Equations of Lines Chapter 8 Sections 8.1-8.4

Copyright © Cengage Learning. All rights reserved. 8.1 Linear Equations with Two Variables Copyright © Cengage Learning. All rights reserved.

Linear Equations with Two Variables We have studied linear equations with one variable, such as 2x + 4 = 10 and 3x – 7 = 5. We found that most linear equations in one variable have only one root. In this section, we study equations with two variables, such as 3x + 4y = 12 or x + y = 7

Linear Equations with Two Variables How many solutions does the equation x + y = 7 have? Any two numbers whose sum is 7 is a solution–for example, 1 for x and 6 for y, 2 for x and 5 for y, –2 for x and 9 for y, for x and for y, and so on. Most linear equations with two variables have many possible solutions. Since it is very time consuming to write pairs of replacements in this manner, we use ordered pairs in the form (x, y) to write solutions of equations with two variables.

Linear Equations with Two Variables Therefore, instead of writing the solutions of the equation x + y = 7 as above, we write them as (1, 6), (2, 5), (–2, 9), and so on. These correspond to points on a linear grid we can graph and connect to make the line. A point is always listed as (x, y) Linear Equation with Two Variables A linear equation with two variables can be written in the form ax + by = c where the numbers a, b, and c are all real numbers such that a and b are not both 0.

Example 1 Determine whether the given ordered pair is a solution of the given equation. a. (5, 2); 3x + 4y = 23 To determine whether the ordered pair (5, 2) is a solution to 3x + 4y = 23, substitute 5 for x and 2 for y as follows: 3x + 4y = 23 3(5) + 4(2) = 23 Substitute x = 5 and y = 2.

Example 1 cont’d 15 + 8 = 23 23 = 23 The result is true, so (5, 2) is a solution of 3x + 4y = 23. b. (–5, 6); 2x – 4y = –32 2(–5) – 4(6) = –32 –10 – 24 = –32 –34 = –32 The result is false, so (–5, 6) is not a solution of 2x – 4y = –32. True Substitute x = –5 and y = 6. False

Linear Equations with Two Variables Solutions to linear equations with two variables may be shown visually by graphing them in a number plane. Points in the number plane are usually indicated by an ordered pair of numbers written in the form (x, y), where x is the first number in the ordered pair and y is the second number in the ordered pair. The numbers x and y are also called the coordinates of a point in the number plane. Figure 8.1 is often called the rectangular coordinate system or the Cartesian coordinate system. Rectangular coordinate system Figure 8.1

Linear Equations with Two Variables Plotting Points in the Number Plane To locate the point in the number plane which corresponds to an ordered pair (x, y): Step 1: Count right or left (right if positive, left if negative), from 0 (the origin) along the x axis, the number of spaces corresponding to the first number of the ordered pair.

Linear Equations with Two Variables Step 2: Count up or down(up if positive, down if negative), from the point reached on the x axis in Step 1, the number of spaces corresponding to the second number of the ordered pair. Step 3: Mark the last point reached with a dot. (On the GED Exam bubble in the open dot that sits at that space.)

Example 6 Plot the points corresponding to the ordered pairs in the number plane in Figure 8.3: A(1, 2), B(3, –2), C(–4, 7), D(5, 0), E(–2, –3), F(–5, –1), G(–2, 4).

Copyright © Cengage Learning. All rights reserved. 8.2 Graphing Linear Equations Copyright © Cengage Learning. All rights reserved.

Graphing Linear Equations We have learned that a linear equation with two variables has many solutions. We see that three of the solutions of 2x + y = 5 are (4, – 3), (– 2, 9), and (0, 5). Now plot the points corresponding to these ordered pairs and connect the points, as shown in Figure 8.4. Figure 8.4

Graphing Linear Equations You can see from the figure that the three points lie on the same straight line. If you find another solution of 2x + y = 5—say, (1, 3)—the point corresponding to this ordered pair also lies on the same straight line. The solutions of a linear equation with two variables always correspond to points lying on a straight line.

Graphing Linear Equations Therefore, the graph of the solutions of a linear equation with two variables is always a straight line. Only part of this line can be shown on the graph; the line actually extends without limit in both directions.

Graphing Linear Equations To draw the graph of a linear equation with two variables: Step 1: Find any three solutions of the equation. (Note: Two solutions would be enough, since two points determine a straight line. However, a third solution gives a third point as a check. If the three points do not lie on the same straight line, you have made an error.)

Graphing Linear Equations Step 2: Plot the three points corresponding to the three ordered pairs that you found in Step 1. Step 3: Draw a line through the three points. If the line is not straight, check your solutions.

Graphing Linear Equations We will show two methods for finding the three solutions of the equation. The first method involves solving the equation for y and then substituting three different values of x to find each corresponding y value. The second method involves substituting three different values of x and then solving each resulting equation for y. You may use either method. Depending on the equation, one method may be easier to use than the other.

Example 1 Draw the graph of 3x + 4y = 12. Step 1: Find any three solutions of 3x + 4y = 12. First, solve for y: 3x + 4y – 3x = 12 – 3x 4y = 12 – 3x Subtract 3x from both sides. Divide both sides by 4.

Example 1 cont’d Choose any three values of x and solve for y. Here we have chosen x = 4, x = 0, and x = –2. Note: We often choose a positive number, 0, and a negative number for x to obtain a range of points in the graph. Although finding and plotting any two points will allow you to graph the straight line, the third point provides an excellent check.

Example 1 cont’d Three solutions are (4, 0), (0, 3), and

Example 1 Step 2: Plot the points corresponding to (4, 0), (0, 3), and cont’d Step 2: Plot the points corresponding to (4, 0), (0, 3), and Step 3: Draw a straight line through these three points.

The x Intercept and the y Intercept of a Line

The x Intercept and the y Intercept of a Line The line in Figure 8.6 crosses the x axis at the point (3, 0). The number 3 is called the x intercept—the x coordinate of the point where the graph crosses the x axis. To find the x intercept of a line, replace y in the equation by 0 and solve for x. Figure 8.6

The x Intercept and the y Intercept of a Line The line in Figure 8.6 crosses the y axis at the point (0, –2). The number –2 is called the y intercept—the y coordinate of the point where the graph crosses the y axis. To find the y intercept of a line, replace x in the equation by 0 and solve for y.

Example 3 Find the x and y intercepts of the graph of 3x – 5y = 30 and then graph the equation. To find the x intercept, replace y in the equation by 0 and solve for x as follows: 3x – 5y = 30 3x – 5(0) = 30 3x = 30 x = 10 So, the x intercept is (10, 0). Divide both sides by 3.

Example 3 cont’d To find the y intercept, replace x in the equation by 0 and solve for y as follows: 3x – 5y = 30 3(0) – 5y = 30 – 5y = 30 y = – 6 So, the y intercept is (0, – 6). Divide both sides by – 5.

Example 3 cont’d Let’s find a third point by letting x = 5 and solve for y as follows: (you can choose any value you wish for x!) 3x – 5y = 30 3(5) – 5y = 30 15 – 5y = 30 – 5y = 15 y = – 3 The third point is (5, – 3).

Example 3 cont’d Plot these three points and draw a line through them. (See Figure 8.7.) Figure 8.7

The x Intercept and the y Intercept of a Line Finding the x and y intercepts is an excellent method for graphing a linear equation or for checking the graph of a linear equation. Two special cases of linear equations have a graph of a horizontal line or a vertical line. The equation y = 5 is a linear equation with an x coefficient of 0. (This equation may also be written as 0x + 1y = 5.) This equation have graph that is vertical straight line, as shown in the next example.

Example 4 Draw the graph of y = 5. Set up a table and write the values you choose for x—say, 3, 0, and –4. As the equation states, y is always 5 for any value of x that you choose.

Example 4 cont’d Plot the points from the table: (3, 5), (0, 5), and (–4, 5). Then draw a straight line through them, as in Figure 8.8. Figure 8.8

The x Intercept and the y Intercept of a Line Horizontal Line The graph of the linear equation y = k, where k is a constant, is the horizontal line through the point (0, k). That is, y = k is a horizontal line with a y intercept of k. Vertical Line The graph of the linear equation x = k, where k is a constant, is the vertical line through the point (k, 0). That is, x = k is a vertical line with an x intercept of k.

Copyright © Cengage Learning. All rights reserved. 8.3 The Slope of a Line Copyright © Cengage Learning. All rights reserved.

The Slope of a Line The slope of a line or the “steepness” of a roof (see Figure 8.11) can be measured by the following ratio: A straight line can also be graphed by using its slope and knowing one point on the line. Slope Figure 8.11

The Slope of a Line If two points on a line (x1, y1) (read “x-sub-one, y-sub-one”) and (x2, y2) (read “x-sub-two, y-sub-two”) are known (see Figure 8.12). Slope of a line through two points Figure 8.12

The Slope of a Line The slope of the line is defined as follows. Note slope is abbreviate “m”

Example 1 Find the slope of the line passing through the points (– 2, 3) and (4, 7). (See Figure 8.13.) Figure 8.13

Example 1 cont’d If we let x1 = – 2, y1 = 3, x2 = 4, and y2 = 7, then

Example 1 cont’d Note that if we reverse the order of taking the differences of the coordinates, the result is the same:

The Slope of a Line Note that in Example 1, the line slopes upward from left to right. In general, the following is true. General Statements About the Slope of a Line 1. If a line has positive slope, then the line slopes upward from left to right. 2. If a line has negative slope, then the line slopes downward from left to right.

The Slope of a Line 3. If the slope of a line is zero, then the line is horizontal. 4. If the slope of a line is undefined, then the line is vertical. The slope of a straight line can be found directly from its equation as follows: 1. Solve the equation for y. 2. The slope of the line is given by the coefficient of x.

Example 5 Find the slope of the line 4x + 6y = 15. First, solve the equation for y. 4x + 6y = 15 -4x -4x 6y = –4x + 15 Note: write the x term first always followed by the number 6 6 6 y = -4/6 x + 15/6 (We can reduce -4/6 to -2/3 and 15/6 to 5/2) y = - 2/3 x + 5/2 The slope of the line is given by the coefficient of x, or m = Subtract 4x from both sides. Divide every term by 6. slope

The Slope of a Line Two lines in the same plane are parallel if they do not intersect even if they are extended. Two lines in the same plane are perpendicular if they intersect at right angles. Since parallel lines have the same steepness, they have the same slope. Parallel lines Perpendicular lines

The Slope of a Line The slopes of perpendicular lines are negative reciprocals of each other; that is, Ex. If line 1 has a slope of 5 then line 2 perpendicular to it would have a slope of - 1/5 Since perpendicular lines run in opposite directions it makes sense their slopes are true opposites.

Example 7 Determine whether the lines given by the equations 2x + 3y = 6 and 6x – 4y = 9 are parallel, perpendicular, or neither. First, find the slope of each line by solving its equation for y. 2x + 3y = 6 (subtract 2x from both sides) 3y = – 2x + 6 y = -2/3 x + 6/3 (divide each term by 3)

Example 7 Since the slopes are not equal, the lines are not parallel. cont’d Since the slopes are not equal, the lines are not parallel. Since - 2/3 is the negative opposite of 3/2 the lines are perpendicular. Another way to think of this is that -2/3 * 3/2 = -1 so any two lines whose product of their slopes is -1 are perpendicular.

Copyright © Cengage Learning. All rights reserved. 8.4 The Equation of a Line Copyright © Cengage Learning. All rights reserved.

The Equation of a Line We have learned to graph the equation of a straight line and to find the slope of a straight line given its equation or any two points on it. In this section, we will use the slope to graph the equation and to write its equation. A fast and easy way to draw the graph of a straight line when the slope and y intercept are known is to first plot the y-intercept point on the graph and consider this to be the starting point.

The Equation of a Line Then from this starting point and using the slope (rise over run), move up or down the number of units indicated by the rise and then move right or left the number of units indicated by the run to a second point. Mark this second point. Then draw a straight line through these two points. Note: If the slope is negative, you must move in an opposite direction for one part of it only. Ex. -1/2 would be plotted by either moving: (-1/2) down 1 and to the right 2 or (1/ -2) up 1 and to the left 2. Also note that if a slope is a whole number you must write it as a fraction over 1 to plot it. Ex. M= 5 means 5/1 (up five and to the right 1)

Example 1 Draw the graph of the line with slope and y intercept 4. The slope corresponds to From the y intercept 4 [the point (0, 4)], move 2 units up and then 3 units to the right. Then draw a straight line through (0, 4) and (3, 6).

The Equation of a Line Slope-Intercept Form When the equation of a straight line is written in the form y = mx +(or -) b the slope of the line is m and the y intercept is b.

Example 2 Draw the graph of the equation 8x + 2y = –10 using its slope and y intercept. First, find the slope and y intercept by solving the equation for y as follows. 8x + 2y = –10 2y = –8x – 10 y = –4x – 5 Subtract 8x from both sides. Divide each term by 2. y intercept slope

Example 2 cont’d The slope is –4, and the y intercept is –5. The slope –4 corresponds to When the slope is an integer, write it as a ratio with 1 in the denominator. From the y intercept –5 (0, -5), move 4 units down and 1 unit to the right, as shown Draw a straight line through the points (0, –5) and (1, –9).