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Review solving one-step equations with integers, fractions, and decimals. One-step equations Vocabulary equation solve solution inverse operation isolate the variable Addition Property of Equality Subtraction Property of Equality An equation uses an equal sign to show that two expressions are equal. All of these are equations. 3 + 8 = 11 24 = x – 7 100 = 50 2 To solve an equation, find the value of the variable that makes the equation true. This value of the variable is called the solution of the equation. Additional Example 1: Determining Whether a Number is a Solution of an Equation Determine which value of x is a solution of the equation. r + 6 = 14 Additional Example 1 Continued x + 8 = 15; x = 5, 7, or 23 Determine which value of x is a solution of the equation. x + 8 = 15; x = 5, 7, or 23 Substitute each value for x in the equation. Substitute each value for x in the equation. ? ? x + 8 = 15 ? 5 + 8 = 15 ? 13= 15 x + 8 = 15 Substitute 5 for x. So 5 is not solution. ? 7 + 8 = 15 ? 15= 15 Substitute 7 for x. So 7 is a solution. 1 Additional Example 1 Continued Determine which value of x is a solution of the equation. x + 8 = 15; x = 5, 7, or 23 Substitute each value for x in the equation. ? x + 8 = 15 ? 23 + 8 = 15 ? 31= 15 Substitute 23 for x. So 23 is not a solution. Try This: Example 1 Determine which value of x is a solution of the equation. x – 4 = 13; x = 9, 17, or 27 Substitute each value for x in the equation. ? x – 4 = 13 ? 9 – 4 = 13 ? 5= 13 Substitute 9 for x. So 9 is not a solution. Try This: Example 1 Continued Try This: Example 1 Continued Determine which value of x is a solution of the equation. x – 4 = 13; x = 9, 17, or 27 Determine which value of x is a solution of the equation. x – 4 = 13; x = 9, 17, or 27 Substitute each value for x in the equation. Substitute each value for x in the equation. ? x – 4 = 13 ? 17 – 4 = 13 ? Substitute 17 for x. ? 13 = 13 So 17 is a solution. Addition and subtraction are inverse operations, which means they “undo” each other. To solve an equation, use inverse operations to isolate the variable. This means getting the variable alone on one side of the equal sign. x – 4 = 13 ? 27 – 4 = 13 Substitute 27 for x. ? 23 = 13 So 27 is not a solution. To solve a subtraction equation, like y 15 = 7, you would use the Addition Property of Equality. ADDITION PROPERTY OF EQUALITY Words You can add the same number to both sides of an equation, and the statement will still be true. Numbers Algebra 2+3=5 +4 +4 2+7=9 x=y x+ z = y + z 2 There is a similar property for solving addition equations, like x + 9 = 11. It is called the Subtraction Property of Equality. SUBTRACTION PROPERTY OF EQUALITY Words You can subtract the same number from both sides of an equation, and the statement will still be true. Numbers Algebra 4 + 7 = 11 3 3 4+4= 8 x=y x z = y z Additional Example 2A: Solving Equations Using Addition and Subtraction Properties Solve. A. 10 + n = 18 10 + n = 18 –10 –10 0+n= 8 n= 8 Check 10 + n = 18 ? 10 + 8 = 18 Subtract 10 from both sides. Identity Property of Zero: 0 + n = n. ? 18 = 18 Additional Example 2B: Solving Equations Using Addition and Subtraction Properties Additional Example 2C: Solving Equations Using Addition and Subtraction Properties Solve. Solve. B. p – 8 = 9 p–8=9 +8 +8 C. 22 = y – 11 22 = y – 11 + 11 + 11 Add 8 to both sides. p + 0 = 17 p = 17 Check p–8=9 ? 17 – 8 = 9 Identity Property of Zero: p + 0 = p. ? 33 = y + 0 Identity Property of Zero: y + 0 = 0. 33 = y Check 22 = y – 11 ? 22 = 33 – 11 ? 9 = 9 22 = 22 Try This: Example 2A Solve. A. 15 + n = 29 15 + n = 29 –15 –15 0 + n = 14 n = 14 Check 15 + n = 29 ? 10 + 14 = 29 ? 29 = 29 Add 11 to both sides. Subtract 15 from both sides. Identity Property of Zero: 0 + n = n. Try This: Example 2B Solve. B. p – 6 = 7 p–6=7 +6 +6 p + 0 = 13 p = 13 Check p–6=7 ? 13 – 6 = 7 Add 6 to both sides. Identity Property of Zero: p + 0 = p. ? 7 = 7 3 Try This: Example 2C Solve. C. 44 = y – 23 44 = y – 23 + 23 + 23 Add 23 to both sides. Learn to solve equations using multiplication and division. 67 = y + 0 Identity Property of Zero: y + 0 = 0. 67 = y Check 44 = y – 23 ? 44 = 67 – 23 ? 44 = 44 Vocabulary Division Property of Equality Multiplication Property of Equality Additional Example 1: Solving Equations Using Division Solve 8x = 32. 8x = 32 8x = 32 8 8 1x = 4 x=4 You can solve a multiplication equation using the Division Property of Equality. DIVISION PROPERTY OF EQUALITY Words Numbers Algebra You can divide both sides of an equation by the same nonzero number, and the equation will still be true. 4 • 3 = 12 4 • 3 = 12 2 2 12 = 6 2 x=y x=y z z Try This: Example 1 Solve 9x = 36. 9x = 36 Divide both sides by 8. 1•x=x Check 8x = 32 ? 32 8(4) = Substitute 4 for x. ? 32 = 32 9x = 36 9 9 1x = 4 x=4 Divide both sides by 9. 1•x=x Check 9x = 36 ? 36 9(4) = Substitute 4 for x. ? 36 = 36 4 Additional Example 2: Solving Equations Using Multiplication You can solve a division equation using the Multiplication Property of Equality. MULTIPLICATION PROPERTY OF EQUALITY Words Numbers You can multiply both sides of an equation by the same number, and the statement will still be true. 2•3=6 4 • 2 • 3 =4 • 6 Algebra Solve n = 7. 7 n 7• = 7•7 7 Multiply both sides by 7. n = 49 x=y zx =zy Check n =7 7 ? 49 = 7 7 ? 7 = 7 8 • 3 = 24 Substitute 49 for n. Try This: Example 2 Solve n = 16 4 n 4 • 4 = 4 • 16 Learn to solve equations with integers. Multiply both sides by 4. n = 64 Check n = 16 4 64 =? 16 4 ? 16 16 = Substitute 64 for n. When you are solving equations with integers, your goal is the same as with whole numbers: isolate the variable on one side of the equation. + – + – + – Additional Example 1A & 1B: Adding and Subtracting to Solve Equations 0 3 + (–3) = 0 Recall that the sum of a number and its opposite is 0. When you add the opposite to a + (–a) = 0 get 0, you can isolate the variable. Solve. A. x–3=–6 x–3=–6 x–3+3=–6+3 x=–3 B. –5 + r = 9 –5 + r = 9 –5 + 5 + r = 9 + 5 r = 14 Add 3 to both sides. Commutative Property x–3+3=–3 0 Add 5 to both sides. 5 Additional Example 1C & 1D: Adding and Subtracting to Solve Equations Continued Solve. C. –6 + 8 = n –6 + 8 = n 2=n D. z+6= z+6= –6 z= –3 –3 –6 –9 Solve. A. The variable is already isolated. B. –2 + g = 5 Add –6 to each side. Try This: Example 1C & 1D –1 + 7 = r –1 + 7 = r 6=r D. a + 9 = –9 a + 9 = –9 –9 –9 a = –18 p–7=–9 p–7=–9 p–7+7=–9+7 p = –2 Add integers. Solve. C. Try This: Example 1A & 1B The variable is already isolated. –2 + g = 5 –2 + 2 + g = 5 + 2 g=7 Add 2 to both sides. Solve. A. –5x = 35 –5x = 35 –5 –5 Divide both sides by –5. x = –7 Add –9 to each side. Additional Example 2B: Multiplying and Dividing to Solve Equations Continued B. 0 Additional Example 2A: Multiplying and Dividing to Solve Equations Continued Add integers. Solve. Add 7 to both sides. Commutative Property p–7+7=–2 Try This: Example 2A Solve. z =5 –4 z –4 = –4 5 Multiply both sides by –4. –4 z = –20 A. –7x = 42 –7x = 42 –7 –7 Divide both sides by –7. x = –6 6 Try This: Example 2 Solve. z =9 –3 B. –3 z = –3 9 Multiply both sides by –3. –3 z = –27 Additional Example 1A: Solving Equations with Decimals. One-Step Equations with Rational Numbers (fractions and decimals) Additional Examples 1B: Solving Equations with Decimals Solve. B. 8.2p = –32.8 Solve. A. m + 4.6 = 9 m + 4.6 = 9 Subtract 4.6 from – 4.6 = – 4.6 both sides. m = 4.4 8.2p = –32.8 8.2 8.2 Divide both sides by 8.2 p = –4 Remember! Once you have solved and equation it is a good idea to check your answer. To check your answer, substitute your answer for the variable in the original equation. Additional Examples 1C: Solving Equations with Decimals Solve. x C. 1.2 = 15 1.2 • x = 1.2 • 15 1.2 x = 18 Multiply both sides by 1.2 Try This: Example 1A & 1B Solve. A. m + 9.1 = 3 m + 9.1 = 3 –9.1 = –9.1 Subtract 9.1 from both sides. m = –6.1 B. 5.5b = 75.9 75.9 5.5 b = 5.5 5.5 Divide both sides by 5.5 b = 13.8 7 Try This: Examples 1C Additional Examples 2A: Solving Equations with Fractions Solve. Solve. y C. 4.5 = 90 y = 4.5 • 90 4.5 • 4.5 A. n + 2 = – 3 7 7 Multiply both sides by 4.5 n– 2 2 3 2 + =– – 7 7 7 7 y = 405 n=– Additional Examples 2B: Solving Equations with Fractions Solve. Solve. C. 5 x = 5 Add 1 to both sides. 6 Find a common denominator; 6. Simplify. 5 6 5 63 • x= 8 • 5 6 5 4 3 4 Try This: Example 2A 9 9 1 1 5 1 n– + =– – 9 9 9 9 2 n=– 3 1 Subtract 9 from both sides. Simplify – 6 . 9 5 7 8 6 x = Solve. A. n + 1 = – 5 2 from both sides. 7 Additional Examples 2C: Solving Equations with Fractions B. y – 1 = 2 6 3 1 1 2 1 +y– = + 6 6 3 6 4 1 y= + 6 6 y= 5 6 Subtract Multiply both sides by 6 . 5 Simplify. Try This: Example 2B Solve. 1 3 B. y – 2 = 4 1 y 1 3 1 + – = + 2 2 4 2 3 2 + 4 4 1 y=1 4 y= Add 1 to both sides. 2 Find a common denominator; 4. Simplify. 8 Try This: Examples 2C Solve. 6 3 C. 8 x = 19 6 3x = 19 8 2 6 8 3 8 • x = • 19 3 1 8 3 x = 16 19 Multiply both sides by 8 . 3 Simplify. 9