Chapter 2 Section 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

More on Solving Linear Equations Learn and use the four steps for solving a linear equation. Solve equations with fractions or decimals as coefficients. Solve equations with no solution or infinitely many solutions. Write expressions for two related unknown quantities

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Slide Learn and use the four steps for solving a linear equation.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Learn and use the four steps for solving a linear equation. Slide We solve more complicated equations using the following four-step method. Step 1: Simplify each side separately. Clear parentheses, using the distributive property if needed, and combine all like terms. Step 3: Isolate the variable. Use the multiplication property if necessary to get the equation in the form x = a number. (Other letters may be used for variables.) Step 2: Isolate the variable term on one side. Use the addition property if necessary so that the variable term is on one side of the equation and a number is on the other. Step 4: Check. Substitute the proposed solution into the original equation to see if a true statement results.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Slide Solve. Solution: Using the Four Steps to Solve an Equation Check: The solution set of the equation is {1}.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Slide Solve. Solution: Using the Four Steps to Solve an Equation Check: The solution set of the equation is {−1}.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Slide Solve. Solution: Using the Four Steps to Solve an Equation Check: The solution set of the equation is

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Slide Solve. Using the Four Steps to Solve an Equation Be very careful with signs when solving an equation like the one in the previous example. When clearing parentheses in the expression remember that the − sign acts like a factor of −1 and affects the sign of every term within the parentheses.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Slide Solve. Solution: Using the Four Steps to Solve an Equation Check: The solution set of the equation is {−1}.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Slide Solve equations with fractions or decimals as coefficients.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 to avoid messy computations. Solve equations with fractions or decimals as coefficients. Slide When clearing an equation of fractions, be sure to multiply every term on each side of the equation by the LCD. Make sure to multiply each term of the equation by the LCD. It is best to identify how many terms are in the equation before multiplying.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Slide Solve. Solution: Solving an Equation with Fractions as Coefficients Check: The solution set of the equation is {−7}.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Slide Solve. Solution: Solving an Equation with Decimals as Coefficients

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Slide Solving an Equation with Decimals as Coefficients (Check) Check: The solution set of the equation is {−2}.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide Solve equations with no solution or infinitely many solutions.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Each equation that we have solved so far has had exactly one solution. An equation with exactly one solution is a conditional equation because it is only true under certain conditions. Sometimes equations may have no solution or infinitely many solutions. Solve equations with no solution or infinitely many solutions. Slide DO NOT write { 0 } as the solution set. There are infinitely many other solutions.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Slide Solve. Solution: Solving an Equation That Has Infinitely Many Solutions The variable has “disappeared.” Since the last statement is true, any real number is a solution. An equation with both sides exactly the same, is called an identity. An identity is true for all replacements of the variables. As shown above, write the solution set as {all real numbers}.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Slide Solve. Solution: Solving an Equation That Has No Solution. Again, the variable has “disappeared,” but this time a false statement results. When this happens in solving an equation, it indicates that the equation has no solution and is called a contradiction. Its solution set is the empty set, or null set, symbolized Ø. DO NOT write { Ø } to represent the empty set.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The following table summarizes the solution sets of the three types of equations presented so far. Solve equations with no solution or infinitely many solutions. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Slide Write expressions for two related unknown quantities.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Slide Solution: Translating a Phrase into an Algebraic Expression One number is 5 more than twice another. If the first number is represented by x, write an expression for the second number.