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We are now in a position to study some applications of rational equations. Some of these problems will have practical applications while others are intended as logic developers.
We will apply the five-step method for solving word problems.
Remember, step 1 is very important: always
Introduce a variable.
When the same number is added to the numerator and denominator of the fraction
35, the result is
79. What is the number that is added?
Step 1: Let
x= the number being added.
Step 2:3+x5+x=79.Step 3:3+x5+x=79.An excluded value is −5.The LCD is 9(5+x). Multiply each term by 9(5+x).9(5+x) · 3+x5+x=9(5+x) · 799(3+x)=7(5+x)27+9x=35+7x2x=8x=4Check this potential solution.Step 4:3+45+4=79.Yes, this is correct.Step 5:The number added is 4.
The same number is added to the numerator and denominator of the fraction 49. The result is 23. What is the number that is added?
Step 1: Let x =
Step 2:
Step 3:
Step 4:
Step 5: The number added is
The number added is 6.
Two thirds of a number added to the reciprocal of the number yields
256. What is the number?
Step 1: Let
x = the number.
Step 2: Recall that the reciprocal of a number
x is the number
1x .
23 · x+1x=256
Step 3:23 · x+1x=256The LCD is6x.Multiply each term by6x.6x · 23x+6x · 1x=6x · 2564x2+6=25xSolve this nonfractional quadratic equation to obtain thepotential solutions. (Use the zero-factor property.)4x2−25x+6=0(4x−1)(x−6)=0x=14,6Check these potential solutions.
Step 4: Substituting into the original equation, it can be that both solutions check.
Step 5: There are two solutions:
14 and 6.
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