Problem 36 - Elementary Algebra

Question

In the following exercises, solve each number word problem. One number is eight more than another. If their sum is increased by \(17,\) the result is \(95 .\) Find the numbers.

Step-by-Step Solution

Answer

The numbers are 35 and 43.

Step 1 - Define Variables

Let's define the two unknown numbers. Let the first number be denoted as \(x\). Since one number is eight more than another, the second number will be \(x + 8\).

Step 2 - Set Up the Equation

According to the problem, the sum of these two numbers increased by 17 equals 95. Therefore, we set up the equation: \(x + (x + 8) + 17 = 95\)

Step 3 - Simplify the Equation

Combine like terms in the equation: \(x + x + 8 + 17 = 95\) This simplifies to: \(2x + 25 = 95\)

Step 4 - Isolate the Variable

Subtract 25 from both sides of the equation to isolate the term with the variable: \(2x = 70\)

Step 5 - Solve for \(x\)

Divide both sides by 2 to find the value of \(x\): \(x = 35\)

Step 6 - Find the Second Number

The second number is eight more than the first number, so: \(x + 8 = 35 + 8 = 43\)

Step 7 - Verify the Solution

Check the solution by verifying the conditions of the problem. The sum of the two numbers is \(35 + 43 = 78\). Increasing this sum by 17 gives: \(78 + 17 = 95\). Therefore, the solution is correct.

Key Concepts

algebraic equation

An algebraic equation is a mathematical statement that shows the equality between two expressions. In this number word problem, we used an algebraic equation to represent the relationship between two unknown numbers. By translating the word problem into a mathematical form, we created the equation:

• \( x + (x + 8) + 17 = 95 \)

Here, the left side of the equation shows the relationship described in the problem, and the right side shows the final result after increasing the sum by 17. We often use algebraic equations to model real-life situations, making it easier to find unknown values by manipulating the equation.

variable isolation

Variable isolation is the process of rearranging an equation to get the unknown variable by itself on one side. This is one of the essential steps in solving algebraic equations. In our example, we work through these steps:

• First, we simplified the equation: \( x + x + 8 + 17 = 95 \) to \( 2x + 25 = 95 \)
• Next, we isolated the term with the variable: \( 2x = 70 \) by subtracting 25 from both sides.
• Finally, we solved for \( x \) by dividing both sides by 2: \( x = 35 \).

By isolating the variable, we pinpoint the exact value of the unknown number, making it simpler to find the solution.

problem-solving steps

Solving algebraic word problems involves a series of logical steps. These steps help break down the problem into manageable parts, making finding the solution easier. Let's review our steps:

• **Define Variables**: Start by letting one unknown number be \( x \), and the second number be \( x + 8 \), based on the problem statement.
• **Set Up the Equation**: Translate the word problem into an algebraic equation: \( x + (x + 8) + 17 = 95 \).
• **Simplify the Equation**: Combine like terms to make the equation simpler: \( 2x + 25 = 95 \).
• **Isolate the Variable**: Rearrange the equation to solve for \( x \): \( 2x = 70 \), and then \( x = 35 \).
• **Find the Second Number**: Use the value of \( x \) to find the second number: \( 35 + 8 = 43 \).

Following these steps ensures a logical progression from the problem statement to the final solution.

verification of solutions

Verifying your solutions is a crucial last step in problem-solving. It ensures accuracy and builds confidence in your solution. Let's verify our solution:

• We found the numbers to be 35 and 43.
• Sum of these numbers: \( 35 + 43 = 78 \).
• Increasing the sum by 17 should give us 95: \( 78 + 17 = 95 \).

The conditions of the problem are satisfied, confirming that our solution is correct. Verification helps confirm that all the logical steps we took were sound and that we've correctly understood and solved the problem.