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The difference of two numbers is 3 and the difference of their squares is 39, then the larger number is
- a)8
- b)9
- c)12
- d)13
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
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Problem:
The difference of two numbers is 3 and the difference of their squares is 39. Find the larger number.
Solution:
Let's assume the two numbers as x and y, where x is the larger number.
Given information:
1. Difference of two numbers: x - y = 3
2. Difference of their squares: x^2 - y^2 = 39
Step 1: Simplifying the equations
We can simplify the equation x^2 - y^2 = 39 using the difference of squares formula: (x + y)(x - y) = 39.
Step 2: Substituting the value of x - y
Since x - y = 3, we can substitute this value into the equation (x + y)(x - y) = 39.
(x + y)(3) = 39
3x + 3y = 39
Step 3: Solving the system of equations
We now have a system of equations:
x - y = 3
3x + 3y = 39
We can solve this system by eliminating one variable. Let's multiply the first equation by 3 to make the coefficients of y in both equations equal:
3(x - y) = 3(3)
3x - 3y = 9
Now we have:
3x - 3y = 9
3x + 3y = 39
Adding these two equations eliminates the y variable:
(3x - 3y) + (3x + 3y) = 9 + 39
6x = 48
x = 8
Step 4: Finding the larger number
Since we assumed x as the larger number, the larger number is 8.
Therefore, the correct answer is option 'A' (8).
The difference of two numbers is 3 and the difference of their squares is 39. Find the larger number.
Solution:
Let's assume the two numbers as x and y, where x is the larger number.
Given information:
1. Difference of two numbers: x - y = 3
2. Difference of their squares: x^2 - y^2 = 39
Step 1: Simplifying the equations
We can simplify the equation x^2 - y^2 = 39 using the difference of squares formula: (x + y)(x - y) = 39.
Step 2: Substituting the value of x - y
Since x - y = 3, we can substitute this value into the equation (x + y)(x - y) = 39.
(x + y)(3) = 39
3x + 3y = 39
Step 3: Solving the system of equations
We now have a system of equations:
x - y = 3
3x + 3y = 39
We can solve this system by eliminating one variable. Let's multiply the first equation by 3 to make the coefficients of y in both equations equal:
3(x - y) = 3(3)
3x - 3y = 9
Now we have:
3x - 3y = 9
3x + 3y = 39
Adding these two equations eliminates the y variable:
(3x - 3y) + (3x + 3y) = 9 + 39
6x = 48
x = 8
Step 4: Finding the larger number
Since we assumed x as the larger number, the larger number is 8.
Therefore, the correct answer is option 'A' (8).
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