While it is possible to solve these equations as is, usually it is preferable to clear all of the fractions prior to solving.
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1 MODULE. FRACTIONS c Solving Equations While it is possible to solve these equations as is, usually it is preferable to clear all of the fractions prior to solving. Clearing Fractions from the Equation Clearing Fractions from the Equation. To clear all fractions from an equation, multiply both sides of the equation by the least common denominator of the fractions that appear in the equation. Let s put this idea to work. EXAMPLE. Solve the following equation for x: x 5 6 =. Solution. First clear all fractions from the equation. To achieve this, multiply both sides of the equation by the least common denominator of all the fractions appearing in the equation. x 5 6 = 6 x 5 ) ) =6 6 ) ) 5 6x 6 =6 6 6x 5= Multiply both sides by 6. Distribute the 6. On each side, multiply first. ) ) 5 6 =5and6 =. 6 Note that the equation is now entirely clear of fractions, making it a much simpler equation to solve. 6x 5+5=+5 6x = 7 6x 6 = 7 6 x = 7 6 Add 5 to both sides. Divide both sides by 6. Simplify.
2 C. SOLVING EQUATIONS 9 Checking the Solution. Substitute 7/6 for x in the original equation and simplify. x 5 6 = = 6 = = Substitute 7/6 for x. Subtract. Reduce. Because the last statement is true, we conclude that 7/6 is a solution ofthe equation x 5/6 = /. Answer: /5 Let s solve another one. EXAMPLE. Solve the following equation for x. 9 x = 5 As before, solve this equation by first clearing all fractions. Solution. Multiply both sides of the equation by the least common denominator. 9 x = 5 9 ) ) 5 x = 6x = 5 Multiply both sides by. On each side, cancel and multiply. ) ) 5 = 6 and =5. 9 Note that the equation is now entirely free of fractions. Continuing, 6x 6 = 5 6 x = 5 6 Divide both sides by 6. Simplify.
3 0 MODULE. FRACTIONS Checking the Solution. Substitute 5/6 for x in the original equation and simplify. 9 9 x = 5 5 ) = = 5 5 = 5 Substitute -5/6 for x. Multiply numerators; multiply denominators. Reduce both sides to lowest terms. Answer: 6/5 Because this last statement is true, we conclude that 5/6 is a solution of the equation /9)x =5/. A couple of key points to keep in mind are: Both sides of the equation need to be multiplied by the LCD. We must use the distributive property when ever addition or subtraction show up on in an equation. Let s work another example. Solve for s: s 5 = s 5 EXAMPLE. Solve for x: x 4 = x. Solution. Multiply both sides of the equation by the least common denominator. 4 x 4 = x 4 x ) x =4 4 ) ) ) x x ) 4 =4 4 4 ) Multiply both sides by 4. On both sides, distribute 4. ) ) x 6 x =x Left: 4 =6,4 =x. 4 x ) ) Right: 4 =x, 4 =.
4 C. SOLVING EQUATIONS Note that the equation is now entirely free of fractions. We need to isolate the terms containing x on one side of the equation. 6 x x =x x Subtract x from both sides. 6 0x = Left: x x = 0x. Right: x x =0. 6 0x 6 = 6 Subtract 6 from both sides. 0x = 9 Left: 6 6 = 0. Right: 6 = 9. 0x 0 = 9 0 Divide both sides by 0. x = 9 0 Readers are encouraged to check this solution in the original equation. Answer: 5/ Applications We close out this review with an application problem. EXAMPLE 4. In the third quarter of a basketball game, announcers informed the crowd that attendance for the game was,50. If this is two-thirds of the capacity, find the full seating capacity for the basketball arena. Solution. We follow the Requirements for Word Problem Solutions.. Set up a Variable Dictionary. Let F represent the full seating capacity. Note: It is much better to use a variable that sounds like the quantity that it represents. In this case, letting F represent the full seating capacity is much more descriptive than using x to represent the full seating capacity. Attendance for the Celtics game was 9,50. If this is /4 of capacity, what is the capacity of the Celtics arena?. Set up an Equation. Two-thirds of the full seating capacity is,50. Two-thirds Hence, the equation is of Full Seating Capacity is,50 F =,50 F =50.
5 MODULE. FRACTIONS. Solve the Equation. Multiply both sides by to clear fractions, then solve. F =50 ) F = 50) Multiply both sides by. F = 6750 F = 6750 F = 75 Divide both sides by. 4. Answer the Question. The full seating capacity is, Look Back. The words of the problem state that / of the seating capacity is,50. Let s take two-thirds of our answer and see what we get. 75 = 75 = 65 = 65 =50 Answer:,60 This is the correct attendance, so our solution is correct.
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