Now, we can talk about slope intercept form. In other words y equals mx plus b four. The equation of any line can be stated in several different, algebraically equivalent forms. We could have x & y on the same side, or solve for x, or 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 representations, mathematicians have selected one as particularly useful. If we solve the equation of a line for y, getting y on one side by itself, we put 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 the 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 slop 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, one unit to the right, up m slopes or one end to the left down m slops, etc. Here's a practice question, pause the video and then we'll talk about this.

Okay, find all the points (j, k) on the line. Y = -4/3x + 2 such that 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 -4/3, the y intercept is positive 2. And so that's the point 0, -2.

So that's actually one of the points, so 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 the 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 three up four. So if we start at the point two, zero comma two. Then if we move right three, down four, we go from zero to three and down from 2 to -2. If we go right three down four, again, we'll get to six and then down four more down to negative 6.

Right three, down four, again, we'll get to 9, and negative 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 going to go the other way. We're gonna go.

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 we have to stop there because again, we've reached an absolute value of 10. And so there are six possible points. Here are the six possible points that satisfy this condition.

They're 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 = was eight. 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 three X from both sides. Now, we're gonna divide everything by five. Remember when we divide by five, all three terms get divided by five. And so, we get a slope of negative -3/5 in fact, we also find out that the y intercept is eight fifth 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 negative four, then m equals zero, and b equals four. And so the slope intercept form for this line would be y equals zero x X + 4 in other words, just white was for the default form for a horizontal line can be interpreted as a kind of slope intercept form.

It's what happens when slope equals 0. Now vertical lines are very different. First of all because vertical lines have undefined slope because the slope fraction is always something over zero. So you can say they have infinite slope or undefined slope. Furthermore, vertical lines run parallel to 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 equals 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 axiss right here, and we have a run of 5 and a rise of -2. And so we have b equals two rise of negative to run a five and we get a slope of negative two fifths.

So now we can write the equation of the line y = mx + b, y equals negative two fifths x -2. All right, very good, but now we go back to the answer choices we see that's not listed. We have integers times x and y, and the x and the 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 5x plus 5y + 2x equals -10. Now we go back to the answer choices, and we recognize answer choice D.

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

Read full transcriptNo matter how we rearrange the algebra, it's always the same line that's represented. That's an important idea. Of all possible algebraic representations, mathematicians have selected one as particularly useful. If we solve the equation of a line for y, getting y on one side by itself, we put 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 the 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 slop 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, one unit to the right, up m slopes or one end to the left down m slops, etc. Here's a practice question, pause the video and then we'll talk about this.

Okay, find all the points (j, k) on the line. Y = -4/3x + 2 such that 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 -4/3, the y intercept is positive 2. And so that's the point 0, -2.

So that's actually one of the points, so 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 the 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 three up four. So if we start at the point two, zero comma two. Then if we move right three, down four, we go from zero to three and down from 2 to -2. If we go right three down four, again, we'll get to six and then down four more down to negative 6.

Right three, down four, again, we'll get to 9, and negative 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 going to go the other way. We're gonna go.

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 we have to stop there because again, we've reached an absolute value of 10. And so there are six possible points. Here are the six possible points that satisfy this condition.

They're 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 = was eight. 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 three X from both sides. Now, we're gonna divide everything by five. Remember when we divide by five, all three terms get divided by five. And so, we get a slope of negative -3/5 in fact, we also find out that the y intercept is eight fifth 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 negative four, then m equals zero, and b equals four. And so the slope intercept form for this line would be y equals zero x X + 4 in other words, just white was for the default form for a horizontal line can be interpreted as a kind of slope intercept form.

It's what happens when slope equals 0. Now vertical lines are very different. First of all because vertical lines have undefined slope because the slope fraction is always something over zero. So you can say they have infinite slope or undefined slope. Furthermore, vertical lines run parallel to 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 equals 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 axiss right here, and we have a run of 5 and a rise of -2. And so we have b equals two rise of negative to run a five and we get a slope of negative two fifths.

So now we can write the equation of the line y = mx + b, y equals negative two fifths x -2. All right, very good, but now we go back to the answer choices we see that's not listed. We have integers times x and y, and the x and the 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 5x plus 5y + 2x equals -10. Now we go back to the answer choices, and we recognize answer choice D.

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