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To solve an equation with fractions

To solve an equation with fractions. Procedure: To solve a fractional equation. Step 1. Factor all denominators. Step 2. Determine any restrictions for the fractional equation. Step 3. Determine LCM of denominators. Step 4. Multiply all terms by LCD (clearing denominators).

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To solve an equation with fractions

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  1. To solve an equation with fractions Procedure: To solve a fractional equation Step 1. Factor all denominators. Step 2. Determine any restrictions for the fractional equation. Step 3. Determine LCM of denominators. Step 4. Multiply all terms by LCD (clearing denominators). Step 5. Solve equation as in previous lessons. Step 6. Compare answer to restriction. If answer is the restricted value, write no solution. If not, go on to the next Step. Step 7. Check

  2. 2 2 Solution: 3 3 Your Turn Problem #1  1. Find the LCD LCD=18 2. Multiply all fractions by the LCD. 3. Cancel the denominators. 4. Simplify and solve. 5. Check.

  3. Rational Expressions have restrictions. Restrictions are values for the variable which would make the fraction undefined. In this section, we will have fractions where the denominator may contain the variable. If the denominator contains a variable, the equation is said to have restrictions. Procedure: To solve a fractional equation (for current section) Step 6. Check all answers which are not restrictions. Step 1. Determine any restrictions for the fractional equation. Step 2. Determine LCM of denominators (LCD). Step 3. Multiply all terms by LCD (clearing denominators). Step 4. Solve equation as in previous lessons. Step 5. Compare answer to restriction. If answer is the restricted value, write no solution. If not, go on to the next Step.

  4. 3 4 6 Solution: Your Turn Problem #2  1. State restrictions 2. Find LCD. • Multiply all terms by LCD. 4. Simplify and solve. 5. Answer not equal to restriction. LCD=12x 6. Check.

  5. Solution: 5. Answer not equal to restriction. 6. Check: (Not Shown) Your Turn Problem #3 1. State restrictions 2. Find LCD. • Multiply all terms by LCD. 4. Simplify and solve. LCD=3x

  6. Solution: Your Turn Problem #4 This example was solved by multiplying both sides of the equation by the LCD. Cross multiplication is another method for solving fractional equations where there is only one fraction on each side. Using the Cross-multiplication property: Next Slide 1. State restrictions 2. Find LCD. • Multiply all terms by LCD. 4. Simplify and solve. 5. Answer not equal to restriction. 6. Check (not shown).

  7. Solution: 5. Answer not equal to restriction. 6. Check.  Your Turn Problem #5 1. State restrictions 2. Find LCD. • Multiply all terms by LCD. 4. Simplify and solve.

  8. Proportion Application Problems To recognize a problem is a proportion problem, a sentence will be given which can be written as a fraction. Examples: 3 out of 20 smoke; 4 panels generate 1500 watts; Procedure: To solve a proportion problem Examples: , 4. Solve and answer the question. 1. Define the variable (what you are looking for?) 2. Find the recognizable ratio and write it as a fraction. 3. After the fraction is written write an equal sign. Then write a fraction using the remaining information with the defined variable. Make sure the fraction is written so both of the numerators have the same units and the denominators have the same units.

  9. Make units match: Equation: Your Turn Problem #6 A carpenter estimates that he uses 8 studs for every 10 linear feet of framed walls. At this same rate, how many studs are needed for a garage with 35 linear feet of framed walls? Example 6. A chef estimates that 50 lbs of vegetables will serve 130 people. Using this estimate, how many pounds will be necessary to serve 156 people? Solution: Let x = number of pounds of vegetables for 156 people. Solve by cross multiplying: Answer: 28 studs are needed

  10. Example 7: The federal income tax on $50,000 of income is $12,500. At this rate, what is the federal income tax on $125,000 of income? Solution: Make units match: 1 4 Your Turn Problem #7 If a home valued at $120,000 is assessed $2160 in real estate taxes, then how much, at the same rate, are the taxes on a home valued at $200,000? Let x = federal income tax on $125,000 Reduce, then solve by cross multiplying: Answer: The property would be $3600.

  11. Solution: Let x = amount first person receives Let 1400 – x = amount second person receives Your Turn Problem #8 A sum of $5000 is to be divided between two people in the ratio of 3 to 7. How much does each person receive? Example 8. A sum of $1400 is to be divided between two people in the ratio of 3 to 5. How much does each person receive? Solve by cross multiplying: 525 = amount for first person. To find amount for second person, 1400 – 525 = 875. Answer: The first person receives $1500 and the second person receives $3500.

  12. ( ) 7. Check x = 1 (not shown). Your Turn Problem #9 1. Factor Denominators 2. State restrictions 3. Find LCD. 4. Multiply all terms by LCM. 5. Simplify and solve. 6. Since x can not equal 3, cross it out as a solution.

  13. Solution: 7. Check both –11 and 5 (not shown). Your Turn Problem #10 1. Factor Denominators 2. State restrictions 3. Find LCD. 4. Multiply all terms by LCD. 5. Simplify and solve. • Answers not equal to restrictions.

  14. Work Problems Work problems are common in algebra because of the use of the rate concept as well as their application to real-life situations. Rate of work: The portion of a task completed per one unit of time. Important: In Work Problems, the time it takes a person (or machine) to do a task is the reciprocal of his/her rate! Example: If a carpenter can build a shed in 8 hours, then his rate of work is of the task completed per hour. Also, if this carpenter works for 5 hours, then his part of task completed is Basic Work Formula: (Rate of work)  (Time Worked) = (Part of Task completed) In most work problems, there will always be two individuals or devices who will contribute to the completion of the task. The formula for work problems will be to add the “part of task completed” for each individual or device and set it equal to “1” (for 1 completed job). Next Slide

  15. Procedure: To Solve Work Problems Step 7. Solve and answer question. Next Slide Step 1. Write down the “let statements” (what you are looking for) and any information given in the problem. Step 2. Create a chart with columns “Rate of work”, “Time worked”, and “Part of task completed.” Step 3. To find rate of work, make a fraction with a numerator equal to 1; and a denominator equal to the time it would have taken the individual or device to complete the task alone. Step 4. To find time worked, write in the actual time the individual or device worked or will work on the task. Step 5. To find Part of task completed, use the formula: (rate of work)  (time worked) = (part of task completed) Step 6. To write equation, add the “part of task completed” and set equal to 1.

  16. Rate of work Time worked Part of Task completed carpenter assistant x x Your Turn Problem #11 Maria can do a certain job in 50 minutes, whereas it takes Kristin 75 minutes to do the same job. How long would it take then to do the job together? Example 11. A carpenter can build a shed working alone in 5 hours. His assistant working alone would take 10 hours to build a similar shed. If the two work together, how long will it take them to build the shed? 1. Define variable (Let statements) Let x = time to build shed together Carpenter: 5hrs., Assistant: 10 hrs. • Create chart. 3. Write in rate of work for each. 4. Write in time worked for each. • Write in part of task completed • by multiplying rate  time. • Obtain equation by adding the • column and setting it equal to 1. 7. Solve and answer question. (Multiply all terms by LCD) The End. B.R. 12-30-06 Answer: It would take them 30 minutes to do the job together.

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