Problem 35 - 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 difference between two numbers is \(5 .\) Four times the larger number is 6 times the smaller number. Find the numbers.

Step-by-Step Solution

Answer

The smaller number is 10 and the larger number is 15.

Step 1: Make equations from the problems

We'll first make equations from the conditions given. We'll call the smaller number \(x\), and the larger number \(y\). The gap between the two numbers is five, so that gives us the equation \(y = x + 5\). The second condition mentions that four times of the larger number equals to six times of the smaller number. Therefore, we get the equation \(4y = 6x\).

Step 2: Use the substitution method

Substitute the expression for \(y\) in the equation from Step 1, to the second equation to solve for \(x\). This gives \(4(x + 5) = 6x\). Simplify it to \(4x + 20 = 6x\). Isolate \(x\) to one side and we get \(x = 10\). Now we will then substitute \(x = 10\) into the equation \(y = x + 5\) to find the value of \(y\). We get, \(y = 10 + 5\), therefore, \(y = 15\) .

Step 3: Check the solution

Substitute \(x = 10\) and \(y = 15\) into both original equations to make sure they both hold true. For the first equation, \(15 - 10 = 5\) which is true. For the second equation, \(4 * 15 = 6 * 10\) which is also true.