Question

A line has an x-intercept of 3 and a y-intercept of -6.The slope-intercept form of its equation is:y = 2x + 3y = 2x - 6y = -2x + 3y = -2x - 6Which is the point-slope form of this equation?

Answer 1

The slope-intercept form of an equation of the line is:

$y=mx+b$

Where m is known as the slope and b as the y-intercept. We are being told that the y-intercept is -6 so we already have b=-6.

We still need to find the slope. In order to find it we can use the information about the x-intercept. The x-intercept is the x value that meets:

$mx+b=0$

We know that the x-intercept is so we have an equation for m (we also use the fact that b=-6):

$m\cdot3-6=0$

We add 6 at both sides of the equation:

$\begin{gathered} m\cdot3-6=0 \\ m\cdot3-6+6=0+6 \\ m\cdot3=6 \end{gathered}$

And we divide both sides by 3:

$\begin{gathered} m\cdot\frac{3}{3}=\frac{6}{3} \\ m=2 \end{gathered}$

Now that we found the slope we can write the complete equation of the line:

$y=2x-6$

Which means that the answer is the second option.

The slope-point form of an equation of the line is:

$y-d=m\cdot(x-c)$

Where m is the slope and (c,d) is a point through which the line passes.

We already know the slope, it's m=2 so we have:

$y-d=2(x-c)$

We need a point (c,d). If the line passes through that point then taking x=c and y=d is a solution to the equation in the slope-intercept form:

$\begin{gathered} y=2x-6 \\ d=2c-6 \end{gathered}$

If we take a random value for c, for example c=1 we get:

$\begin{gathered} d=2\cdot1-6 \\ d=-4 \end{gathered}$

So we know the line passes through point (c,d)=(1,-4). Then the point-slope form can be written as:

$y+4=2(x-1)$