Write the point-slope form of the equation of a line that passes through the points (2,1)\left(2,1\right)(2,1) and (−4,3)\left(-4,3\right)(−4,3) . Then graph the equation.

y − 1 = − 1 3 ( x − 2 ) y-1=-\frac13\left(x-2\right) y − 1 = − 3 1 ( x − 2 )

Write the point-slope form of the equation of a line that passes through the points ( 2 , 1 ) \left(2,1\right) ( 2 , 1 ) and ( − 4 , 3 ) \left(-4,3\right) ( − 4 , 3 ) . Then graph the equation .

Hey, everyone. Let's take a look at this practice problem. Hopefully, you had a chance to try it on your own. Let's check it out. So we're gonna find the point slope form of an equation that passes through 2 points. So we've got 2 comma 1 and negative 4 comma 3. Just gonna plot those out really quickly. That's 2 comma 1 and negative 4 comma 3 is over here. So I wanna write the point slope form, which is just this form over here. Let's get let's go let's go ahead and do that. So I've got y minus y one is equal to mx minus x one. Remember, for this equation, I need 3 things. I need x one and y one. Those are the just the values that I plug into from the points, and then I also need my m, the slope. In most cases, that slope is already given to us, but in this situation, it's not. So I'm gonna have to go find it. So let's go ahead and take care of that first. How do I calculate m? Well, I could just use rise over run, or because I already have these two points over here, I can just use sort of the more sort of fancy version, which is x 2, you know, y y two minus y one, x two minus x one, so the with a more complete equation. And so if you're ever, you know, sort of struggling to figure out which one you should pick as your x one y one x two y two, it actually doesn't matter. Remember that the order of these points doesn't matter. You can pick either one of them to be x 1, y 1. But I always like to pick the leftmost points to be my x one, y 1. So in other words, the point that has the smallest or the most negative x value. So in this case, I'm gonna pick my x one, y one to be this and my x two, y two to be 2 comma 1. However, again, it would have worked totally fine if you just did it backwards. You'll end up getting the same answer. Alright. So this is going to be my x2y2x1y1. So according to this equation, I'm going to have 1 minus 3. So 1 minus 3 is y2 minus y1 divided by x2 minuteusx1, which is 2 minus negative 4. So what this simplifies to is it simplifies to negative 2 over positive 6, which ends up just being a slope of negative 1 third. I can't reduce that any further, so that's the simplified version of this. So my slope is negative 1 third. So I'm almost done here. Now I just basically have to plug in the x one and y one in this equation. So because we use negative 4 comma 3, in this case, as my x one y one, then what my equation ends up being is it ends up being y minus 3 equals negative 1 third x minus, and then this is my negative 4 over here. So this simplifies to y minus 3 equals negative 1 third x oops. Parentheses x plus 4. That is the point slope form of this equation that we sort of derived from these two points over here. So this is one of your answer choices. However, I also want to point out that you also could have picked the other points. If you pick this to be your x one y one, then you would actually end up with a slightly different equation, because yours would look like this. Yours would look like y minus 1 equals parenthesis or equals negative 1 third parenthesis x minus 2. So which of these two equations is actually the same or or or is is actually the correct one? It actually doesn't matter. They're both equal to each other. Both of these, if you were to write these on a homework or a test or something like that, would be correct. Because if you actually solve for these and isolate y and get this into y equals mx plus b, what you're actually gonna find here is that both of these things actually simplify down to negative 1 third x plus 5 thirds. And you can go ahead and pause the video to make sure that you actually get that. So both of these things, if you isolate for y, you're actually just gonna get these this equation over here. So these 2, even though they look different, actually represent the same exact line. So now for graphing it, all you have to do is just connect these two points over here with our line, and that is the equation of this line over here. Alright? So thanks for watching. Let me know if you have any questions. Let me know if that made sense. Let's move on to the next one.

Write the point-slope form of the equation of a line that passes through the points ( 2 , 1 ) \left(2,1\right) ( 2 , 1 ) and ( − 4 , 3 ) \left(-4,3\right) ( − 4 , 3 ) . Then graph the equation .

y − 1 = − 1 3 ( x − 2 ) y-1=-\frac13\left(x-2\right) y − 1 = − 3 1 ( x − 2 )

y − 3 = − 1 3 ( x − 2 ) y-3=-\frac13\left(x-2\right) y − 3 = − 3 1 ( x − 2 )

y = 1 3 x − 4 y=\frac13x-4 y = 3 1 x − 4

y − 2 = − 1 3 ( x − 1 ) y-2=-\frac13\left(x-1\right) y − 2 = − 3 1 ( x − 1 )

2. Graphs of Equations

Lines

College Algebra

2. Graphs of Equations

Lines

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Multiple Choice

Write the point-slope form of the equation of a line that passes through the points $\left(2,1\right)$ and $\left(-4,3\right)$ . Then graph the equation.

$y-1=-\frac13\left(x-2\right)$

$y-3=-\frac13\left(x-2\right)$

$y=\frac13x-4$

$y-2=-\frac13\left(x-1\right)$

Verified step by step guidance

Identify the two points given: (2, 1) and (-4, 3).

Calculate the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1). Substitute the coordinates of the points into the formula: m = (3 - 1) / (-4 - 2).

Simplify the slope calculation to find the value of m.

Use the point-slope form of a line equation: y - y1 = m(x - x1). Choose one of the points, for example, (2, 1), and substitute the slope and the coordinates into the equation.

Simplify the equation to express it in point-slope form. This will give you the equation of the line that passes through the given points.

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