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Chapter 42: Chapter 42

Problem 1203

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The sum of two numbers is 25 and the difference of their squares is 225 . Find the numbers.

Short Answer

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The two numbers are 8 and 17.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Assign variables to the unknowns

Let x be the first number and y be the second number.

Step 2: Write the given information as equations

We are given that the sum of the two numbers is 25, which can be written as an equation: \(x + y = 25\) We are also given that the difference of their squares is 225, which can be written as another equation: \(x^2 - y^2 = 225\)

Step 3: Solve the equations

We will first use the first equation and solve for y: \(y = 25 - x\) Now, we will substitute this expression of y in the second equation to solve for x: \(x^2 - (25 - x)^2 = 225\) Expand the equation and simplify: \(x^2 - (625 - 50x + x^2) = 225\) Combine like terms: \(-50x = -400\) Now, divide by -50 to find the value of x: \(x = 8\) Now that we have found the value of x, we can plug it into the expression we found for y: \(y = 25 - 8 = 17\)

Step 4: State the final answer

The two numbers are 8 and 17.

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