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sum and difference of reciprocals

by Moises Evardo
(Cebu City)











































the sum and the difference of the reciprocals of the two numbers are 7 and 3 respectively . Find the numbers

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Jan 11, 2011
Math - Equations
by: Staff

The question:
by Moises Evardo
(Cebu City)


the sum and the difference of the reciprocals of the two numbers are 7 and 3 respectively. Find the numbers




The answer:

This is case of two equations with two unknowns.

x = 1st unknown
y = 2nd unknown

Step 1: translate the verbal statement into equations

Step 1A:

Verbal statement

The sum of the reciprocals of two numbers equals 7

Equation

(1/x) + (1/y) = 7

Step 1B:

Verbal statement

The difference of the reciprocals of two numbers equals 3

Equation

(1/x) - (1/y) = 3


Step 2: solve the two equations for x & y

(1/x) + (1/y) = 7

(1/x) - (1/y) = 3


Step 2A:

Solve for x using the addition method (add the two equations together to eliminate the fraction 1/y)

(1/x) + (1/y) = 7

(1/x) - (1/y) = 3
_____________
(2/x) + 0 = 10

2/x = 10

Multiply each side of the equation by x to eliminate the fraction 2/x

(2/x)*x = 10*x

2*(x/x) = 10*x

2*(1) = 10*x

2 = 10*x

Divide each side of the equation by 10 to eliminate the 10 on the right side of the equation

2/10 = (10*x)/10

2/10 = x*(10/10)

2/10 = x*(1)

1/5 = x

The first unknown: x = 1/5


Step 2B:

Multiply the 2nd equation by -1 to reverse the signs

(-1)*(1/x) – (-1)*(1/y) = (-1)*3

(-1/x) + (1/y) = -3

-(1/x) + (1/y) = -3


Solve for y using the addition method (add the two equations together to eliminate the fraction 1/x)

(1/x) + (1/y) = 7

-(1/x) + (1/y) = -3
___________________
0 + (2/y) = 4

2/y = 4

Multiply each side of the equation by y to eliminate the fraction 2/y

(2/y)*y = 4*y

2*(y/y) = 4*y

2*(1) = 4*y

2 = 4*y

Divide each side of the equation by 4 to eliminate the 4 on the right side of the equation

2/4 = (4*y)/4

2/4 = y*(4/4)

2/10 = y*(1)

1/2 = y

The second unknown: y = 1/2


The final answer: x = 1/5, y = ½



Step 3: Check the answer

Substitute 1/5 for the x in both of the original equations, and ½ for the y in both the original equations

First equation

(1/x) + (1/y) = 7

[1/(1/5)] + [1/(1/2)] = 7

5 + 2 = 7

7 = 7

Second equation

(1/x) - (1/y) = 3

[1/(1/5)] - [1/(1/2)] = 3

5 - 2 = 3

3 = 3

The values of 1/5 for x & ½ for y satisfy the original equations. Therefore, the final answers (x = 1/5 & y = ½) are correct.



Thanks for writing.


Staff
www.solving-math-problems.com


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