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Problem 1203

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

Expert verified
The two numbers are 8 and 17.
See the step by step solution

## 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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