Slope-intercept equation from slope & point: fractions (old)
About this video
Sal writes the equation of a line with slope 1/3 that passes through (-12, -14/3). He substitutes the point into y = (1/3)x + b, solves for b using a common denominator, and gets y = (1/3)x - 2/3.
You'll be able to
- substitute a point into y = mx + b when the slope is a fraction.
- solve for b by rewriting whole numbers as fractions with a common denominator.
- write the final equation in slope-intercept form.
Key words
- slope-intercept form
- The form y = mx + b, where m is the slope and b is the y-intercept.
- y-intercept
- The value of y where the line crosses the y-axis (when x = 0).
- common denominator
- A shared bottom number that lets you add or subtract fractions, like writing 4 as 12/3.
Transcript
Write the equation of the line that has a slope of 1/3 and contains the point negative 12, negative 14/3. So the equation of a line we can write as y is equal to mx plus b. This is in slope-intercept form, where m is our slope and b right over here is our y-intercept.
And they have actually given us our slope. Our slope, they tell us, is 1/3. So we know that m is going to be 1/3. So we know that y is equal to 1/3 x plus b. And so now we just have to solve for b. And the way we can solve for b is that we know this line contains the point negative 12, negative 14/3.
So when x is equal to negative 12, y is equal to negative 14/3. So let's write that down. Negative 14/3 is going to be equal to 1/3-- our slope-- times our x value, which is negative 12-- times negative 12-- and then plus b. So when x is negative 12, we know that y is equal to negative 14/3.
And we also know that the slope is 1/3. That's the m in mx plus b. So now we can just solve for b. The left-hand side-- I'll just leave it as negative 14/3-- is equal to 1/3 times negative 12. 1/3 times negative 12 is negative 4, so this is negative 4, plus b.
Let me write the plus b in that same color that I had started it with. Plus b. It looks like a yellow color. And now, to isolate b on the right-hand side, we can add 4 to both sides. I want to get rid of this negative 4. So let's add 4 to both sides. So plus 4, plus 4-- on the right-hand side, we're left with just a b.
b is equal to-- on the left-hand side, we have 4. 4 I can rewrite. Because I'm going to have to add 4 to negative 14/3. So I want a common denominator. 4 is the same thing as 12/3. So 4 is 12/3, minus 14/3. So I just want to be clear. I just rewrote the 4 as 12/3.
So 12/3 minus 14/3, or negative 14/3 plus 12/3, is going to give us-- well, we have the same denominator of 3. So we're going to have 3. 12 minus 14 is negative 2. So we have negative 2/3 is equal to b. So the equation of the line we figured out. They gave us the slope.
And now we just figured out the y-intercept using this information right over here. So the equation of our line is going to be y is equal to 1/3 x-- we got the 1/3 from the problem-- plus b. b is negative 2/3. So I could just write minus 2/3. And we are done.
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