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Lesson
Consider an equation that contains one variable, such as 2 + x = 6. A solution to an equation is a value of the variable that makes the equation true. For 2 + x = 6, 4 is a solution because 2 + 4 = 6.
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Solving with a Table and a Graph
In this lesson, you will learn two ways to solve equations: using a table and using a graph. We apply each way to solve 11 = 4 x + 3.
Example 1
Solve 11 = 4 x + 3 using a table. Solution Replace y with 11 and make a table of values for y = 4 x + 3. Look in the table to the right to see what value of x gives a value of 11 for y. Looking at the table, you can see that when x = 2, y = 11. Based on the table, you can conclude that the solution to the equation 11 = 4x + 3 is x = 2.
Example 2
Solve 11 = 4 x + 3 using a graph. Solution As in Example 1, replace y with 11. Then graph y = 4 x + 3 and look for the value of x that corresponds with the value of 11 for y. To do this, start at 11 on the y-axis. Move right to the line, and then move down to the x-axis. As in Example 1, x = 2 is the solution.
Vocabulary solution to an equation
Solving Equations with
Tables and Graphs
BIG IDEA Tables and graphs lead to exact or approximate solutions to equations.
QY Is 5 a solution to |–12m + 19| = 41?
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Solving Equations with Tables and Graphs 135
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Lesson 3-
x 4 x + 3 y 1 4(1) + 3 7 2 4(2) + 3 11 3 4(3) + 3 15 4 4(4) + 3 19 5 4(5) + 3 23
Evaluate. a. 50% of 120P b. 5% of 120P c. 50% of 12P d. 5% of 12P
Mental Math
136 Linear Equations and Inequalities
You can also see this on a graphing calculator. Most graphing calculators have a trace option. The trace option allows you to move a cursor along the graph. As the cursor moves, it lists the coordinates on the graph. The calculator screen at the right shows the cursor at (–0.8510638, –0.4042553). Move the cursor until the y-coordinate is as close as possible to 11. Remember that the x value is only an approximation for the solution if the y value is also an approximation.
Situations Leading to Equations
Real-world situations can often be translated into relationships among numbers. If you can represent these relationships with an equation, then the methods of this and the next lesson can lead to a solution.
Example 3
Khalid lives in Alaska where there is no sales tax. He bought 3 tacos with a $5 bill and received $0.77 in change. Define the variable and represent the relationship with an equation. Solution Let t represent the unknown cost of one taco. Because $5 minus the cost of 3 tacos equals $0.77 in change, 5 - 3t = 0.77 is the equation.
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Questions
COVERING THE IDEAS
- Determine if the value of x is a solution for the equation. a. 11 - 9 x = 47, x = 4 b. 3 x^2 + 4 x = 55, x = – c. – x^3 = 64, x = –
In 2 and 3, solve by making a table for each equation and circling the row on the table that contains the solution.
- 6 = 2 x + 2 3. –5 a + 7 = –
In 4 and 5, solve by making a graph for each equation and drawing lines from the y-axis to the graph, and then to the x-axis to indicate the solution.
- y = –3 x - 4, y = 2 5. y = 5 x - 18, y = –
- a. With a graphing calculator, make a table for y = 5(5 x - 2) using the integers from –5 to 0. b. Adjust the table to solve 5(5 x - 2) = –26.
Chapter 3
QY Todd has $250 in his savings account. If he has a job that pays $11 per hour, how many working hours will it take for him to have enough money to buy a $481 surfboard?
In 2006, the states without a sales tax were Alaska, Delaware, Montana, New Hampshire, and Oregon. Source: Federation of Tax Administrators
138 Linear Equations and Inequalities
- Sierra redwood trees are known to be among the world’s tallest trees, growing an average of 3.5 feet each year until they mature. If a redwood tree is now 28 feet tall, its height h after y years is described by h = 28 + 3.5 y. (Lesson 3-1) a. Make a table of values for this relationship. b. Draw a graph of this situation. c. How tall will the tree be in 21 years? d. How many years will it take the tree to grow to a height of 259 feet?
- Internet World Stats reported in the summer of 2005 that there were about 220 million Internet users in North America. That number was growing by an average of 23 million users per year. Suppose this rate continues between 2005 and 2010. (Lessons 3-1, 1-5) a. Make a table showing the total number of millions of Internet users 0, 1, 2, 3, 4, and 5 years after 2005. b. Let x represent the number of years after 2005. What window on your graphing calculator would be most appropriate to view the graph? c. Draw the graph using your graphing calculator.
In 21 and 22, a number is given. a. Find its opposite. b. Find its reciprocal. (Lessons 2-8, 2-5)
- 8.3 22. – __^265
In 23 and 24, evaluate the expression _____a^ + 3^ b + 3(a - b) for the given values of a and b. (Lesson 1-1)
- a = 11 and b = –4 24. a = 11 x and b = 55 x
EXPLORATION
- Use the graph below to solve 11 = 7 + 2 √ x + 1. Draw dotted lines to show your method.
Chapter 3
Many Sierra redwoods are between 250 and 300 feet tall, the tallest being about 325 feet high. Source: California Department of Parks & Recreation
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QY ANSWERS
- m = 5 is a solution because
|–60 + 19| = |–41| = 41.
- 21 hr
Years after 2005
Users (millions)
