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### Course: Algebra (all content)>Unit 3

Lesson 8: Point-slope form

# Point-slope & slope-intercept equations

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). We can rewrite an equation in point-slope form to be in slope-intercept form y=mx+b, to highlight the same line's slope and y-intercept.

## Want to join the conversation?

• Why did Sal use the point (4,9)? Does it matter if you use (6,1) instead?
• No, it doesn't matter. You can use either point to find the equation of the line. Once the equation is to changed to slope-intercept form, the equation would match what Sal came up with.
• At , Sal says that slope is change over x/change over y. I have learned this over and over, and now I finally get to ask my question: Why is it not change over x/change over y?
• It is not the change in X over the change in Y because X is always the independent variable in the situation, and Y is always the dependent variable in the situation. So when you are finding slope, you are trying to find the rate of change of the independent variable. So say for example that I am looking for christmas tree ornaments. Every ornament I buy increases the amount of money I spend by \$2. The dependent variable is the amount of money because it changes depending on the amount of ornaments. So you already know what the dependent variable is. That is why you always do change in Y over change in X. You are searching for X. Otherwise, you would be searching for Y, and you already know what it is. So now I will solve this problem. I bought 2 ornaments, so my points for the graph would be the following:
( 1, 2 ) , ( 2, 4 ).
So now you need to find the slope. Remember, it is change in Y over change in X because you need to find the independent variable for the slope. To find slope you would do the following:
4-2
----- = 2/1 = 2.
2-1
So now you know the slope. Now you must find the y-intercept.
2=2(1) + b
2=2 + b
0=b
So now you know that there is no y-intercept in this problem. So you would set up the slope equation like so:
y = 2x + 0 or just y = 2x.

I hope that this helped! I have never answered a question before, so I'm sorry if this answer seemed to go on forever.
Sincerely - Anna F. 8th grade.
• Sal keeps saying arbitrary, what does it mean?
• why didn't he plug in x=6 and y=1 for the point slope?
• You can absolutely do that! In fact, you can plug any point on the line and it will be correct. If you convert it to slope intercept form, you're gonna get the same answer in all cases--try it out! :)
• What if you have three points?
• If they are on the same line, you can use any two of the three and you should get the same answer.
• When the two points are (4, 9) and (6, 1), then to obtain m, how do we know whether to
(9-1)/(4-6)
or
(1-9)/(6-4)
?
• Both are equivalent! Simplify the fractions to see it!
Can you figure out why the slopes turn out to be the same as long as we subtract both coordinates from each other in the same order regardless of the order we choose?
• What if you have fractions in the problem as your points and you have one zero as a y value?
• All of the rules still apply, no matter what numbers you have. So just do it the same as you would if you had whole numbers. Hope this helps.
• Sir, everything is cleared but I have been still thinking that what is the difference btw two types SLOPE INTERCEPT and point slope form, what is the actual difference suppose if we have given one point (3,8) and (8,3) find the equation, it is cleared that is Point means we have to use point solve formula, but I saw somewhere many of the people using both types where is not giving and not asking about y-intercept, kindly response me on my this confusion?
• Point slope form and slope intercept form are both ways of expressing the equation of a straight line. Point slope form emphasizes the slope and ANY point on the line. Slope intercept form just shows the slope and the y-intercept of a line.

Finding the equation of the line using the points (3,8) and (8,3) [SLOPE INTERCEPT]
1. Find the slope between these points which is -1.
Equation then becomes y=-x+b. Solve for b by using one of the points above (I'll use (3,8))
y=-x+b
8=-3+b
b=11
Equation is y=-x+11

Finding the equation of the line using (3,8) and (8,3) [POINT SLOPE]
1. Find the slope which is -1. Input any point (I'll use 3,8)) into the equation.
y-8=-1(x-3)
2. That's the equation of the line in point slope form but you could put it into slope intercept form by performing distributive property.
y-8=-1(x-3)
y-8=-x+3
y=-x+11
Equation is y=-x+11