`
`y=2x+14`

As you can see, both methods give the same results by substitution.

For the standard form, you use the format of the equation of the point-slope but rearrange it so that the `x` and `y` terms are on one side of the equal sign and the constants are on the other, `Ax+By=C`:

`y-4=2(x+5)`
`y-4=2x+10`
`-2x+y=14`

This is almost in the format of `Ax+By=C` but `Ax` cannot have a negative sign in front of it. To get rid of this negative, divide both sides of the equation by `-1`:

`(-2x+y)/-1=14/-1`
`2x-y=-14`

Which can also be solved by taking the results from one of the other equations and rearranged to follow the format of `Ax+By=C`.

Hope this helps.