Finding Two Numbers

Video Explanation

Question

The sum of two numbers is 8. If their sum is four times their difference, find the numbers.

Solution

Step 1: Let the Variables

Let first number = \(x\)

Let second number = \(y\)

Step 2: Form the Equations

Sum of the numbers is 8:

\[ x + y = 8 \quad (1) \]

Sum is four times their difference:

\[ x + y = 4(x – y) \quad (2) \]

Step 3: Simplify Equation (2)

\[ x + y = 4x – 4y \]

\[ x + y – 4x + 4y = 0 \]

\[ -3x + 5y = 0 \quad (3) \]

Step 4: Solve the Equations

From equation (1):

\[ x = 8 – y \]

Substitute in equation (3):

\[ -3(8 – y) + 5y = 0 \]

\[ -24 + 3y + 5y = 0 \]

\[ 8y = 24 \]

\[ y = 3 \]

Step 5: Find the Value of x

\[ x = 8 – 3 \]

\[ x = 5 \]

Conclusion

First number:

\[ \boxed{5} \]

Second number:

\[ \boxed{3} \]

Final Answer (For Exam)

The two numbers are 5 and 3.