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Slope-Intercept Form

Mike McGarry
Lesson by Mike McGarry
Magoosh Expert
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Now, we can talk about Slope-Intercept Form. In other words, y = mx + b form. The equation of any line can be stated in several different algebraically equivalent forms. We could have x and y on the same side, or solve for x, solve for y. These algebraic changes don't change the underlying mathematical information.

No matter how we rearrange the algebra, it's always the same line that's represented, that's an important idea. Of all possible algebraic representation, mathematicians have selected one as particularly useful. If we solve the equation of a line for y, getting y on one side by itself equal to everything else then we automatically put the equation into what's called slope-intercept form or y = mx + b form.

In this equation, y = mx + b, m is the slope of the line, and b is the y intercept. If we solve the algebra for y, we automatically get these two very important quantities, the slope and the y intercept. We can see that m must be the slope, because when x increases by 1, y has to increase by m.

We can see that b must be the y intercept because if we set x = 0 then y = b. It's easy to put any equation into slope intercept form, and once it's in that form, it's easy to understand where the graph will go. Starting from the y intercept, we can take slope steps that we discussed in the slope lesson. 1 unit to the right up, m slope are 1 unit to the left down, n slope etc.

Here's a practice question, pause the video and then will talk about this. Okay, find all the points j, k on the line y = -four-thirds x + 2. Subset j and k are both integers with absolute values less than or equal to 10. So this is in y = mx + b form, the slope is -four-thirds, the y intercept is positive 2, and so that's the point (0,-2). So that's actually one of the points.

That's one point, that satisfies this condition. Both the coordinates are integers and they both have absolute values less than or equal to 10. Now, what does a slope of four-thirds mean? Here, we're gonna take the numerical perspective on slope, slope means move right 3, down 4, or it could mean move left 3, up 4.

So we start at the point to (0, 2). Then if we move, left 3, down 4, we go from 0 to 3 and down from 2 to -2. If we go left 3, down 4 again, we'll get to 6 and then down 4 more down to -6. Left 3, down 4 again, we'll get to 9 and -10. So we have to stop there because we've reached an absolute value of 10, but all of those points work.

Now go back to (0, 2), now we're gonna go the other way. We're gonna go backwards 3, to the left 3, and up 4. So that would bring us to (-3,6), and then again, (-6,10) and then we have to stop there because again we've reached an absolute value of 10. And so there are 6 possible points. Here are the 6 possible points that satisfy this condition.

There are on that line, and they have x and y coordinates that are integers and both the x and y coordinates have absolute values less than 10. Here's another practice question, what is the slope of the line with the equation 3x + 5y = 8, so that's one way to write it. So all we have to do is solve for y. So we're gonna start here.

First, we'll subtract 3x from both sides. Now, we're gonna divide everything by 5. Remember when we divide by 5, all three terms get divided by 5. And so we get a slope of -three-fifths. In fact, we also find out that the y intercept is eight-fifths with that wasn't passed, but that information is kind of tossed in as a bonus.

Notice that horizontal lines have a slope of zero. Because they are all run with no rise, the horizontal line has a y intercept of -4, m = 0 and b = 4. And so the slope intercept form for this line would be y = 0 times x + 4, or in other words just y = 4. The default form for a horizontal line can be interpreted as a kind of slope intercept form.

It's what happens when slope equals zero. Now, vertical line are very different. First of all, because vertical lines, they have undefined slope because the slope fraction is always something over zero. So you could say they have infinite slope or undefined slope. Furthermore, vertical lines run paralleled with the y axis, and typically, they never intersect the y axis, except for the y axis itself.

Any vertical line that is not identical to the y axis never intersects the y axis. So the standard form for vertical lines, x = k is completely unrelated to slope intercept form, it's its own thing. Here's a practice problem, pause the video and then we'll talk about this. A couple videos ago, we talked about how, if you were given the x and y coordinates, the x intercept, the y intercept of a line, we could find the equation of the line.

And that's exactly what this problem is asking us to do. So first thing we're gonna say is we're given the x intercept, so we know that. We have a little slope triangle drawn right here, it's formed by the axis right here, and we have a run of 5 and a rise of -2. And so we have b = 2. A rise of -2, a run of 5, we get a slope of -two-fifths.

So now we can write the equation of the line y = mx + b, y = -two-fifths x-2. Very good, but now we go back to the answer choices that we see that's not listed. We have integers times x and y, and the x and y are on the same side of the equation. All right, so how do we do this?

Well first, we're just gonna multiply everything on both sides by 5 to get rid of the denominator. So we get a 5y = -2x-10, then what we're gonna do is add 2x to both sides. So we get 5y + 2x = -10. Now, we go back to the answer choices and we recognize as choice D.

In summary, if we solve the equation of any line for y, we automatically put the equation into slope intercept form, y = mx + b, where m is the slope and b is the y intercept. This form makes it easier to graph the line and understand where it goes.

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