Find the slope of the following line. 1 2 x − 2 3 y = 4 \frac12x-\frac23y=4

this problem, we are asked to find the slope of the following lines, and we'll go ahead and start with this line, which is one half x minus two thirds y equals four. Now this is kind of a confusing problem right at first, because notice we have an equation right here. Now we know how to find the slope. The slope is just rise over run. But it's clear that we do not have a graph where we can compare the rise and the run, so we're going to need some other strategy to figure this out. Well, another version of this equation I know is the rise, which is y two minus y one divided by the run, which is x two minus x one. But again, we don't really have a lot to go off of here because we don't have any points given to us either, but I can figure out some points by finding the x and y intercepts. So what I'm going to do is find the x intercept, and I'm also going to find the y intercept. Now remember, if I wanna find the x intercept, I just need to set y equal to zero. And if I wanna find the y intercept, I just need to set x equal to zero. So let's start by finding the x intercept. Well, to find the x intercept, I'm just going to set y equal to zero in this equation. So we'll get one half x minus two thirds zero since we set y to zero, and then we'll say that whole thing equals four. Now this whole piece just goes to zero, leaving us with one half x minus zero equals four, which is the same thing as one half x equals four. Now what I can do is multiply both sides of this equation by two, because the two and the one half will reduce to one, so those will just cancel each other out, leaving me with just x. And two times four is eight, meaning that x equals eight. So So we have our x is eight, we know that our y is zero, so that means we get the point eight zero. So now we have our x intercept. Now next I need to find my y intercept, which will be done by just setting x equal to zero in the same equation we have right about here. So what I'm going to do is set this x equal to zero, and then we'll have minus two thirds y is equal to four. Now anything times zero just gives us zero as we know, so we'll end up with negative two thirds y equals four. Now at this point, I need to isolate this y by itself. Now recall there's a way that we can do this. What I need to do is multiply by the reciprocal. But I can't just multiply both sides by three halves because there's still a negative sign out front here that will remain. So I need to actually multiply both sides by negative three halves. So we'll go negative three halves on the left side and negative three halves on the right side. I want you to watch carefully as to what happens here. Notice the three and the three cancel each other, the two and the two cancel each other, and two negatives cancel out to give you a positive. So all we end up with on the left side here is just y. And y is equal to four times negative three halves. Well, what I can do is recognize that four divided by two will give us two. So we get two times negative three, and two times negative three is just equal to negative six. So notice now we have our y, and we know our x is zero, so we end up with the point zero negative six. So we now have our y intercept and our x intercept, and it looks like we now have a couple of points we can use to find our slope. So the way I'm going to find our slope is I'm going to make this top point point one, and I'm going to make this bottom point point two. It doesn't matter which one I pick as point one or two, as long as I'm consistent with these points. So So now let's go ahead and use our equation for the slope. We're going to have y two minus y one. Now y two, well, that's going to be my y value for point two, which is negative six. Then we're going to have minus y one, which is going to be my y value for point one, which is zero. Now I'm going to divide that by x two minus x one. Well, I can see that my x two is going to be zero, then that's gonna be minus x one, which I can see is eight. So now we're just calculating our slope. Negative six minus zero is just negative six, and zero minus eight is negative eight. Now from here, what I can do is reduce this fraction since negative six is the same thing as negative two times three, and negative eight is the same thing as negative two times four. Again, cancel a negative two, leaving me with three fourths as my slope and the solution to this problem. That is how we can find the the slope for this line. Hope you found this video helpful, and let's move on and get some more practice.

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4. Graphing Linear Equations in Two Variables

Slope of a Line

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Find the slope of the following line.
$\frac{1}{2} x - \frac{2}{3} y = 4$

$\frac{3}{4}$

$- \frac{3}{4}$

$negative 6$

$negative 8$

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Start with the given equation: $\frac12x - \frac23y = 4$.

Isolate the $y$-term by moving the $x$-term to the other side: $-\frac23y = -\frac12x + 4$.

Multiply both sides of the equation by $-1$ to make the $y$-term positive: $\frac23y = \frac12x - 4$.

Solve for $y$ by dividing both sides by $\frac23$, which is the same as multiplying by its reciprocal $\frac32$: $y = \left(\frac12x - 4\right) \times \frac32$.

Distribute $\frac32$ to both terms inside the parentheses to write $y$ in slope-intercept form $y = mx + b$, where $m$ is the slope.