Question

FIND THE NUMBERS The sum of three numbers is \(123 .\) The second number is 9 less than two times the first number. The third number is 6 more than three times the first number. Find the three numbers.

Short answer

The three numbers are 21, 33, and 69.

Step-by-step solution

Identify and set up the equations

From the problem, we can set up the following equations: \(x + y + z = 123 \), \(y = 2x - 9 \), \(z = 3x + 6 \)

Substitute equations into the first equation

We can substitute the expressions for \( y \) and \( z \) from the second and third equations into the first equation, resulting in: \(x + (2x - 9) + (3x + 6) = 123\). Simplify this to get an equation in terms of \(x\): \(6x - 3 = 123\)

Solve for x

By rearranging the equation, add 3 on both sides, we get: \(6x = 126\). Now solve for \(x\) by dividing both sides by 6 to yield \(x = 21\).

Find the values for y and z

Substitute the value of \(x = 21\) into the equations for \( y \) and \( z \). Therefore, \( y = 2(21) - 9 = 33 \) and \(z = 3(21) + 6 = 69\).

Validation

Check if these numbers meet the conditions of the original problem. The numbers do indeed add up to 123, so the answer makes sense.