* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Ch2-Sec 2.3
Survey
Document related concepts
Debye–Hückel equation wikipedia , lookup
Unification (computer science) wikipedia , lookup
Two-body problem in general relativity wikipedia , lookup
Schrödinger equation wikipedia , lookup
BKL singularity wikipedia , lookup
Maxwell's equations wikipedia , lookup
Itô diffusion wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Calculus of variations wikipedia , lookup
Equation of state wikipedia , lookup
Equations of motion wikipedia , lookup
Schwarzschild geodesics wikipedia , lookup
Differential equation wikipedia , lookup
Transcript
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 1 Chapter 2 Equations, Inequalities, and Applications Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 2 2.3 More on Solving Linear Equations Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 3 2.3 More on Solving Linear Equations Objectives 1. 2. 3. 4. Learn and use the four steps for solving a linear equation. Solve equations with fractions or decimals as coefficients. Solve equations that have no solution or infinitely many solutions. Write expressions for two related unknown quantities. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 4 2.3 More on Solving Linear Equations 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. 2.3 – Slide 5 2.3 More on Solving Linear Equations 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. 2.3 – Slide 6 2.3 More on Solving Linear Equations 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. 2.3 – Slide 7 2.3 More on Solving Linear Equations 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. 2.3 – Slide 8 2.3 More on Solving Linear Equations 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. 2.3 – Slide 9 2.3 More on Solving Linear Equations 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. 2.3 – Slide 10 2.3 More on Solving Linear Equations Using the Four Steps for Solving a Linear Equation Example 3 (continued) Solve the equation. Step 4 Check by substituting 4 for y in the original equation. 5 15y – ( 10y – 2 ) = 2 ( 5y + 7 ) – 16 15 45 – ( 10 45 – 2 ) = 2 ( 5 45 + 7 ) – 16 ? Let y = 12 – ( 8 – 2 ) = 2 ( 4 + 7 ) – 16 12 – 6 = 2 ( 11 ) – 16 12 – 6 = 22 – 16 6 = 6 The solution to the equation is 4 . 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. 4 5 . ? Multiply. ? Subtract, add. ? Multiply. True 2.3 – Slide 11 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients 1) We clear an equation of fractions by multiplying each side by the least common denominator (LCD) of all the fractions in the equation. It is a good idea to do this in Step 1 to avoid messy computations. 2) When clearing an equation containing decimals, choose the smallest exponent on 10 needed to eliminate the decimals. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 12 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients Example 4 Solve the equation. 5 m – 10 = 8 8 8 5 m 8 3 m + 1 m 4 2 5 m – 10 8 = 8 3 m + 1 m 4 2 – 8 10 = 8 3 m 4 + 5m – 80 = 6m + 4m 8 Multiply by LCD, 8. 1 m 2 Distribute. Multiply. Now use the four steps to solve this equivalent equation. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 13 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients Example 4 (continued) Solve the equation. 5m – 80 = 6m + 4m Step 1 5m – 80 = 10m Step 2 5m – 80 – 5m = 10m – 5m – 80 = 5m Step 3 – 80 = 5m 5 5 – 16 = m Copyright © 2010 Pearson Education, Inc. All rights reserved. Combine terms. Subtract 5m. Combine terms. Divide by 5. 2.3 – Slide 14 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients Example 4 (continued) Solve the equation. Step 4 Check by substituting –16 for m in the original equation. 5 m – 10 = 8 5 (–16) – 10 = 8 3 m + 1 m 4 2 3 (–16) 1 (–16) + 4 2 –10 – 10 = –12 – 8 –20 = –20 ? Let m = –16. ? Multiply. True The solution to the equation is –16. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 15 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients Note Multiplying by 10 is the same as moving the decimal point one place to the right. Example: 1.5 ( 10 ) = 15. Multiplying by 100 is the same as moving the decimal point two places to the right. Example: 5.24 ( 100 ) = 524. Multiplying by 10,000 is the same as moving the decimal point Answer: 4 places. how many places to the right? Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 16 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients Example 5 Solve the equation. 0.2v – 0.03 ( 11 + v ) = – 0.06 ( 31 ) 20v – 3 ( 11 + v ) = – 6 ( 31 ) Step 1 20v – 33 – 3v = – 186 17v – 33 = – 186 Copyright © 2010 Pearson Education, Inc. All rights reserved. Multiply by 100. Distribute. Combine terms. 2.3 – Slide 17 2.3 More on Solving Linear Equations Solving Equations with Fraction or Decimal Coefficients Example 5 (continued) Solve the equation. 17v – 33 = – 186 Step 2 Step 3 From Step 1 17v – 33 + 33 = – 186 + 33 Add 33. 17v = – 153 Combine terms. 17v – 153 = 17 17 Divide by 17. v = –9 Copyright © 2010 Pearson Education, Inc. All rights reserved. Check to confirm that – 9 is the solution. 2.3 – Slide 18 2.3 More on Solving Linear Equations Solving Equations with No or Infinitely Many Solutions Example 6 Solve the equation. 4 ( 2n + 6 ) = 2 ( 3n + 12 ) + 2n 8n + 24 = 6n + 24 + 2n Distribute. 8n + 24 = 8n + 24 Combine terms. 8n + 24 – 24 = 8n + 24 – 24 8n = 8n 8n – 8n = 8n – 8n 0 = 0 Subtract 24. Combine terms. Subtract 8n. True Solution Set: {all real numbers}. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 19 2.3 More on Solving Linear Equations Solving Equations with No or Infinitely Many Solutions An equation with both sides exactly the same, like 0 = 0, is called an identity. An identity is true for all replacements of the variables. We write the solution set as {all real numbers}. CAUTION In example 6, do not write {0} as the solution set of the equation. While 0 is a solution, there are infinitely many other solutions. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 20 2.3 More on Solving Linear Equations Solving Equations with No or Infinitely Many Solutions Example 7 Solve the equation. 6x – 1 ( 4 – 3x ) 6x – 4 + 3x 9x – 4 9x – 4 – 9x –4 = = = = = 8 + 8 + –19 –19 –19 3 ( 3x – 9 ) 9x – 27 + 9x + 9x – 9x Distribute. Combine terms. Subtract 9x. False There is no solution. Solution set: 0 Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 21 2.3 More on Solving Linear Equations Solving Equations with No or Infinitely Many Solutions Again, the variable has disappeared, but this time a false statement (– 4 = – 19) results. This is the signal that the equation, called a contradiction, has no solution. Its solution set is the empty set, or null set, symbolized 0 . CAUTION Do not write {0 } to represent the empty set. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 22 2.3 More on Solving Linear Equations Writing Expressions for Two Related Unknown Quantities Example 8 Two numbers have a sum of 32. If one of the numbers is represented by c, write an expression for the other number. Given: c represents one number. The sum of the two numbers is 32. Solution: 32 – c represents the other number. Check: One number + the other number = 32 c + (32 – c) = 32 Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 – Slide 23 2.3 More on Solving Linear Equations Writing Expressions for Two Related Unknown Quantities Example 9 Andrew arrived at work with x dollars. How much money will he have if he finds $15 in his desk? How much money will he have if he spends $10 on his lunch? Given: x represents the amount of money Andrew has currently. He found $15 in his drawer. He then spent $10 on lunch. Solution for Part A: x + 15 Solution for Part B: x + 15 – 10 or Copyright © 2010 Pearson Education, Inc. All rights reserved. x + 5 2.3 – Slide 24