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Worked example: intercepts from an equation

Let's find the x- and y-intercepts of the equation 2y + 1/3x = 12, where a line crosses the x-axis (the x-intercept) and the y-axis (the y-intercept) on a graph. To find the x-intercept, we make y equal to 0 and solve the equation for x. To find the y-intercept, we make x equal to 0 and solve for y. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

We're told to find the x- and y-intercepts for the graph of this equation: 2 y plus 1/3x is equal to 12. And just as a bit of a refresher, the x-intercept is the point on the graph that intersects the x-axis. So we're not above or below the x-axis, so our y value must be equal to 0. And by the exact same argument, the y-intercept occurs when we're not to the right or the left of the y-axis, so that's when x is equal to 0. So let's set each of these values to 0 and then solve for what the other one has to be at that point. So for the x-intercept, when y is equal to 0, let's solve this. So we get 2 times 0, plus 1/3x is equal to 12. I just set y is equal to 0 right there, right? I put 0 for y. Well, anything times 0 is just 0, so you're just left with 1/3x is equal to 12. To solve for x, you can think of it as either dividing both sides by 1/3, or we can multiply both sides by the reciprocal of 1/3. And the reciprocal of 1/3 is 3, or you can even think of it as 3 over 1. So times 3 over 1. And so we're left with 3 times 1/3, that just cancels out, so you're left with x is equal to 12 times 3, or x is equal to 36. So when y is equal to 0, x is 36. So the point 36 comma 0 is on the graph of this equation. And this is also the x-intercept. Now, let's do the same thing for the y-intercept. So let's set x equal 0, so you get 2y plus 1/3, times 0 is equal to 12. Once again, anything times 0 is 0. So that's 0, and you're just left with 2y is equal to 12. Divide both sides by 2 to solve for y, and you're left with y is equal to 12 over 2, is 6. So the y-intercept is when x is equal to 0 and y is equal to 6. So let's plot these two points. I'll just do a little hand-drawn graph, and make it clear what the x- and the y-intercepts are. So let me draw-- that's my vertical axis, and that is my horizontal axis-- and we have the point 36 comma 0. So this is the origin right here, that's the x-axis, that's the y-axis. The point 36 comma 0 might be all the way over here. So that's the point 36 comma 0. And if that's 36, then the point 0, 6 might be right about there. So that's the point 0, 6. And the line will look something like this. I'm trying my best to draw a straight line. And notice where the line intercepted or intersected the y-axis, that's the y-intercept, x is 0, because we're not to the right or the left of it. Where the line intersected the x-axis, y is 0, because we're not above or below it.