Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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. Using the Addition Property of Equality

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Note Equations can be thought of in terms of a balance. Thus, adding the same quantity to each side does not affect the balance. Using the Addition Property of Equality

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Example 1 Solve each equation. Our goal is to get an equivalent equation of the form x = a number. (a) x – 23 = 8 x – = Using the Addition Property of Equality x = 31 Check: 31 – 23 = 8 (b) y – 2.7 = –4.1 y – = – y = – 1.4 Check: –1.4 – 2.7 = –4.1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 (a) –12 = z + 5 –12 – 5 = z + 5 – 5 Using the Addition Property of Equality –17 = z Check: –12 = – (b) 4a + 8 = 3a 4a – 4a + 8 = 3a – 4a 8 = –a Check: 4(–8) + 8 = 3(–8) ? –8 = a –24 = –24 Example 2 Solve each equation. Our goal is to get an equivalent equation of the form x = a number.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Example 3 Solve. 5(2b – 3) – (11b + 1) = 20 10b – 15 – 11b – 1 = 20 Simplifying and Using the Addition Property of Equality –b – 16 = 20 Check: 5((2 · –36) –3) – (11(–36) + 1) = –b – = –b = 36 b = –36 5(–72 –3) – (– ) = 5(–75) – (–395) = – = 20

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Solving a Linear Equation Step 1Simplify each side separately. Clear (eliminate) parentheses, fractions, and decimals, using the distributive property as needed, and combine like terms. Step 2Isolate 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 3Isolate 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 4Check. Substitute the proposed solution into the original equation to see if a true statement results. Solving a Linear Equation

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Using the Four Steps for Solving a Linear Equation 5w + 3 – 2w – 7 = 6w + 8 3w – 4 = 6w + 8 Combine terms. 3w – = 6w Step 1 Step 2Add 4. 3w = 6w + 12 Combine terms. 3w – 6w = 6w + 12 – 6w Subtract 6w. Combine terms. – 3w = 12 – 3 Divide by –3. – 3w 12 = w = – 4 Step 3 Example 1 Solve the equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Using the Four Steps for Solving a Linear Equation 5w + 3 – 2w – 7 = 6w + 8 ? Let w = – 4. Step 4 5(– 4) + 3 – 2(– 4) – 7 = 6(– 4) + 8 Check by substituting – 4 for w in the original equation. – – 7 = – ? Multiply. – 16 = – 16 True The solution to the equation is – 4. Example 1 (continued) Solve the equation.

2h 14 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Using the Four Steps for Solving a Linear Equation 5 ( h – 4 ) + 2 = 3h – 4 5h – = 3h – 4 Distribute. 5h – 18 = 3h – 4 Step 1 Step 2 Combine terms. 5h – = 3h – Add 18. 5h = 3h + 14Combine terms. Subtract 3h.5h – 3h = 3h + 14 – 3h2 Combine terms.2h = 14 = h = 7 Step 3 Divide by 2. Example 2 Solve the equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Using the Four Steps for Solving a Linear Equation Check by substituting 7 for h in the original equation.Step 4 5 ( h – 4 ) + 2 = 3h – 4 5 ( 7 – 4 ) + 2 = 3(7) – 4 5 (3) + 2 = 3(7) – = 21 – 4 17 = 17 ? Let h = 7. ? Subtract. True ? Multiply. The solution to the equation is 7. Example 2 (continued) Solve the equation.

–5y – 4 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Using the Four Steps for Solving a Linear Equation 15y – ( 10y – 2 ) = 2 ( 5y + 7 ) – 16 15y – 10y + 2 = 10y + 14 – 16 Distribute. 5y + 2 = 10y – 2 Step 1 Step 2 Combine terms. 5y + 2 – 2 = 10y – 2 – 2Subtract 2. 5y = 10y – 4Combine terms. Subtract 10y.5y – 10y = 10y – 4 – 10y –5 Combine terms.–5y = – 4 = y = Step 3 Divide by – Example 3 Solve the equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression The sum of a number and 2 Mathematical Expression (where x and y are numbers) Addition 3 more than a number 7 plus a number 16 added to a number A number increased by 9 The sum of two numbers x + 2 x x x + 16 x + 9 x + y

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression 4 less than a number Mathematical Expression (where x and y are numbers) Subtraction 10 minus a number A number decreased by 5 A number subtracted from 12 The difference between two numbers x – 4 10 – x x – 5 12 – x x – y

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression 14 times a number Mathematical Expression (where x and y are numbers) Multiplication A number multiplied by 8 Triple (three times) a number The product of two numbers 14x 8x8x 3x3x xy of a number (used with fractions and percent) 3 4 x 3 4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Applications of Linear Equations Translating from Words to Mathematical Expressions Verbal Expression The quotient of 6 and a number Mathematical Expression (where x and y are numbers) Division A number divided by 15 The ratio of two numbers or the quotient of two numbers (x ≠ 0) 6 x (y ≠ 0) x y x 15

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 number divided by 15” and “15 divided into x ” both translate as. Similarly, “the quotient of x and y ” is translated as. 2.3 Applications of Linear Equations Caution x 15 x y

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 Applications of Linear Equations Translating Words into Equations Verbal SentenceEquation 16x – 25 = 87 If the product of a number and 16 is decreased by 25, the result is 87. = 48 The quotient of a number and the number plus 6 is 48. x + 6 x + x = 54 The quotient of a number and 8, plus the number, is x Twice a number, decreased by 4, is 32. 2x – 4 = 32