LESSON 8.3 EQUATIONS WITH FRACTIONS

Transcription

1 LESSON 8. EQUATIONS WITH FRACTIONS LESSON 8. EQUATIONS WITH FRACTIONS

2 OVERVIEW Here is what you'll learn in this lesson: Solving Equations a. Solving equations with rational epressions b. Solving for an unknown in a formula involving a rational epression Suppose you want to figure out a baseball pitcher's earned run average, or estimate the population of fish in a lake, or figure out how tall a building is based on a scale model. For each of these eamples, you can figure out the answer by setting up an equation that involves fractions or ratios. However, even with all the techniques you have for solving equations, solving an equation with fractions can be tricky. There may not be any solution at all, or the solution you find might not check when you plug it back into the original equation. In this lesson you will learn how to solve equations that have fractions in them, and you will learn how to identify etraneous, or false solutions. 6 TOPIC 8 RATIONAL EXPRESSIONS

3 EXPLAIN SOLVING EQUATIONS Summary Solving Equations with Rational Epressions When you solve an equation that contains a rational epression, it helps to clear the fraction in the equation. To do this, multiply both sides of the equation by the least common denominator (LCD) of the fractions. To solve an equation that contains rational epressions:. Clear the fractions by multiplying both sides of the equation by the LCD of the fractions.. Distribute the LCD and simplify.. Finish solving for the variable. For eample, to solve = + for :. Multiply by the LCD of the fractions,. ( ) = +. Distribute the LCD and simplify. = + 0 = = 6 Here s how to find the LCD of,, and : = = =. Finish solving for. = = = = = Checking for Etraneous Solutions If an equation contains a fraction with a variable in the denominator, the solution of the equation might be etraneous (false). To check for an etraneous solution:. Solve the equation for the variable.. Substitute the solution into the original equation and simplify.. Look at the denominators of the fractions. If any denominator is zero, the solution is etraneous. LESSON 8. EQUATIONS WITH FRACTIONS EXPLAIN

4 + For eample, to determine if = has an etraneous solution:. Solve the equation for. = + ( + ) ( + ) The LCD of,, and is + ( + ) ( + ). ( + ) = ( + ) + ( + ) + ( + ) = ( + ) ( + ) = + ( + ) = 9 =. Substitute for = +. in the original equation.. Check the denominators. = 0. When you substitute =,. two of the fractions have a. denominator of zero. So, the solution is etraneous. This equation has no solution. ( + ) 0 Proportions have eactly one term on each side of the equation. Here are some eamples of proportions: 0 y = = = 0 0 Using Cross Multiplication to Solve Proportions An equation that sets one fraction equal to another fraction is called a proportion. An easy way to solve a proportion is to cross multiply. To solve a proportion using cross multiplication:. Multiply the numerator of the first fraction by the denominator of the second fraction.. Multiply the denominator of the first fraction by the numerator of the second fraction.. Set the two products equal to each other.. Finish solving for the variable. 6 For eample, to solve the proportion = :. Multiply the numerator of by the denominator of Multiply the denominator of by the numerator of.. Set the products equal to each other. 6 = ( ) 6 6 ( ) 6. Finish solving for. = 6 = = 8 TOPIC 8 RATIONAL EXPRESSIONS

5 Sample Problems Answers to Sample Problems 6 6. Solve the equation + = for. Determine if the solution is etraneous. a. Multiply both sides by the + = 6 6 a. LCD of the fractions. b. Distribute the LCD b. and simplify. + = + = 0 c. Finish solving for. = = = d. Substitute the solution for d. in the original equation. = 6 6 a., b.,,, ( ), 6 c = 0 = 0 = d. + = 6 6 e. Is the solution etraneous? e. No + ( + ). Solve the equation = for. Determine if the solution is etraneous. a. Multiply both sides by ( + ) + a. the LCD of the fractions. b. Distribute the ( + ) ( + ) + b. LCD and simplify. = ( + ) = ( + ) = ( + ) ( + ) b. ( + ),, c. Finish solving for. = = c. +, = d. Substitute the solution for c. in the original equation. = e. Is the solution etraneous? d. = e. Yes + ( +) 0. Use cross multiplication to solve the proportion = for. a. Multiply the numerator of 0 a. denominator of. b. Multiply the denominator of by the by the a. numerator of. c. Set the products equal to each other. = 0 d. Finish solving for. = b., (in either order) c. 0 d. or LESSON 8. EQUATIONS WITH FRACTIONS EXPLAIN 9

6 HOMEWORK Homework Problems Circle the homework problems assigned to you by the computer, then complete them below. Eplain Solving Equations with Rational Epressions In problems through solve for the variable. Be sure to check whether your answer is an etraneous solution. y y + 9. Solve for : =. Solve for y : y =. Solve for : + =. Solve for y : =. Solve for : = y Solve for : =. Solve for y : = y 6 6 8y 6 9. A person who weighs 00 pounds on Earth would weigh 8 pounds on Mars. Use the proportion below to figure out how much someone who weighs 60 pounds on Earth would weigh on Mars. weight on Mars weight on Earth 8 = = A person who weighs 00 pounds on Earth would weigh pounds on Jupiter. Use the proportion below to figure out how much someone who weighs 60 pounds on Earth would weigh on Jupiter. weight on Jupiter weight on Earth 6. Solve for : = 6. Solve for : = = = Solve for : = 6 80 TOPIC 8 RATIONAL EXPRESSIONS

7 APPLY Practice Problems Here are some additional practice problems for you to try. Be sure to check whether your answer is an etraneous solution. Solving Equations with Rational Epressions. Solve for : + = 8. Solve for : =. Solve for : + = 0 +. Solve for : =. Solve for : + = 6. Solve for : + =. Solve for : = 8. Solve for : = 9. Solve for : + = 6 0. Solve for : = +. Solve for : + =. Solve for : + = 6 +. Solve for : = 9 0. Solve for : + = 6. Solve for : =. Solve for : = 8. Solve for : + = 9. Solve for : = ( ) 6 ( ) 8 ( ) 0 0. Solve for : + =. Solve for : =. Solve for : =. Solve for : =. Solve for : =. Solve for : + = 6. Solve for : + =. Solve for : = 8. Solve for : + = 6 +. Solve for : + = LESSON 8. EQUATIONS WITH FRACTIONS APPLY 8

8 Practice Test EVALUATE Take this practice test to be sure that you are prepared for the final quiz in Evaluate.. Solve = for. Is the solution etraneous? y y + 9 y +. Solve = for y. Is the solution etraneous?. The volume V of a right circular cone is V = r h, where r is the radius, and h is the height. Solve this formula for h Solve = for. Is the solution etraneous?. The surface area, S, of a right circular cylinder is S = rh + r, where r is the radius, and h is the height. Solve this formula for h. h r h r 8 +. Solve this proportion for : = 6y 8. Solve this proportion for y : = 6y y y 6. Solve = for y. Is the solution etraneous? 8 TOPIC 8 RATIONAL EXPRESSIONS