Question

6. Write an equation for the line that passes through (-4,4) and (8,8) in point-slope

form, slope-intercept form and standard form.
Point-Slope Form:
Slope-Intercept Form:
Standard Form:

Help!!

Answer 1

Point-slope form:

`y-4=\displaystyle(1)/(3)(x+4)`

`y-8=m(x-8)`

Slope-intercept form:

`y=\displaystyle(1)/(3)x+\displaystyle(16)/(3)`

Standard form:

`-x+3y=16`

Explanation:

Hi there!

1) Point-slope form

Point-slope form:
`y-y_1=m(x-x_1)` where
`(x_1,y_1)` is a point on the line and m is the slope

First, determine the slope:

`m=\displaystyle(y_2-y_1)/(x_2-x_1)` where two points on the line are
`(x_1,y_1)` and
`(x_2,y_2)`

Plug in the points (-4,4) and (8,8):

`m=\displaystyle(8-4)/(8-(-4))\\\\m=\displaystyle(8-4)/(8+4)\\\\m=\displaystyle(4)/(12)\\\\m=\displaystyle(1)/(3)`

Therefore, the slope of the line is
`\displaystyle(1)/(3)`. Plug this into
`y-y_1=m(x-x_1)`:

`y-y_1=\displaystyle(1)/(3)(x-x_1)`

Now, for
`(x_1,y_1)`, we can either plug in (-4,4) or (8,8):

`y-4=\displaystyle(1)/(3)(x-(-4))\\\\y-4=\displaystyle(1)/(3)(x+4)`

or

`y-8=m(x-8)`

2) Slope-intercept form

Slope-intercept form:
`y=mx+b` where m is the slope and b is the y-intercept (the value of y when the line crosses the y-axis)

`y=mx+b`

Plug in the slope

`y=\displaystyle(1)/(3)x+b`

Now, to determine the y-intercept, plug in one of the points (-4,4) or (8,8) and solve for b:

`8=\displaystyle(1)/(3)(8)+b\\\\8=\displaystyle(8)/(3)+b\\\\8-\displaystyle(8)/(3)=b\\\\b=(16)/(3)`

Therefore, the y-intercept is
`\displaystyle(16)/(3)`. Plug this back into
`y=\displaystyle(1)/(3)x+b`:

`y=\displaystyle(1)/(3)x+\displaystyle(16)/(3)`

3) Standard form

Standard form:

`Ax+By=C` where A, B, and C are numbers which are typically integers

`y=\displaystyle(1)/(3)x+\displaystyle(16)/(3)`

Organize this into standard form:

`-\displaystyle(1)/(3)x+y=\displaystyle(16)/(3)`

Multiply both sides by 3 so A, B, and C are integers:

`-x+3y=16`

I hope this helps!