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Let's do some examples dealing with equations of lines in

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standard form.

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So, so far we've had two other forms. We've had

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slope-intercept, which is of the form, y is

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equal to mx plus b.

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That's actually this right here.

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This is in slope-intercept form.

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We've seen point-slope form in the last video.

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That's of the form, y minus some y-value on the line being

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equal to the slope times x minus some x-value on the

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line, when you have that y-value.

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So the point x1, y1 is on the line.

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This right here is an example of point-slope form.

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And now we're going to talk about the standard form.

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And the standard form-- let me write it here-- standard form

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is essentially putting all of the x and y terms onto the

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left-hand side of the equation.

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So you get ax plus by is equal to c.

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I want to really emphasize that all of these are just

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different ways of writing the same equation.

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If you're given this, you can out algebraically manipulate

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it to get to that or to that.

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If you're given that, you can get to that or that.

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These are all different ways of writing the exact same

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relationship, the exact same line.

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So let's do a couple of examples of this.

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So here we have a line right here.

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We have an equation written in slope-intercept form.

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The slope is 3, the y-intercept is negative 8.

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Let's put it into standard form.

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So we just have to get the 3x onto the

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other side of the equation.

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And the best way I can think of doing that-- let me rewrite

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the equation, y is equal to 3x minus 8-- let's some subtract

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3x from both sides of the equation.

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So if you subtract 3x from both sides-- so you subtract

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3x, subtract 3x-- what do the left- and right-hand sides of

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the equation become?

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The left-hand side becomes negative 3x plus y being equal

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to-- the 3x and the negative 3x cancel out-- being equal to

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negative 8.

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We're done.

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That's standard form right there.

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Standard form, I guess people like it because it has both

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the coefficients on the left-hand side.

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But it's kind of useless in trying to figure out slope and

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y-intercept.

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I don't know what the slope and y-intercept is when I look

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at it in standard form.

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My favorite is slope-intercept form, because it tells you

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exactly the slope and an intercept.

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Point-slope, easy to get to, and you can look at it and

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figure out the slope.

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But y-intercept, you have to do a little bit of work to

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figure it out.

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But at least you can just go immediately from the slope and

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a point to it.

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But anyway, let's go from this equation, which is written in

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point-slope form, and get it to the standard form.

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So we want to get it to the standard form, to the same

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type of standard form.

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So a good thing to do, let's just distribute things out. y

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minus 7 is equal to negative 5 times x, negative 5x, plus

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negative 5, times negative 12, which is positive 60.

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Now, we want all of the variable terms on the left,

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all of the constant terms on the right.

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So let's add 7 to both sides of this equation.

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So plus 7 to both sides of this equation.

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What does it become?

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Well, the minus 7 disappears, because negative 7 plus 7.

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So you're just left with a y being equal to

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negative 5x plus 67.

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Now, if we want this x term on the left-hand side, we could

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add 5x to both sides.

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So let's add 5x to both sides of this equation.

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And we will get y plus 5x is equal to-- these

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cancel out-- 67.

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Now, this is pretty much standard form.

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If you really want to be a stickler for it, you can

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rearrange these two.

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So it'd be 5x plus y is equal to 67.

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And you are done.

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Let's do one more of these.

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So this is in neither point-slope nor in

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slope-intercept form.

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It's just in some type of intermediary

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mixed form right there.

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This looks like some type of point-slope, but this looks

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like something different.

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So it's really not point-slope.

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Let's see if we can algebraically manipulate it to

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the standard form.

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So we get 3y plus 5.

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Let's distribute out this 4.

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So it's equal to 4x minus 36.

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Let's do exactly what we did in the last.

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I'm using different notation on purpose, to expose you to

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different things.

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So instead of doing it this way, I'm going to subtract 5

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from both sides, but I'm going to do it on the same line.

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So I'm going to subtract 5 from both sides.

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And so the left-hand side of this equation becomes 3y,

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because these two guys cancel out, and that is equal to 4x.

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And then what is minus 36 minus 5?

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That's minus 41.

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And now we want the x terms of the left-hand side.

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So let's subtract 4x from both sides of this equation.

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So negative 4x plus, and then minus 4x.

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What does our equation become?

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Well, the left-hand side just stays negative 4x plus 3y.

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And the right-hand, the reason why we subtracted 4x is so it

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cancels out with that.

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You just have a negative 41.

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And we're done.

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We are in standard form.

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Now, let's go the other way.

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Let's start with some equations in standard form and

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figure out their slope and y-intercept.

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And the best way I know to figure out the slope and

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y-intercept is to put it into slope-intercept form.

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So we want to put these equations right here into the

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form, y is equal to mx plus b.

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So we're essentially solving for y.

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Let's do that.

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So the best thing to do here-- so let me rewrite it.

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5x minus 2y is equal to 15.

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Let's subtract 5x from both sides.

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So minus 5x plus, you have a minus 5x.

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These cancel out.

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And so you're left with negative 2y is

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equal to 15 minus 5x.

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And now, let's divide everything by negative 2.

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If you divide everything by negative 2, what do we get?

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The left-hand side just becomes a y.

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y is equal to-- 15 divided by negative 2 is negative 7.5.

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And then negative 5 divided by negative 2-- you can imagine

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I'm distributing the negative 1/2 if you will.

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I'm dividing both of these by negative 2.

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So negative 5 divided by negative 2 is positive 2.5x.

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And if you really wanted to put it in the slope-intercept

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form, you could say that y is equal to-- you could just

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rearrange these-- 2.5x minus 7.5.

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You want the slope.

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It's right here.

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That is our slope.

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You want the y-intercept.

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Actually, let me be careful.

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It is right there.

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It is negative 7.5.

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That is the y-intercept.

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And now this would be a form that's actually pretty

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straightforward to graph it in.

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Let's do this one.

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So once again, we just need to solve for y.

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So let's subtract 3x from both sides.

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So you get 6y is equal to 25 minus 3x.

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And then you can divide both sides by 6.

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So you're left with y is equal to 25 over 6 minus 3 over 6,

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or minus 1/2x.

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If you really want it in this from, you just rearrange this.

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y is equal to negative 1/2x plus 25 over 6.

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Where is the slope?

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Here is the slope.

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Negative 1/2, that is the slope.

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Where is the y-intercept?

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That's the y-intercept.

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That is our b.

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The point 0, 25 over 6 is on the line.

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Let's do one more of these.

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So we get 9x minus 9y is equal to 4.

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Just for fun, let's just start off by dividing both sides of

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the equation by 9.

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You don't have to do it that way.

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But this is kind of a fun way to do it, because the

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coefficients here will immediately become 1.

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So if you divide both sides of the equation by 9, if you

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divide everything by 9, it becomes-- actually, well,

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let's divide everything.

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Let's divide everything by negative 9, even better.

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I'm just doing this for fun.

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So this first term will become negative x.

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The second term, you have a negative 9 divided by a

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negative 9, it will be a plus y.

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And then this last term will just become a

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negative 4 over 9.

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Actually, let me write this out here.

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Negative 4 over 9.

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I'm giving some space there.

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Now, we want the x on the right-hand side, so let's add

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x to both sides of this equation.

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These cancel out.

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And then the equation becomes y is equal to x minus 4/9.

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Where is the slope?

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The slope is the coefficient on the x term.

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The slope is equal to 1.

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Where is the y-intercept?

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The y-intercept is right there.

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It is negative 4/9.

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