Lesson #14 Graphing Lines

 

A linear equation is an equation in two variables (usually x and y) which involves only multiplication, addition, and/or subtraction. There can be no exponents on the variables, and the variables are not used in the denominator of any fraction. Since such an equation has two variables, we must specify the values for both variables which, taken together, make the equation true.

Example 1.  Show that x = 3 and y = 5 is a solution of the equation  4x - 2y = 2.

Solution:  Substitute (3) for x and (5) for y to get 4(3) - 2(5) = 2, or 12 - 10 = 2, which is true.

The pair of values (x = 3 and y = 5) is written as an ordered pair (3,5). In this notation, the values 3 and 5 are given for the two variables. The value for x is always listed first in the pair, then y. So Example 1 could have been (and will be) written as “Show that (3,5) is a solution to 4x - 2y = 2.

Example 2.  In the equation  y = 3x + 1, find the ordered pair solution that has x = -2.

Solution: Substitute the given value for x into the equation.  So y = 3(-2) + 1. Simplify (or solve for y if necessary) to find the corresponding value for y. In this case, y = -6 + 1 = -5.  So the solution with x = -2 is (-2,-5).

Each equation with two variables has infinitely many ordered pair solutions (one for each choice of x). Knowing any one value (x or y) is enough to solve for the other. If we make a list of several different ordered pairs that are solutions to an equation, we can plot them as points on a graph. Fact: the graph of every linear equation is a straight line. Determine any two points, and you can graph the equation as a straight line through those two points. I recommend that you find three solutions; the extra point serves as a check on your arithmetic.

Example 3. Find three different solutions to the equation y = 3x + 1, and graph the equation.

Solution: One solution is (-2,-5)  (from Example 2). We could use this as the start of an x-y table of values.

 

 

 

x

y = 3x + 1

 

 

2

3(-2) + 1 = -5

 

 

1

3(1) + 1 = 4

 

0

3(0) + 1 = 1

 

 

 

The points (-2, -5), (1, 4), and (0, 1) are solutions. We mark these on the graph and connect them with a straight line.

 

You are free to choose any three points to graph. It is best to keep your choices for x small so that you can fit the solutions easily onto the graph. Try x = 0 for one easy calculation. Try one positive value for x, and then try one negative value for x. This plan will give you a good sample of possible solutions for points on the line.

Example 4. Graph the equation  y = -2x + 3.

 

 

x

y = -2x + 3

 

 

0

-2(0) + 3 = 3

 

 

2

-2(2) + 3 = -1

 

-1

-2(-1) + 3 = 5

 

 

 

We plot the points (0,3), (2, -1) and (-1, 5), and then connect them with a straight line.

 

 

 

Slope of a line

An important feature of a line is its slant, or slope. The slope of a line is a ratio of the vertical distance to the horizontal distance from one point to another on the line. This ratio is a fixed constant for the line. That is, the slope is the same no matter which two points you chose. An easy way to remember how to compute the slope is “the change in y over the change in x”.

 

 

 

In Example 3, with y = 3x + 1, we found the line went through the points (-2, -5) and (0, 1). The change in y (from -5 to +1) is up 6 units. The change in x (from -2 to 0) is right 2 units. Thus the slope is given by  .

(the letter m is always used to designate the slope).

 

 

 

 

In Example 4, the equation y = -2x + 3 passed through the points (0,3) and (2,-1). The change in y is -4 (from 3 to -1 is down 4), and the change in x is 2 (from 0 to 2 is right 2).

So the slope is  .

 

 

 

There are a couple easy things we can say about slopes.

1)      A line with positive slope rises as you move to the right.

2)      A line with negative slope falls as you move to the right.

3)      The slope of a line is always the same as the coefficient of the x term when the equation is solved for y. That is, in the equation y = mx + b, the slope of the line is m.

4)      In the equation y = mx + b, the solution corresponding to x = 0 is (0,b).

The point (0,b) is called the y-intercept. In general, an intercept on a graph is any point where one value is 0. The x-intercept has y = 0, and the y-intercept has x = 0.

Example 5. Find the slope and the y-intercept of the line y = 9x + 2.

Solution: Since this equation is solved for y, compare it to y = mx + b. Here m = 9 (the coefficient of the x term) and b = 2 (the constant term). So the slope is m = 9, and the y-intercept is (0,2). To confirm these answers, let’s check a couple points. First, when x = 0, we should get the y-intercept from the equation. When x = 0, y = 9(0) + 2 = 0 + 2 = 2. So the y-intercept is in fact (0,2). For another point, take x = 1. With x = 1, y = 9(1) + 2 = 11. Then we have the two points (0,2) and (1, 11) on the graph. The slope is

.

 

Example 6. Find the x-intercept in the equation y = 9x + 2.

Solution: Substitute y = 0 into the equation to get  0 = 9x + 2. Solve for x by separating the variable and constant terms. By subtracting 9x from both sides, we get -9x = 2.  Then divide both sides by  -9  to get .

Thus the x-intercept is .

Exercises

 Check whether the given point is on the line.

1)  Is (2,5) on the line y = 3x - 1?                                2) Is (-3,1) on the line y = 2x + 8?

 Find the ordered pair solution corresponding to the given x value for each equation:

3.  y = 2x - 5;  x = 2                                                   4.  y = 6x - 3;  x = 1

5.  y = -4x - 1;  x = 5                                                 6.  y = x + 7; x = -10

  Find three solutions and graph each line:

 

7.  y = x + 2

8.  y = 3x - 2


9.  y = 4x + 1

10.  y = -2x + 5

11.  y = -3x - 4

12.  y = 7x – 8

13.  y = -x + 5

14.  y = -2x - 1

15.  y = 3x + 2

16.  y = -5x

17.  y = -3x + 4

18.  y = -x - 4

19.  y = 3x

20.  y = 6x + 1

21.  y = 4x - 10

22.  y = -2x – 2

  Determine the slope between each pair of points: 

(Use the slope formula )

23.  (2,6) and (5,10)                                               24.  (0,3) and (5, -1)

25.  (-2,-3) and (4,-1)                                            26.  (7,0) and (0,6)

27.  (-1,5) and (6,2)                                               28.  (3,-4) and (1,-8)

29.  (2, 5) and (7,5)                                                30.  (1,-5) and (3,17)

  From each linear equation, determine the slope, the y-intercept, and the x-intercept.

31.  y = 3x + 2                                                           32.  y = 3x - 12

33.  y = 2x + 3                                                           34.  y = -2x + 4

35.  y = -10x - 4                                                         36.  y =5x - 3

37.  y = -x + 4                                                            38.  y = -2x - 1

39.  y = 8x + 2                                                           40.  y = -7x

41.  y = -2x + 14                                                        42.  y = -7x - 4

 

Check the answers to Lesson 14 exercises

This completes the series of Beginning Algebra lessons.

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