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Section 8.2 Solving Linear Equations: ax + b = c HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Solve equations of the form ax + b = c. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations of the Form ax + b = c Procedure for Solving Linear Equations that Simplify to the Form ax + b = c 1. Combine like terms on both sides of the equation. 2. Use the addition principle of equality and add the opposite of the constant b to both sides. 3. Use the multiplication (or division) principle of equality to multiply both sides by the reciprocal of the coefficient of the variable (or divide both sides by the coefficient itself). The coefficient of the variable will become +1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations of the Form ax + b = c Procedure for Solving Linear Equations that Simplify to the Form ax + b = c (cont.) 4. Check your answer by substituting it for the variable in the original equation. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Linear Equations of the Form ax + b = c Solve the equation 3x + 3 = -18. Solution 3x + 3= -18 3x + 3 - 3 = -18 - 3 3x = -21 3x -21 = 3 3 x = -7 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Add -3 to both sides. Simplify. Divide both sides by 3. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Linear Equations of the Form ax + b = c (cont.) Check 3x + 3= -18 ? 3 -7 + 3 = - 18 ? -21 + 3 = - 18 -18 = -18 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Substitute x = -7. Simplify. true statement Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Linear Equations of the Form ax + b = c Solve the equation –26 = 2y – 14 – 4y. Solution -26 = 2y - 14 - 4 y Write the equation. -26 = -2y - 14 Combine like terms. -26 + 14 = -2y - 14 + 14 Add 14 to both sides. -12 = -2y -12 -2y = -2 -2 6=y HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify. Divide both sides by -2. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Linear Equations of the Form ax + b = c (cont.) Check -26 2y - 14 - 4 y ? -26 2 6 - 14 - 4 6 ? -26 12 - 14 - 24 Simplify. -26 -26 true statement HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Substitute y = 6. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations involving Decimals Solve the equation 16.53 – 18.2z – 7.43 = 0. Solution Write the equation. 16.53 - 18.2z - 7.43 0 100 16.53 - 18.2z - 7.43 100 0 1653 - 1820z - 743 0 910 - 1820z 0 910 - 1820z - 910 0 - 910 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Multiply both sides by 100. (This results in integer coefficients.) Simplify. Combine like terms. Add -910 to both sides. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations involving Decimals (cont.) -1820 z -910 -1820 -1820 1 z 0.5 or z 2 16.53 - 18.2z - 7.43= 0 ? 16.53 - 18.2 0.5 - 7.43 = 0 Divide both sides by -1820. Simplify. Check ? 16.53 - 9.10 - 7.43 = 0 0=0 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Substitute z = 0.5. Simplify. true statement Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Linear Equations involving Decimals Solve the equation 5.1x + 7.4 – 1.8x = –9.1. Solution Because the decimal numbers are accurate to tenths, we could multiply both sides by 10 to get integer coefficients. However, we can also work with the decimal numbers directly as shown here. 5.1x + 7.4 - 1.8 x = -9.1 3.3x + 7.4 = -9.1 3.3x + 7.4 - 7.4 = -9.1 - 7.4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Combine like terms. Add -7.4 to both sides. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Linear Equations involving Decimals (cont.) 3.3x = -16.5 3.3x -16.5 = 3.3 3.3 x = -5 Simplify. Divide both sides by the coefficient 3.3. Simplify. 5.1x + 7.4 - 1.8 x -9.1 ? 5.1 -5 + 7.4 - 1.8 -5 - 9.1 ? -25.5 + 7.4 + 9 - 9.1 Check -9.1 -9.1 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Substitute x = -5. Simplify. true statement Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solving Linear Equations with Fractional Coefficients 5 5 10 Solve the equation x - = - . 6 2 9 Solution 5 5 10 x- =6 2 9 5 5 10 18 x - = 18 6 9 2 5 5 10 18 x - 18 = 18 6 2 9 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Multiply both sides by 18 (the LCM of the denominator). Apply the distributive property. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solving Linear Equations with Fractional Coefficients (cont.) 3 9 2 18 5 18 5 18 10 x - = - 1 6 1 2 1 9 Multiply. 15x - 45= -20 Simplify. 15x - 45 + 45 = -20 + 45 15x = 25 15x 25 = 15 15 5 x= 3 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Add 45 to both sides. Simplify. Divide both sides by 15. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solving Linear Equations with Fractional Coefficients (cont.) Check 5 5 10 x- 6 2 9 5 5 5 ? 10 - 6 3 2 9 25 45 ? 20 - 18 18 18 20 20 - 18 18 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 5 Substitute x = . 3 Simplify. true statement Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations of the Form ax + b = c Notes About Checking: Checking can be quite time consuming and need not be done for every problem. This is particularly important on exams. You should check only if you have time after the entire exam is completed. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Solving Linear Equations with Fractional Coefficients 1 3 7 2 Solve the equation x + x + - x = 0. 2 4 2 3 1 3 7 2 x+ x+ - x=0 2 4 2 3 Solution Write the equation. 3 7 2 1 12 x + x + - x = 12 0 2 4 2 3 Multiply both sides by 12 (the LCM of the denominator). 1 3 7 2 12 x + 12 x + 12 - 12 x = 12 0 2 4 2 3 Apply the distributive property. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Solving Linear Equations with Fractional Coefficients (cont.) 6x + 9 x + 42 - 8 x = 0 7x + 42 = 0 7x + 42 - 42 = 0 - 42 Simplify. Combine like terms. Add -42 to both sides. 7 x = -42 Simplify. 7 x -42 = 7 7 Divide both sides by 7. x = -6 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify. Checking will show that -6 is the solution. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 7: Solving Linear Equations with Fractional Coefficients 3 1 5 Solve the equation y + = . 4 2 8 3 1 5 y+ = Solution 4 2 8 1 3 5 _8 y + = 8_ 4 8 2 3 1 5 _8 y + 8 _ = 8_ 4 2 8 6_y + _4 = 5_ HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. 8 Multiply both sides by__. Apply the distributive property. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 7: Solving Linear Equations with Fractional Coefficients (cont.) 6_y + 4 - 4_ = 5 - _4 6y = _1 6y 1 = _6 6_ 1 y = 6_ HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 4 Subtract __from both sides. Simplify. 6 Divide both sides by__. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Solve the following linear equations. 1. x + 14 - 8x = -7 2. 2.4 = 2.6y - 5.9y - 0.9 2n 1 1 3. n - - = 3 2 6 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 3x 1 x 4. + - =0 14 2 7 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1. x = 3 2. y = -1 3. n = 2 4. x = -7 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.
