# 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

### Comments for sum and difference of reciprocals

 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