Objectives:
1. to write the equation of a line using the point-slope form of a line.
2. to understand and be able to explain the different part of the point-slope equation.
3. to be able to differentiate between when to use the different forms of a line.
Definition:
Point-Slope Form: the point-slope form of a line uses a point (x, y) and a slope (m), to write the equation of a line. This form is given by the equation:
y - y1 = m(x - x1) where m is the slope and the given point is substituted into the equation for x1 and y1.
This equation is very similar to the equations that we learned on day 5 and day 6. Just like on those days, today we are again learning another way to write the equation of a line. So, after today, we will have three different methods of writing the equation of a line, and one should be able to differentiate beteween when each method should be used.
This differentiation should be easy because the name tells one which equation to use. In order to use the slope-intercept form, one needs a slope and the y-intercept. If you don't have or can't find these two pieces of information, you can't use the slope-intercept form.
In order to write the equation of a line in standard form, one must already have the equation of line. So, if you don't have the equation of a line, you can't convert it into standard form. Here, one must have already used either the slope-intercept equation or the point-slope equation to find the equation of a line.
Now, it is time to discuss the final equation, the point-slope form of a line. As stated in the definition, the point slope form of a line is given by the equation:
This equation is probably the most useful of the three because it uses the information that it is most common to have about a line, the slope and a point on the line. Examples 1 - 3 will explain how to use this equation.
Example1: Write the equation of a line that goes through the point (2, 1) with a slope of 4.
Here, we want to be able to make three substitutions. We need to replace m, x1 and y1 with a numerical value. In this class, we will solve this equation for y, or in other words, we will change point-slope form into slope intercept form. So, the first thing that we have to decide is if we have values for these three variables. In this case, the answer is yes because m = 4, x1 = 2, and y1 = 1.
Step 1: Substitute.
We substitute into the equation to get: y - 1 = 4(x - 2)
Step 2: Distribute.
Next, distribute the 4 to get: y - 1 = 4x - 8
Step 3: Solve for y.
Finally, solve for y by adding 1 to both sides to get our answer of: y = 4x - 7.
So, we now have the equation of a line through the given point with the given slope. The process started with the point-slope equation that was then changed into slope-intercept or y = mx + b form. This change is made because it is much easier to work with the slope-intercept form than any of the others.
Example2: Write the equation of a line that passes through the point (4, -3) with a slope of -2.
First, do we have enough information to use the point-slope equation? Yes, we have m, x1, and y1 which are the three values that we need. So, we just follow the same three steps as in example1.
Step 1: Substitute.
y - -3 = -2(x - 4). (notice that there are 2 negatives so we will get y + 3 on the left)
Step 2: Distribute the -2.
y + 3 = -2x + 8
Step 3: Solve for y by subtracting 3 from both sides.
y = -2x + 5.
Sometimes, we will not have all of the information that we need to use the point-slope equation given to us. When this happens, we will have to calculate the missing information in order to write the equation of a line. Usually, the missing information is the slope of the line. This should not be a problem to calculate because we learned how to find slope on day 2.
Example3: Write the equation of a line that passes through the points (1, 2) and (4, 8).
Here, we want to use the point-slope form, but we do not have slope. That means that the first thing we must do is find slope.
Part 1: Finding Slope.
m = (8-2)/(4-1) = 6/3= 2.
Do we have enough information to write the equation of a line now? Yes, we actually have more than enough information. To use the point-slope form, we need a point on the line and the slope. Well, we have 2 points on the line and the slope. So, does it matter which point we use? Will one point give us a different equation than the other. No, each line has a unique equation. So, any point on the line will yield the same equation. We will write the equation of the line using both points to illustrate this.
Case 1: Finding the equation of the line using the point (1, 2).
Step 1: Substitute. (m = 2, x1 = 1, and y1 = 2).
y - 2 = 2(x - 1)
Step 2: Distribute the 2.
y - 2 = 2x - 2
Step 3: Solve for y by adding 2 to both sides.
y = 2x
Case 2: Finding the equation of the line using the point (4, 8).
Step 1: Substitute. (m = 2, x1 = 4, and y1 = 8).
y - 8 = 2(x - 4)
Step 2: Distribute the 2.
y - 8 = 2x - 8
Step 3: Solve for y by adding 8 to both sides.
y = 2x
So, as claimed, it did not matter which point we used. In both cases, we found the equation of the line to be y = 2x.
When we are given two points and asked to find the equation of the line, we must find the slope first. After that, it is the same three steps: substitute, distribute, solve for y.
You should now be able to write the equation of a line in: point-slope form, slope-intercept form, or standard form.