Question

The difference between two numbers is 35. Six times the smaller is 9 less than 3 times the larger. Find the two numbers

Answer 1

Let the two numbers be \( x \) (the larger number) and \( y \) (the smaller number). According to the problem, we have the following two equations:

1. The difference between the two numbers is 35:
\[
x - y = 35
\]

2. Six times the smaller number is 9 less than three times the larger number:
\[
6y = 3x - 9
\]

Now we can solve the system of equations. First, from the first equation, solve for \( x \):
\[
x = y + 35
\]

Next, substitute this expression for \( x \) into the second equation:
\[
6y = 3(y + 35) - 9
\]

Now simplify the equation:
\[
6y = 3y + 105 - 9
\]
\[
6y = 3y + 96
\]

Subtract \( 3y \) from both sides:
\[
6y - 3y = 96
\]
\[
3y = 96
\]

Now divide both sides by 3:
\[
y = 32
\]

Now that we have \( y \), we can find \( x \) using the first equation:
\[
x = y + 35
\]
\[
x = 32 + 35 = 67
\]

The two numbers are:
\[
x = 67 \quad \text{(larger number)}
\]
\[
y = 32 \quad \text{(smaller number)}
\]

Thus, the larger number is **67** and the smaller number is **32**.