Problem 34 - Introductory Algebra for College Students

Question

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The sum of two numbers is \(62 .\) One number is 12 more than the other. Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 37 and 25.

Step 1: Setup the System of Equations

Let's call the two numbers \(x\) and \(y\). Given that the sum of the two numbers is 62, this can be represented as the equation \(x + y = 62 \). Given that one number is 12 more than the other, this can be represented as the equation \(x = y + 12\). Thus, the system of equations to solve is \(\{x + y = 62, x = y + 12\}\)

Step 2: Substitute \(y + 12\) for \(x\) in the First Equation

Replace \(x\) in the first equation with the expression from the second equation (\(y + 12\)) to get the new equation: \(y + 12 + y = 62\).

Step 3: Solve for \(y\)

Simplify and solve the equation for \(y\). \(2y + 12 = 62\) Subtract 12 from both sides to get \(2y = 50\). Then, divide by 2 to find that \(y = 25\).

Step 4: Substitute \(y = 25\) Into the Second Equation

Substitute \(y = 25\) into the second equation \(x = y + 12\) to get \(x = 25 + 12\), hence \(x = 37\).

Step 5: Check the Solution

Verify that these values work in the original equations. Substituting \(x = 37\) and \(y = 25\) into both initial equations shows that they indeed are solutions.