Problem 57 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. The difference of two numbers is 67 . The larger number is three less than six times the smaller number. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 14 and 81.

Step 1: Define Variables

Let the smaller number be denoted as \( x \). The larger number can then be expressed in terms of \( x \).

Step 2: Express the Larger Number in terms of the Smaller Number

According to the problem, the larger number is three less than six times the smaller number. Therefore, we can express the larger number as \( 6x - 3 \).

Step 3: Set Up the Equation Based on the Given Difference

The problem states that the difference between the two numbers is 67. So, we set up the equation \((6x - 3) - x = 67\).

Step 4: Solve for the Smaller Number

Simplify the equation: \(6x - 3 - x = 67\) becomes \(5x - 3 = 67\). Add 3 to both sides to get \(5x = 70\). Then, divide by 5 to find \(x = 14\).

Step 5: Find the Larger Number

Substitute \(x = 14\) back into the expression for the larger number: \(6(14) - 3 = 84 - 3 = 81\).

Step 6: Verify the Solution

Check that the difference is 67: \(81 - 14 = 67\). This confirms that our solution is correct.

Key Concepts

Defining Variables

In algebraic problem solving, the first crucial step is defining variables. This involves choosing letters to represent unknown values that you're trying to find. Here, we begin by identifying the smaller of the two numbers in the problem. We use "\( x \)" to denote the smaller number. Using variables allows us to create equations, which are necessary for finding unknown values in algebraic expressions.

Setting Up Equations

Once variables are defined, the next step is constructing equations that represent the problem's conditions. In our problem, the larger number was described as three less than six times the smaller number. Thus, we use the expression "\( 6x - 3 \)" to symbolize the larger number. Then, we translate the stated condition about their difference being 67 into a mathematical equation: \((6x - 3) - x = 67\). Creating accurate equations is essential for effectively solving any algebraic problem.

Solving Linear Equations

After setting up the equation, it's time to solve it. This equation, \(5x - 3 = 67\), is a linear equation. To solve linear equations, we typically simplify and isolate the variable. Here, we add 3 to both sides, simplifying to \(5x = 70\). Then, we divide both sides by 5, eventually finding \(x = 14\). Solving linear equations generally involves logical and systematic steps to isolate the variable, thereby revealing the unknown number.

Verification of Solution

Finally, verify your solution to ensure it meets the original conditions. Here, with \(x = 14\), substitute back to find the larger number: \(6(14) - 3 = 81\). Check if the difference equals 67: \(81 - 14 = 67\). Verification confirms your answer's correctness and ensures the solution resolves the problem accurately. Always re-check both algebraic calculations and if initial problem conditions are satisfied.