Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 8.3 Solving Linear Equations: ax + b = cx + d 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 = cx + d. o Understand the terms conditional equations, identities, and contradictions. 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 = cx + d General Procedure for Solving Linear Equations that Simplify to the Form ax + b = cx + d 1. Simplify by removing any grouping symbols and combining like terms on each side of the equation. 2. Use the addition principle of equality and add the opposite of a constant term and/or variable term to both sides so that variables are on one side and constants are on the other side. 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 = cx + d General Procedure for Solving Linear Equations that Simplify to the Form ax + b = cx + d (cont.) 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. 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 Equations of the Form ax + b = cx + d Solve the following equation. 5x + 3 = 2x 18 Solution 5x + 3 = 2x 18 Write the equation. 5x + 3 3 = 2x 18 3 Add 3 to both sides. Simplify. 5x = 2x 21 5x 2x = 2x 21 2x Add 2 x to both sides. Simplify. 3x = 21 3x 21 Divide both sides by 3. = 3 3 Simplify. x = 7 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Equations of the Form ax + b = cx + d (cont.) Check 5x + 3 = 2x 18 ? 5 7 + 3 = 2 7 18 ? 35 + 3 = 14 18 32 = 32 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 Equations of the Form ax + b = cx + d Solve the following equation. 4 x + 1 x = 2x 13 + 5 Solution 4 x + 1 x = 2x 13 + 5 3x + 1 = 2x 8 3x + 1 1 = 2x 8 1 3x = 2 x 9 3x 2x = 2x 9 2x x = 9 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Combine like terms. Add 1 to both sides. Simplify. Add 2 x to both sides. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Equations of the Form ax + b = cx + d (cont.) Check 4 x + 1 x = 2x 13 + 5 ? 4 9 + 1 9 = 2 9 13 + 5 ? 36 + 1 + 9 = 18 13 + 5 26 = 26 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Substitute x = 9. Simplify. true statement Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations Involving Decimals Solve the following equation. 6y + 2.5 = 7y 3.6 Solution 6y + 2.5 = 7y 3.6 6y + 2.5 + 3.6 = 7y 3.6 + 3.6 6y + 6.1 = 7y 6y + 6.1 6y = 7y 6y 6.1 = y HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Add 3.6 to both sides. Simplify. Add 6 y to both sides. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations Involving Decimals (cont.) Check 6y + 2.5 = 7y 3.6 ? 6 6.1 + 2.5 = 7 6.1 3.6 ? 36.6 + 2.5 = 42.7 3.6 39.1 = 39.1 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Substitute y = 6.1. Simplify. true statement Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Linear Equations with Fractional Coefficients Solve the following equation. 1 1 2 7 x+ = x 3 6 5 10 1 1 2 7 x+ = x Solution 3 6 5 10 1 7 1 2 30 x + = 30 x 3 5 6 10 1 1 2 7 30 x + 30 = 30 x 30 3 6 5 10 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Multiply both sides by the LCM, 30. Apply the distributive property. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Linear Equations with Fractional Coefficients (cont.) 10x + 5 = 12x 21 10x + 5 5 = 12x 21 5 10x = 12x 26 10 x 12x = 12x 26 12x 2x = 26 2 x 26 = 2 2 x = 13 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify. Add 5 to both sides. Simplify. Add 12x to both sides. Simplify. Divide both sides by 2. Simplify. Checking will show that 13 is the solution. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solving Equations with Parentheses Solve the following equation. 2 y 7 = 4 y + 1 26 Solution 2 y 7 = 4 y + 1 26 2y 14 = 4 y + 4 26 2y 14 = 4 y 22 2y 14 + 22 = 4 y 22 + 22 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Use the distributive property. Combine like terms. Add 22 to both sides. Here we will put the variable on the right side to get a positive coefficient of y . Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solving Equations with Parentheses (cont.) 2y + 8 = 4 y 2y + 8 2y = 4 y 2y 8 = 2y 8 2y = 2 2 4=y HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify. Add 2y to both sides. Simplify. Divide both sides by 2. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Solving Equations with Parentheses Solve the following equation. 2 5x + 13 2 = 6 3x 2 41 Solution 2 5x + 13 2 = 6 3x 2 41 10x 26 2 = 18x + 12 41 10 x 28 = 18 x 29 10x 28 + 18x = 18x 29 + 18x HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Write the equation. Use the distributive property. Be careful with the signs. Combine like terms. Add 18 x to both sides. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Solving Equations with Parentheses (cont.) 8 x 28 = 29 8x 28 + 28 = 29 + 28 8 x = 1 8 x 1 = 8 8 1 x= 8 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify. Add 28 to both sides. Simplify. Divide both sides by 8. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 7: Solving Equations with Parentheses Solve the following equation. Solution 4 x + 3 = 2 3x 1 + 6 4 x + 3 = 2 3x 1 + 6 Write the equation. 4 x + 12 = 6 x 2 + 6 Use the distributive property. 4 x + 12 = 6 x + 4 Combine like terms. 4 x + 12 12 = 6 x + 4 12 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 12 from both sides. Subtract ___ Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 7: Solving Equations with Parentheses (cont.) 4x = 6x 8 4 x 6x = 6 x 8 6x 2 x = 8 2 x = 8 2 2 x= 4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify. 6x from both sides. Subtract ___ Simplify. 2 Divide both sides by ___. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Solutions of Equations Determine whether the equation 3 x + 5 + 1 = 11 is a conditional equation, an identity, or a contradiction. Write the equation. Solution 3 x + 5 + 1 = 11 3x + 15 + 1 = 11 3x + 16 = 11 3x + 16 16 = 11 16 3x = 27 3x 27 = 3 3 x = 9 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Use the distributive property. Combine like terms. Add 16 to both sides. Simplify. Divide both sides by 3. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Solutions of Equations (cont.) The equation has one solution. Therefore it is a conditional equation. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Solutions of Equations Determine whether the equation 3 x 25 + 3x = 6 x + 10 is a conditional equation, an identity, or a contradiction. Solution 3 x 25 + 3x = 6 x + 10 Write the equation. Use the distributive property. 3x 75 + 3x = 6 x + 60 Combine like terms. 6x 75 = 6x + 60 6x 75 6x = 6x + 60 6x Add 6x to both sides. Simplify. 75 = 60 The last equation is never true. Therefore, the original equation is a contradiction and has no solution. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Solutions of Equations Determine whether the following equation is a conditional equation, an identity, or a contradiction. 2 x 7 + x = 14 x Solution 2 x 7 + x = 14 x Write the equation. 2 x + 14 + x = 14 x 14 x = 14 x 14 x 14 = 14 x 14 x = x x + x = x + x 0=0 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Use the distributive property. Combine like terms. Add 14 to both sides. Add x to both sides. Simplify. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Solutions of Equations (cont.) The last equation is always true. Therefore, the original equation is an identity and has an infinite number of solutions. Every real number is a solution. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Solve the following linear equations. 1. x + 14 6x = 2x 7 2. 6.4 x + 2.1 = 3.1x 1.2 2x 1 1 3. = x+ 3 2 6 5. 5 y 3 = 14 4 y + 2 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 3 1 1 1 4. n+ = n 14 4 7 4 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems (cont.) Determine whether each of the following equations is a conditional equation, an identity, or a contradiction. 6. 7 x 3 + 42 = 7 x + 21 7. 2 x + 14 = 2 x + 1 + 10 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Answers 1. x = 3 3. x = 2 2 5. y = 3 7. contradiction HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 2. x = 1 4. n = 7 6. identity Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.