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Lines with positive slope |
Lines with negative slope |
Lines in the Plane |
Contents: This page corresponds to § 1.2 (p. 88) of the text.
Suggested Problems from Text
p. 98 #1, 2, 4, 8, 9, 11, 15, 18, 19, 23, 26, 33, 35, 36, 42, 43, 48, 51, 56, 57, 58, 70, 73, 77-80, 87, 93
Slope of a Line
Equations of Lines
Linear Models
The slope of a line is a number that measures the steepness of the line.
Think of a point moving along a line, always from left to right. (The x coordinate increases.) If the point rises (the y coordinate increases), then the slope is positive. The larger the slope, the faster the point rises. If the point falls (the y coordinate decreases), then the slope is negative.
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Lines with positive slope |
Lines with negative slope |
As a point moves on a horizontal line, the y coordinate does not change. Every horizontal line has slope 0.
Slope is not defined for a vertical line. (A point cannot move left to right on a vertical line!) You can think of a vertical line as being "infinitely steep," and infinity is not a number.
Given any two points (x1, y1) and (x2, y2) on a line, the slope, m, of the line is given by
.
This formula might be easier to remember if you think of it as "Rise over Run." "Rise" is the vertical change, or change in the y coordinate as you move from point 1 to point 2, and "Run" is the horizontal change. Note that rise or run can be negative numbers.
We can see from the slope formula why horizontal lines have zero slope. All the points on a horizontal line are at the same height (the y coordinates are the same), so y1 = y2 and m = 0.
We can also see why slope is not defined for vertical lines. All the points on a vertical line have the same x coordinate, so x1= x2 and the denominator of the slope formula is zero. Therefore, the slope is undefined since division by zero is not allowed.
The Java applet below shows a line, the coordinates of two points on the line and the line's slope. If you drag either of the points to a new location, all of the information is updated. You also may edit the coordinates of the points in the text boxes, then press enter to update the information. Note that some browsers will not read the point correctly if there are spaces. If you have trouble editing coordinates of points, try removing all spaces from the text boxes.
Use the slope formula to compute the slopes of the lines with the following pairs of points. Use the Slope Applet to check your answers.
(a) (-2, 1) and (3,5).
(b) (6, -2) and (-4, 5).
(c) (-2.3, -1.6) and (5.7, 3.9).
Use the Slope Applet to find a line whose slope is -3/4 = -0.75. Apply the slope formula to the points reported by the applet and verify that the slope is -0.75.
Two points determine a line. This means that if A and B are any two distinct points, then there is one and only one line that contains the points A and B.
A single point and a slope also determine a line. So, given a point and a slope we should be able to find an equation whose graph is the line with the given slope that contains the given point.
Point-Slope Form of the Equation of a Line
Let L be a line with slope m and let (x1, y1) be a point on L. Then L is the graph of the equation
y - y1 = m(x - x1).
Put the line through (-3, -5) with slope 2 on the Slope Applet by first dragging one of the points to (-3, -5) and then dragging the other point until the slope is 2. Verify that the coordinates of the other point satisfy the equation
y + 5 = 2(x + 3).
As we have noted, two points determine a line. So if we are given two points we should be able to find an equation of the line determined by the points. We may do this by using the two points to calculate the slope, then using either one of the points to write the Point-Slope form of the equation of the line. Note that the Point-Slope equation obtained using the first point will look different from the equation obtained by using the other point, but the equations are equivalent, which means that they have the same set of solutions points and hence the same graph.
Example 1. Let A = (4,1) , B = (-1, -2) and let L be the line determined by A and B.
The slope of L is m = (-2 - 1)/(-1 - 4) = -3/-5 = 0.6.
Using point A and slope m, we get the equation (1) y - 1 = 0.6(x - 4).
Using point B and slope m, we get the equation (2) y + 2 = 0.6(x + 1).
Solving both of these equations for y yields
Equation (1)
Equation (2)
y - 1
=
0.6(x-4) y - 1
=
0.6x-2.4 y
=
0.6x-1.4
y + 2
=
0.6(x+1) y + 2
=
0.6x +0.6 y
=
0.6x-1.4
Let L be the line through (-5, 6) and (1, 1).
(a) Find the slope of L.
(b) Find the Point-Slope form of the equation of L using (-5, 6).
(c) Find the Point-Slope form of the equation of L using (1, 1).
(d) Solve the equations from parts (b) and (c) for y and note that in this form the equations are the same.
Solving the equation of a line for y is often useful. One advantage of using this form of an equation for a line is that it is easy to substitute any value for x and compute the corresponding value of y. Consider the example y = 7x+5. When we substitute x = 0 we see that the corresponding value of y is 5, so (0,5) is a point on the graph of the equation. A point with first coordinate 0 lies on the y-axis, and the point (0,5) is called the y-intercept of the graph because that is the point where the graph crosses the y-axis. The slope of the graph of y = 7x+5 is 7, the coefficient of x.
Slope-Intercept Form of the Equation of a Line
The line with slope m and y-intercept (0,b) has equation
y = m x + b.
Example 2. Find the slope and y-intercept of the line 3x + 4y - 5 = 0.
When we solve the equation above for y, we get y = (-3/4)x + 5/4, which is the Slope-Intercept form of the equation. The slope of the line is the coefficient of x, namely -3/4, and the y-intercept is the point (0,5/4).
Find the y-intercept of the line through (-6, 1) and (7, 4). Answer: (0, 31/13)
- Videos: b = 0. Animated Gif, MS Avi File, or Real Video File
- Videos: b = 1. Animated Gif, MS Avi File, or Real Video File
- Videos: b = 5. Animated Gif, MS Avi File, or Real Video File
Horizontal Lines
All points on a horizontal line have the same y-coordinate, and the x-coordinate can be any number. The equation of a line spells out the conditions that the coordinates of a point must satisfy to be on the line. In the case of a horizontal line, there is no restriction on the x-coordinate, so x doesn't appear in the equation. The condition on the y-coordinate is that it must equal a fixed number, so the equation of a horizontal line has the form y = b, where b is a number.
Vertical Lines
All points on a vertical line have the same x-coordinate, and the y-coordinate can be any number. Therefore, the equation of a vertical line has the form x = a, where a is a number.
Parallel Lines
Distinct, nonvertical lines are parallel if and only if their slopes are equal. (Note: Statement A if and only if statement B means that if A is true then B is true and if B is true then A is true.)
Perpendicular Lines
Let L1 be a line with slope m1 and L2 a line with slope m2. L1 is perpendicular to L2 if and only if m1 = -1/m2.
Example 3. Let L be the line through (-2, 3) and (4, 0). Find the equation of the line through (1,1) which is perpendicular to L.
First we find the slope of L, m1 = (0-3)/(4-(-2)) = -3/6 = -1/2. The negative reciprocal of m1 = -1/2 is m2 = 2. Therefore, every line perpendicular to L has slope 2. The line, whose equation we want to find, goes through (1,1) with slope 2. So, we use the Point-Slope form to get
y - 1 = 2(x - 1).
In order to graph an equation with the Java Grapher it must be solved for y. So, to graph the equation of a line, we need its Slope-Intercept form.
Find the Slope-Intercept form of the equations of the two lines in example 3 and graph both on the Java Grapher to verify that they are perpendicular.
Example 4.
Joe is a sales clerk in a department store and his gross pay for the month is determined by the value of the merchandise that he sells that month. One month he had $12000 in sales and his gross pay was $1960. The next month he had only $9000 in sales and his gross pay fell to $1720.
Write a linear equation giving Joe's gross pay y in terms of his monthly sales amount x.
We know two points on the graph of the equation: (12000,1960) and (9000,1720).
The slope of the line is m = (1960-1720)/(12000-9000) = 0.08.
Using the second point (9000,1720) and the slope m = 0.08 we find the Point-Slope form of the equation of the line to be
y - 1720 = 0.08 (x - 9000).
Solving for y yields the Slope-Intercept form:
y = 0.08 x + 1000.
If Joe sells $14000 worth of merchandise in a month, what will be his gross pay for the month?
y = 0.08 * 14000 + 1000 = 2120, so his gross pay will be $2120.
The linear equation (equation of a line) found above is an example of a Mathematical Model. Mathematical models are used to predict the value of a variable under certain conditions. For instance, we used our model to predict Joe's gross pay if he were to sell $14000 worth of merchandise. We say that the model above is linear because the graph of the model equation is a line. Modeling is an important aspect of mathematics and we will see many examples of mathematical models in this course.
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