1-2. Solving Equations by Adding or Subtracting Going Deeper Essential question: What are some different methods for solving linear equations?

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1 Name Class Date 1-2 Solving Equations by Adding or Subtracting Going Deeper Essential question: What are some different methods for solving linear equations? The solution of an equation can be given as an equation of the form x = a where a is a solution, as in x = 6, or listed in set notation, as {6}. 1 A-REI.2.3 EXPLORE Solving Equations Using Different Methods Find the solution set for the linear equation. A Use guess and check to find the solution set of the equation x - 5 = 4. Guess x = = 5 > 4, so 10 is too great. Guess x = = 3 < 4, so 8 is too little. Guess x = = 4 = 4, so 9 is correct. B Use a table to find the solution set of the equation y + 7 = 10. y y Sum C Work backward to find the solution set of the equation z - 2 = 8. Start with the number being subtracted,. Working backward, add since it is the inverse of subtracting. You get z = = or z = 10. REFLECT 1a. Could you solve each of the three equations above by all three methods? Explain. Chapter 1 13 Lesson 2

2 Two equations are equivalent equations if they have the same solution set. The two equations below are equivalent because they have the same solution set, {6}. x + 3 = 9 x - 3 = = = 3 To solve an equation algebraically, you perform a series of inverse operations to isolate the variable on one side. When these inverse operations are completed, the other side of the equation is the solution. The Addition and Subtraction Properties of Equality can be used to justify the steps taken to solve an equation. These properties, as well as other useful properties, are listed below. Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - b = b - c. Inverse Property of Addition a + (-a) = -a + a = 0 Identity Property of Addition a + 0 = 0 + a = a Associative Property of Addition (a + b) + c = a + (b + c) 2 A-REI.1.1 EXAMPLE Adding or Subtracting to Find the Solution Set Add or subtract to find the solution set. A x + 5 = 13 x + 5 = 13 - Property of Equality x + = 13 - Property of Addition x = 13 - Property of Addition x = Simplify. B y - 11 = 2 y = 2 + Property of Equality y + = 2 + Property of Addition y = 2 + Property of Addition y = Simplify. REFLECT 2a. Which property of equality would you use to solve x - 47 = 100? Explain. Chapter 1 14 Lesson 2

3 3 A-REI.1.1 EXAMPLE Using the Associative Property Use properties to find the solution set of (x + 5) + 4 = 16. (x + 5) + 4 = 16 Original equation x + (5 + ) = 16 Associative Property x + = 16 Simplify. x = 16 - Property of Equality x + = 16-9 Inverse Property of Addition = 16 - Property of Addition x = Simplify. REFLECT 3a. Solve (x + 5) + 4 = 16 by first subtracting 4 and then subtracting 5. Show your work and justify each step. 3b. Does performing the steps in a different order affect the solution of the equation? Compare the steps in the example with the steps for the question above. How do the two methods differ? Explain. Chapter 1 15 Lesson 2

4 PRACTICE Find the solution set for each equation. State the property you used. 1. m - 7 = r + 12 = p = = q = 8 + z = b - 21 Solve using the Associative Property first. Justify your steps. 7. (y + 8) - 3 = 16 Solve using the Properties of Equality first. Justify your steps. 8. (m - 3) + 5 = 12 Chapter 1 16 Lesson 2

5 Name Class Date Additional Practice 1-2 Solve each equation. Check your answers. 1. g 7 = t + 4 = = m 7 4. x = n 3 8 = p 1 3 = k = = w = r y 57 = b = a + 15 = Marietta was given a raise of $0.75 an hour, which brought her hourly wage to $ Write and solve an equation to determine Marietta s hourly wage before her raise. Show that your answer is reasonable. 14. Brad grew inches this year and is now inches tall. Write and solve an equation to find Brad s height at the start of the year. Show that your answer is reasonable. 15. Heather finished a race in 58.4 seconds, which was 2.6 seconds less than her practice time. Write and solve an equation to find Heather s practice time. Show that your answer is reasonable. 16. The radius of Earth is km, which is km longer than the radius of Mars. Write and solve an equation to determine the radius of Mars. Show that your answer is reasonable. Chapter 1 17 Lesson 2

6 Problem Solving Write the correct answer. 1. Michelle withdrew $120 from her bank account. She now has $3345 in her account. Write and solve an equation to find how much money m was in her account before she made the withdrawal. 3. Earth takes 365 days to orbit the Sun. Mars takes 687 days. Write and solve an equation to find how many more days d Mars takes than Earth to orbit the Sun. 2. Max lost 23 pounds while on a diet. He now weighs 184 pounds. Write and solve an equation to find his initial weight w. 4. In 1990, 53.4% of commuters took public transportation in New York City, which was 19.9% greater than the percentage in San Francisco. Write and solve an equation to find what percentage of commuters p took public transportation in San Francisco. Use the circle graph below to answer questions 5 7. Select the best answer. The circle graph shows the colors for SUVs as percents of the total number of SUVs manufactured in 2000 in North America. 5. The percent of silver SUVs increased by 7.9% between 1998 and If x% of SUVs were silver in 1998, which equation represents this relationship? A x = 14.1 C 7.9x = 14.1 B x 7.9 = 14.1 D 7.9 x = Solve the equation from problem 5. What is the value of x? F 1.8 H 7.1 G 6.2 J The sum of the percents of dark red SUVs and white SUVs was 26.3%. What was the percent of dark red SUVs? A 2.3% C 12.2% B 3.2% D 18% Chapter 1 18 Lesson 2