Section 2.1 Solving Linear Equations Part I: Addition Property of Equality

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1 Section 2.1 Solving Linear Equations Part I: Addition Property of Equality What is a Linear Equation? Definitions A linear equation in one variable can be written in the form A, B and C, with A 0. The solution, or root, of an equation in x is. Equivalent equations are equations that. Tool #1: Addition Property of Equality If A, B, and C are real numbers, then the equations A B and A C B C are Example 1: Solve each equation. a) x 8 9 b) y c) x 7 x d) x

2 Solving Linear Equations Example 2: Solve each equation and check your solution a) 4 3x 6 6x x 3 7x 1 x b) 6 c) x 0 x d) 6 7 x x 1 6 e) 5 1 2x 4 3 x 7 3 x 0

3 Section 2.2 Linear Equations Part II: The Multiplication Property of Equality Tool #2: Multiplication Property of Equality If A, B, and C are real numbers, then the equations are equivalent equations Example 2: Solve each equation and check your solution a) 7x 10 b) 5x 70 1 c). 9x 18 d) x 3 5

4 5 4 1 e) x f) y Example 3: Solve each equation and check your solution. a) 9x 2x 121 b) 11x 5x 6x 168 c) 5x 4x 8x 0 d) 10x 6x 3x 4

5 Section 2.3 Linear Equations Part III: More on Solving Linear Equations Steps for solving a linear equation (An equation where the variable power of 1. only appears to a Step 1: Simplify each side separately. Step 2: Isolate the variable term on one side. Step 3: Isolate the variable. Then check your answer by plugging in your solution into the original equation! Example 1 Solve. (a) (b) (c) (d)

6 If fractions appear in an equation, CLEAR THE FRACTIONS by multiplying both sides of the equation (or every term!) by the least common denominator. If decimals appear in an equation, CLEAR THE DECIMALS by multiplying both sides of the equation (or every term!) by either 10, 100, 1000, etc Example 2 Solve.

7 There are 3 types of linear equations: 1) An equation with exactly one solution is called a. 2) An equation for which every real number is a solution is called an. 3) An equation that has no solution is called a. Example 3 Solve each equation. Then state whether the equation is a conditional equation, an identity, or a contradiction. (a) (b) (c)

8 Algebra in Everyday Life 1. A football player gained y yards on a punt return. On the next return, he gained 4 yd. What expression represents the number of yards he gained altogether? 2. A hockey player scored 42 goals in one season. He scored n goals in one game. What expression represents the number of goals he scored in the rest of the games? 3. Chandler is b years old. What expression represents his age 3 yr ago? 5 yr from now? 4. Jean has y dimes. Express the value of the dimes in cents. 5. A clerk has v dollars, all in $20 bills. What expression represents the number of $20 bills the clerk has? 6. A concert ticket costs c dollars for an adult and f dollars for a child. Find an expression that represents the total cost for 4 adults and 6 children.

9 Section Classwork Solving Linear Equations There are two basic rules to solving linear equations: the addition/subtraction rule and the multiplication/division rule. There are two things to always remember when solving equations: (a) whatever you do to one side, you must do the same to the other side; (b) the purpose of solving equations is to isolate the variable; therefore, the goal is to get the variable on one side of the equal sign and the numbers on the other side. Solve the following equations: Practice Problems 1. 6x 4 5x y 5 11y x 5 5x 4x 8 3x 4. 14x 57 17x 8 2x x x x x 45 x 9. 5x x x 13. 7x 3 4x x x x 2 11 x x 1 5 3x x 3 x x 7 3 x 1 1 x

10 Section 2.4 Introduction: Applications Involving Linear Equations Step 1 found. Step 2 Step 3 Step 4 Step 5 Step 6 Solving an Applied Problem Read the problem carefully until you understand what is given and what is to be Assign a variable to represent the unknown value, using diagrams or tables as needed. Write down what the variable represents. If necessary, express any other unknown values in terms of the variable. Write an equation using the variable expression(s). Solve the equation. State the answer. Does it seem reasonable? Check the answer in the words of the original problem. Example 1: If 5 is added to the product of 9 and a number, the result is 19 less than the number. Find the number. Step 1 Step 2 Read Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check

11 Example 2: In the 2006 Winter Olympics in Torino, Italy, the United States won 6 more medals than Norway. The two countries won a total of 44 medals. How many medals did each country win? Step 1 Read Step 2 Assign a variable Step 3 Write an equation Step 4 Solve Step 5 State the answer Step 6 Check Example 3: The owner of Terry s Coffeehouse found that the number of orders for croissants was 1/6 the number of orders for muffins. If the total number for the two breakfast rolls was 56, how many orders were placed for croissants? Step 1 Read Step 2 Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check

12 Example 4: At a meeting of the local computer user group, each member brought two nonmembers. If a total of 27 people attended, how many were members and how many were nonmembers? Step 1 Read Step 2 Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check Example 5: A piece of pipe is 50 in. long. It is cut into three pieces. The longest piece is 10 in. more than the middle-sized piece, and the shortest piece measures 5 in. less than the middlesized piece. Find the lengths of the three pieces. Step 1 Read Step 2 Assign a variable Step 3 Step 4 Write an equation Solve Step 5 Step 6 State the answer Check

13 Definition: Two integers that differ by 1 are called consecutive integers. Examples: 5 and and and -2 If x represents an integer, then represents the next larger consecutive integer. Consecutive even integers and consecutive odd integers differ by 2. Examples: 4 and 6 (consecutive even integers) 7 and 9 (consecutive odd integers) If x represents an even integer, then represents the next larger even consecutive integer. If x represents an odd integer, then represents the next larger odd consecutive integer. Example 6: Two back-to-back page numbers in this book have a sum of 569. What are the page numbers? Step 1 Step 2 Read Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check

14 Example 7: Find two consecutive even integers such that six times the lesser added to the greater gives a sum of 86. Step 1 Step 2 Read Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check Done!

15 Section 2.4 Classwork 1. If four times a number is added to 8, the result is three times the number added to 5. Find the number. 2. If 3 is added to a number and this sum is doubled, the result is 2 more than the number. Find the number. 3. The total number of Democrats and Republicans in the U.S. House of Representatives during the 109 th Congress was 434. There were 30 more Republicans than Democrats. How many members of each party were there? 4. The Toyota Camry was the top-selling passenger car in the U.S. in 2004, followed by the Honda Accord. Honda Accord sales were 40 thousand less than Toyota Camry sales, and 814 thousand total of the two types of cars were sold. How many of each make of car were sold?

16 5. Arnie Waldman has chosen a workout that combines weight training and aerobics. If he does 30 min of weight training, followed by a 30-min aerobics class, he will burn a total of 374 calories. If he burns 12 5 as many calories doing aerobics as doing weight training, how many calories will he burn in each activity? 6. Find two consecutive odd integers such that twice the greater is 17 more than the lesser. 7. Two houses on the same side of the street have house numbers that are consecutive even integers. The sum of the integers is 58. What are the two house numbers? 8. If the sum of three consecutive odd integers is 69, what is the third of the three odd integers?

17 Section 2.5 Applications from Geometry Example (Perimeter): Find the value of the remaining variable. P 2L 2W P 126, W 25 L W Example (Perimeter): A farmer has 800 m of fencing material to enclose a rectangular field. The width of the field is 50 m less than the length. Find the dimensions of the field. Step 1 Step 2 Read Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check

18 Example (Perimeter): The longest side of a triangle is 1 in. longer than the medium side. The medium side is 5 in. longer than the shortest side. If the perimeter is 32 in., what are the lengths of the three sides? Step 1 Step 2 Read Assign a variable Step 3 Write an equation Step 4 Solve Step 5 Step 6 State the answer Check Example (Area): The area of a triangle is 120 base of the triangle. 2 m. The height is 24 m. Find the length of the Step 1 Step 2 Read Assign a variable Step 3 Write an equation Step 4 Solve

19 Step 5 Step 6 State the answer Check Geometry Notes: Complementary Angles Supplementary Angles Vertical Angles Example: Find the measure of each marked angle in the figures. (a) (6x 29) ( x 11) (b) (4x 19) (6x 5)

20 Solving Literal Equations Example: Solve I prt for t. Example: Solve S rh r for h. Example: Solve A p prt for t. Example: Solve 1 A h ( b B ) for h. 2

21 Section 2.5 Classwork Perimeter = Sum of the Lengths of the Sides 1. The perimeter of a rectangle is 108 cm. The width of the rectangle is 6 cm less than the length. Find the dimensions of the rectangle. 2. The perimeter of a rectangle is 22 ft. The length is 1 ¾ times the width. Find the dimensions of the rectangle. 3. The perimeter of a rectangle is 20 in. The length of the rectangle is 3 in. less than 2 ¼ times the width. Find the dimensions of the rectangle.

22 4. The perimeter of a triangle is 14 ft. One leg is 1 more than twice the length of the shorter leg. The third leg is 3 less than five times the length of the shorter leg. Find the dimensions of the triangle. 5. The perimeter of a triangle is 22 cm. One leg is 4 cm less than the longest leg. The third leg is twice the amount of nine less than the longer leg. Find the dimensions of the triangle. 6. The perimeter of a triangle is 41 m. The longer leg is 4 times the length of the shorter leg. The third leg is six more than twice the length of the shorter leg. Find the dimensions of the triangle.