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.