Question

The given line passes through the points (0,3) and (-4,0).what are 4 equations that describes this line?

Answer 1

We have a line that passes through point (0,3) and (-4,0).

Slope-intercept form:

We can calculate the slope of the line as:

$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=\frac{0-3}{-4-0}=\frac{-3}{-4}=\frac{3}{4}$

The y-intercept can now be calculated using the slope and one of the points:

$\begin{gathered} y=mx+b \\ 3=\frac{3}{4}\cdot0+b=0+b=b \\ b=3 \end{gathered}$

Then, we can express the equation as:

$y=\frac{3}{4}x+3$

Direct variation:

We can not express as a direct variation line as the line do not go through the origin.

Point-slope form:

This form is:

$y-y_1=m(x-x_1)$

We have to know the slope and one point in order to be able to write it.

The slope is 3/4 and one of the points is (0,3). We can write it as:

$y-3=\frac{3}{4}(x-0)$

Two intercept form:

This form is:

$ax+by=c$

We can write this for our case as:

When x=0, y=c/b=3, and when y=0, x=c/a=-4.

We can write that c=1, and then b=1/3 and a=-1/4.

Then, our equation becomes:

$\frac{-1}{4}x+\frac{1}{3}y=1$

If we multiply both sides by 12 (the common factor of 3 and 4), we would get:

$-3x+4y=12$

that is equivalent to the previous equation.