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Using the Addition Property of Equality Addition Property of Equality If A, B, and C are real numbers, then the equations A=B and A+C=B+C are equivalent equations. In words, we can add the same number to each side of an equation without changing the solution. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.1 - 1 Using the Addition Property of Equality Note Equations can be thought of in terms of a balance. Thus, adding the same quantity to each side does not affect the balance. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.1 - 2 Using the Addition Property of Equality Example 1 Solve each equation. Our goal is to get an equivalent equation of the form x = a number. (a) x – 23 = 8 x – 23 + 23 = 8 + 23 x = 31 Check: 31 – 23 = 8 Copyright © 2010 Pearson Education, Inc. All rights reserved. (b) y – 2.7 = –4.1 y – 2.7 + 2.7 = –4.1 + 2.7 y = – 1.4 Check: –1.4 – 2.7 = –4.1 Sec 10.1 - 3 Using the Addition Property of Equality The same number may be subtracted from each side of an equation without changing the solution. If a is a number and –x = a, then x = –a. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.1 - 4 Using the Addition Property of Equality Example 2 Solve each equation. Our goal is to get an equivalent equation of the form x = a number. (a) –12 = z + 5 –12 – 5 = z + 5 – 5 –17 = z Check: –12 = –17 + 5 (b) 4a + 8 = 3a 4a – 4a + 8 = 3a – 4a 8 = –a –8 = a Check: 4(–8) + 8 = 3(–8) ? –24 = –24 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.1 - 5 Simplifying and Using the Addition Property of Equality Check: 5((2 · –36) –3) – (11(–36) + 1) = Example 3 Solve. 5(2b – 3) – (11b + 1) = 20 10b – 15 – 11b – 1 = 20 –b – 16 = 20 5(–72 –3) – (–396 + 1) = 5(–75) – (–395) = –375 + 395 = 20 –b – 16 + 16 = 20 + 16 –b = 36 b = –36 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.1 - 6 Solving a Linear Equation Solving a Linear Equation Step 1 Simplify each side separately. Clear (eliminate) parentheses, fractions, and decimals, using the distributive property as needed, and combine like terms. Step 2 Isolate the variable term on one side. Use the addition property so that the variable term is on one side of the equation and a number is on the other. Step 3 Isolate the variable. Use the multiplication property to get the equation in the form x = a number, or a number = x. (Other letters may be used for the variable.) Step 4 Check. Substitute the proposed solution into the original equation to see if a true statement results. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 7 Using the Four Steps for Solving a Linear Equation Example 1 Solve the equation. Step 1 Step 2 Step 3 5w + 3 – 2w – 7 = 6w + 8 3w – 4 = 6w + 8 3w – 4 + 4 = 6w + 8 + 4 3w 3w – 6w – 3w – 3w –3 w = 6w + 12 = 6w + 12 – 6w = 12 12 = –3 = –4 Copyright © 2010 Pearson Education, Inc. All rights reserved. Combine terms. Add 4. Combine terms. Subtract 6w. Combine terms. Divide by –3. Sec 10.3 - 8 Using the Four Steps for Solving a Linear Equation Example 1 (continued) Solve the equation. Step 4 Check by substituting – 4 for w in the original equation. 5w + 3 – 2w – 7 = 6w + 8 5(– 4) + 3 – 2(– 4) – 7 = 6(– 4) + 8 – 20 + 3 + 8 – 7 = – 24 + 8 – 16 = – 16 ? Let w = – 4. ? Multiply. True The solution to the equation is – 4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 9 Using the Four Steps for Solving a Linear Equation Example 2 Solve the equation. Step 1 Step 2 Step 3 5(h – 4) + 2 5h – 20 + 2 5h – 18 5h – 18 + 18 5h 5h – 3h 2h = = = = = = = 3h 3h 3h 3h 3h 3h 14 2h = 14 2 2 h = 7 Copyright © 2010 Pearson Education, Inc. All rights reserved. – – – – + + 4 4 4 4 + 18 14 14 – 3h Distribute. Combine terms. Add 18. Combine terms. Subtract 3h. Combine terms. Divide by 2. Sec 10.3 - 10 Using the Four Steps for Solving a Linear Equation Example 2 (continued) Solve the equation. Step 4 Check by substituting 7 for h in the original equation. 5 ( h – 4 ) + 2 = 3h – 4 5 ( 7 – 4 ) + 2 = 3(7) – 4 ? Let h = 7. 5 (3) + 2 = 3(7) – 4 ? Subtract. 15 + 2 = 21 – 4 ? Multiply. 17 = 17 True The solution to the equation is 7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 11 Using the Four Steps for Solving a Linear Equation Example 3 Solve the equation. 2 ( 5y + 7 ) – 16 10y + 14 – 16 10y – 2 Step 2 10y – 2 – 2 10y – 4 10y – 4 – 10y –4 = –4 Step 3 –5 = 4 Copyright © 2010 Pearson Education, Inc. All rights reserved. 5 Step 1 15y – 1 ( 10y – 2 ) 15y – 10y + 2 5y + 2 5y + 2 – 2 5y 5y – 10y –5y –5y –5 y = = = = = = = Distribute. Combine terms. Subtract 2. Combine terms. Subtract 10y. Combine terms. Divide by –5. Sec 10.3 - 12 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression Mathematical Expression (where x and y are numbers) Addition The sum of a number and 2 x+2 3 more than a number x+3 7 plus a number 7+x 16 added to a number x + 16 A number increased by 9 x+9 The sum of two numbers x+y Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 13 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression Mathematical Expression (where x and y are numbers) Subtraction 4 less than a number x–4 10 minus a number 10 – x A number decreased by 5 x–5 A number subtracted from 12 12 – x The difference between two numbers x–y Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 14 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression Mathematical Expression (where x and y are numbers) Multiplication 14 times a number 14x A number multiplied by 8 8x 3 of a number (used with 4 fractions and percent) 3 x 4 Triple (three times) a number 3x The product of two numbers xy Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 15 2.3 Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression Mathematical Expression (where x and y are numbers) Division The quotient of 6 and a number 6 (x ≠ 0) x A number divided by 15 x 15 The ratio of two numbers or the quotient of two numbers Copyright © 2010 Pearson Education, Inc. All rights reserved. x (y ≠ 0) y Sec 2.3 - 16 2.3 Applications of Linear Equations Caution CAUTION Because subtraction and division are not commutative operations, be careful to correctly translate expressions involving them. For example, “5 less than a number” is translated as x – 5, not 5 – x. “A number subtracted from 12” is expressed as 12 – x, not x – 12. For division, the number by which we are dividing is the denominator, and the number into which we are dividing is the numerator. For example, “a x number divided by 15” and “15 divided into x” both translate as 15 . Similarly, x “the quotient of x and y” is translated as y . Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 17 2.3 Applications of Linear Equations Indicator Words for Equality Equality The symbol for equality, =, is often indicated by the word is. In fact, any words that indicate the idea of “sameness” translate to =. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.3 - 18 2.3 Applications of Linear Equations Translating Words into Equations Verbal Sentence Equation Twice a number, decreased by 4, is 32. 2x – 4 = 32 If the product of a number and 16 is decreased by 25, the result is 87. 16x – 25 = 87 The quotient of a number and the number plus 6 is 48. The quotient of a number and 8, plus the number, is 54. Copyright © 2010 Pearson Education, Inc. All rights reserved. x = 48 x+6 x + x = 54 8 Sec 2.3 - 19