The Slope and Equation of a Line Help (page 3)
Introduction to The Slope and Equation of a Line
The slope of a line and the meaning of the slope are important in calculus. In fact, the slope formula is the basis for differential calculus. The slope of a line measures its tilt. The sign of the slope tells us if the line tilts up (if the slope is positive) or tilts down (if the slope is negative). The larger the number, the steeper the slope.
We can put any two points on the line, ( x 1 , y 1 ) and ( x 2 , y 2 ), in the slope formula to find the slope of the line.
A slope of means that if we increase the x -value by 2, then we need to increase the y -value by 1 to get another point on the line. For example, knowing that (0, 3) is on the line means that we know (0 + 2, 3 + 1) = (2, 4) is also on the line.
As we can see from Figure 1.4, (−4, −2) and (1, −2) are two points on a horizontal line. We will put these points in the slope formula.
Slopes of Horizontal and Vertical Lines
The slope of every horizontal line is 0. The y -values on a horizontal line do not change but the x -values do.
What happens to the slope formula for a vertical line?
The points (3, 2) and (3, − 1) are on the vertical line in Figure 1.5. Let’s see what happens when we put them in the slope formula.
This is not a number so the slope of a vertical line does not exist (we also say that it is undefined). The x -values on a vertical line do not change but the y -values do.
Any line is the graph of a linear equation. The equation of a horizontal line is y = a (where a is the y -value of every point on the line). Some examples of horizontal lines are y = 4, y = 1, and y = −5 .
The equation of a vertical line is x = a (where a is the x -value of every point on the line). Some examples are x = −3, x = 2, and x = 4.
Slope-Intercept and Point-Slope Form
Other equations usually come in one of two forms: Ax+By = C and y = mx+b . We will usually use the form y = mx + b in this book. An equation in this form gives us two important pieces of information. The first is m , the slope. The second is b , the y -intercept (where the line crosses the y -axis). For this reason, this form is called the slope-intercept form. In the line , the slope of the line is and the y -intercept is (0, 4), or simply, 4.
We can find an equation of a line by knowing its slope and any point on the line. There are two common methods for finding this equation. One is to put m, x , and y (x and y are the coordinates of the point we know) in y = mx + b and use algebra to find b. The other is to put these same numbers in the point-slope form of the line, y − y 1 = m ( x − x 1 ). We will use both methods in the next example.
Find an equation of the line with slope containing the point (8, −2). We will let m , and y = −2 in y = mx + b to find b.
The line is
Now we will let and y 1 = −2 in y − y 1 = m ( x − x 1 ).
- Find an equation of the line with slope 4, containing the point (0, 3). We know the slope is 4 and we know the y -intercept is 3 (because (0, 3) is on the line), so we can write the equation without having to do any work: y = 4 x + 3.
- Find an equation of the horizontal line that contains the point (5, −6). Because the y -values are the same on a horizontal line, we know that this equation is y = −6. We can still find the equation algebraically using the fact that m = 0, x = 5 and y = −6. Then y = mx + b becomes −6 = 0(5) + b. From here we can see that b = −6, so y = 0 x − 6, or simply, y = −6.
- Find an equation of the vertical line containing the point (10, −1). Because the x -values are the same on a vertical line, we know that the equation is x = 10. We cannot find this equation algebraically because m does not exist.
Finding the Equation of a Line Using Slope Formula
We can find an equation of a line if we know any two points on the line. First we need to use the slope formula to find m . Then we will pick one of the points to put into y = mx + b .
Find an equation of the line containing the given points.
- (−2, 3) and (10, 15)
We will use x = −2 and y = 3 in y = mx + b to find b .
3 = 1(−2) + b
5 = b
The equation is y = 1 x + 5, or simply y = x + 5.
- and (4, 3)
Using x = 4 and y = 3 in y = mx + b, we have
- (0, 1) and (12, 1)
The y -values are the same, making this a horizontal line. The equation is y = 1.
Finding Slope from a Graph
If a graph is clear enough, we can find two points on the line or even its slope. If fact, if the slope and y -intercept are easy enough to see on the graph, we know right away what the equation is.
The line in Figure 1.8 crosses the y -axis at 1, so b = 1. From this point, we can go right 2 and up 3 to reach the point (2, 4) on the line. “Right 2” means that the denominator of the slope is 2. “Up 3” means that the numerator of the slope is 3. The slope is , so the equation of the line is .
The y-intercept is not easy to determine, but we do have two points. We can either find the slope by using the slope formula, or visually (as we did above). We can find the slope visually by asking how we can go from (−4, 3) to (2, −1): Down 4 (making the numerator of the slope −4) and right 6 (making the denominator 6). If we use the slope formula, we have
Using x = 2 and y = −1 in y = mx + b, we have . From this, we have . The equation is .
The line in Figure 1.10 is vertical, so it has the form x = a. All of the x-values are −2, so the equation is x = −2.
Solving for y to Find the Slope of an Equation
When an equation for a line is in the form Ax + By = C , we can find the slope by solving the equation for y . This will put the equation in the form y = mx + b.
- Find the slope of the line 6 x − 2 y = 3.
6 x − 2 y = 3
−2 y = −6x + 3
The slope is .
Slopes of a Parallel and Perpendicular Lines
Slopes of Parallel Lines - Equal Slopes
Two lines are parallel if their slopes are equal (or if both lines are vertical).
Slopes of Perpendicular Lines - Slopes with Negative Reciprocals
Two lines are perpendicular if their slopes are negative reciprocals of each other (or if one line is horizontal and the other is vertical). Two numbers are negative reciprocals of each other if one is positive and the other is negative and inverting one gets the other (if we ignore the sign).
- are negative reciprocals
are negative reciprocals
- are negative reciprocals
- 1 and − 1 are negative reciprocals
Determining if Two Lines are Parallel or Perpendicular
We can decide whether two lines are parallel or perpendicular or neither by putting them in the form y = mx + b and comparing their slopes.
Determine whether the lines are parallel or perpendicular or neither.
4 x − 3 y = −15 and 4 x − 3 y = 6
The lines have the same slope, so they are parallel.
3 x − 5 y = 20 and 5 x − 3 y = −15
The slopes are reciprocals of each other but not negative reciprocals, so they are not perpendicular. They are not parallel, either.
x − y = 2 and x + y = −8
x − y = 2 x + y = −8
y = x − 2 y = −x − 8
The slope of the first line is 1 and the second is −1. Because 1 and −1 are negative reciprocals, these lines are perpendicular.
y = 10 and x = 3
The line y = 10 is horizontal, and the line x = 3 is vertical. They are perpendicular.
Finding the Equation of Parallel and Perpendicular Lines
Sometimes we need to find an equation of a line when we know only a point on the line and an equation of another line that is either parallel or perpendicular to it. We need to find the slope of the line whose equation we have and use this to find the equation of the line we are looking for.
Find an equation of the line containing the point (−4, 5) that is parallel to the line y = 2 x + 1.
The slope of y = 2 x + 1 is 2. This is the same as the line we want, so we will let x = −4, y = 5, and m = 2 in y = mx + b. We get 5 = 2(−4) + b , so b = 13. The equation of the line we want is y = 2x + 13.
Find an equation of the line with x -intercept 4 that is perpendicular to x − 3y = 12.
The x -intercept is 4 means that the point (4, 0) is on the line. The slope of the line we want will be the negative reciprocal of the slope of the line x − 3 y = 12. We will find the slope of x − 3 y = 12 by solving for y .
The slope we want is −3, which is the negative reciprocal of . When we let x = 4, y = 0, and m = −3 in y = mx + b , we have 0 = −3(4) + b , which gives us b = 12. The line is y = −3 x + 12.
Find an equation of the line containing the point (3, −8), perpendicular to the line y = 9.
The line y = 9 is horizontal, so the line we want is vertical. The vertical line passing through (3, −8) is x = 3.
Practice problems for this concept can be found at Slope Practice Problems.
More practice problems for this concept can be found at: The Slope and Equation of a Line Practice Test.
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