The slope of a straight line To find the equation of a line Straight lines will be parallel if they have the same slope. There is no limit, then, to the number of lines with slope 2. But there will be only one line with slope 2 that passes through a given point, such as (1, −3). In other words, a line will be determined by its slope and the coördinates of one point on it. How can we find the equation of that line? Problem 1. To write the equation of a line given the slope and the coördinates of one point on it. For example: Write the equation of the line with slope 2 that passes through the point (1, −3).
where We are given that
We must now determine To do that, we must use the information that (1, −3) are the coördinates of a point on the line. Those coördinates then solve the equation. −3 = 2 This implies
The equation of the line is
Then if This is called the point-slope formula for the equation of a straight line. We can use the formula when we know the slope So, returning to our example, we are given that
This is the equation of the line. Problem 2. To find the equation of a line given the coördinates of two points on it. For example: Write the equation of the line that passes through the points (−3, 11) and (−5, 15).
Again, we must find 11 = −2 This implies
The equation is:
The two points are (−3, 11) and (−5, 15). Let each of them solve the equation, 1) 11 = −3 2) 15 = −5 According to the methods of Lesson 35, let us eliminate 2') −15 = 5 On adding, we find −4 = 2 Upon replacing that in Equation 1): 11 = −3(−2) + This implies
The equation is
Let ( This is called the two-point formula for the equation of a straight line. We can use it to determine the equation when we know two points. We may choose ( Thus, in our example, we are given the two points (1, 5) and (4, 17). Let (
This is the equation of the line. Problem 12.
To see the answer, pass your mouse over the colored area. The equation will have the form
where We are given that
Now we must determine We are given that (1, −4) is a point on the line. Those coördinates, then, solve the equation: −4 = 3 This implies
The equation of the line is
We are given that
This is the equation of the line.
Problem 13. Write the equation of the line that passes through (1, −6) and (−1, 2). Use the two-point formula. Let ( According to the formula:
This is the equation of the line. Next Lesson: Simultaneous linear equations Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |