Question

1. the larger of two numbers is two more than five times the smaller number. if the sum of the two numbers is 80, find both numbers.

Explanation:

Step1: Define variables

Let the smaller number be \( x \) and the larger number be \( y \).

Step2: Set up equations

The larger number is two more than five times the smaller number, so \( y = 5x + 2 \).
The sum of the two numbers is 80, so \( x + y = 80 \).

Step3: Substitute and solve

Substitute \( y = 5x + 2 \) into \( x + y = 80 \):
\( x + (5x + 2) = 80 \)
\( x + 5x + 2 = 80 \)
\( 6x + 2 = 80 \)
\( 6x = 80 - 2 \)
\( 6x = 78 \)
\( x = \frac{78}{6} = 13 \)

Step4: Find the larger number

Substitute \( x = 13 \) into \( y = 5x + 2 \):
\( y = 5\times13 + 2 = 65 + 2 = 67 \)

Answer:

The smaller number is 13 and the larger number is 67.