MHB ACT Problem: Finding x-intercept Of LIne Given 2 Points

[816318]

What is the x-intercept of the line that passes through points (-3,7) and (6,4) in the standard (x,y) coordinate plane?

 

[MarkFL]

Re: ACT problem

Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[HallsofIvy]

Re: ACT problem

We really need to know what you do understand and what you can do on this kind of problem so that we know what hints will help you. Do you know that any (non-vertical) line can be written as y= ax+ b for some numbers a and b. Putting x= -3, y= 7 gives you one equation in a and b. Putting x= 6 and y= 4 gives a second equation is a and b. Can you solve those two equations for a and b? Do you know what the "intercept" of a linear equation means?

 

[816318]

Re: ACT problem

Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

I solved the slope for the two points, which is 1/3.
The y-intercept for is y=1/3x+8, I cannot go further past that point.

 

[MarkFL]

Re: ACT problem

I solved the slope for the two points, which is 1/3.
The y-intercept for is y=1/3x+8, I cannot go further past that point.

The slope $m$ of the line passing through the two given points is:

$$m=\frac{\Delta y}{\Delta x}=\frac{7-4}{-3-6}=\frac{3}{-9}=-\frac{1}{3}$$

And so, using the point-slope formula, we find:

$$y=-\frac{1}{3}(x-6)+4=-\frac{1}{3}x+6$$

Here's a plot of the two points and the line through them

[DESMOS=-4,7,0,10]y=-\frac{x}{3}+6;\left(-3,7\right),\left(6,4\right)[/DESMOS]

Now, to find the $x$-intercept, you can let $y=0$ (this is the equation for the $x$-axis) and solve for $x$...:D

 

[816318]

Re: ACT problem

Yes, as you put 0 for the y you get 18.
Which would be (18,0), thanks for the help!

 

[MarkFL]

Re: ACT problem

We can also write the line in the two-intercept form:

$$\frac{x}{18}+\frac{y}{6}=1$$

And this tells us the $x$-intercept is at $(18,0)$ and the $y$-intercept is at $(0,6)$. :D