Problem 33 - 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 \(81 .\) One number is 41 more than the other. Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 61 and 20.

Step 1: Convert Word Problems into Equations

Let the two numbers be \(x\) and \(y\). From the problem, we know that \(x + y = 81\) (because their sum is 81) and \(x = y + 41\) (because one number is 41 more than the other).

Step 2: Substitute the Value from the Second Equation into the First

Substituting \(x\) from the second equation into the first equation, we get \(y + 41 + y = 81\).

Step 3: Solve for y

By simplifying the equation, consolidating like terms, we get \(2y + 41 = 81\). Hence, by subtracting 41 from both sides, we get \(2y = 81 - 41 = 40\). Dividing by 2 we get \(y = 40 / 2 = 20\).

Step 4: Substitute y into the Second Original Equation

By substituting \(y = 20\) into the equation \(x = y + 41\), we get \(x = 20 + 41 = 61\).

Step 5: Verify the Equation

We will verify if \(x = 61\) and \(y = 20\) satisfy the first original equation \(x + y = 81\). By inserting values we find the equation \(81 = 81\) to be true thus verifying the calculated solution.