Problem 10 - College Algebra

Question

One number exceeds another by \(24 .\) The sum of the numbers is \(58 .\) What are the numbers?

Step-by-Step Solution

Answer

The two numbers are 17 and 41.

Step 1 - Represent the problem with algebraic expressions

Let's represent the smaller number as \(x\). Then according to the problem, the larger number would be \(x + 24\).

Step 2 - Set up the equation

Now the problem states that the sum of these numbers is 58. So let's formulate an equation based on this: \(x + (x + 24) = 58\).

Step 3 - Solve the equation

Rearrange and merge like terms to solve above equation: \(2x + 24 = 58\). Now, subtract 24 from both sides to get \(2x = 34\). Lastly, divide both sides by 2 to solve for \(x\), \(x = 17\).

Step 4 - Identify the two numbers

Substitute \(x = 17\) into the two expressions to find out the two numbers: the smaller number is \(x = 17\) and the larger number is \(x + 24 = 17 + 24 = 41\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that combine numbers, variables, and operators to represent a quantity. In our example, we used an algebraic expression to represent two numbers.

• The smaller number is defined as \( x \).
• The larger number is expressed as \( x + 24 \), where 24 is the difference between the two numbers.

Understanding how to represent real-world situations with algebraic expressions is crucial. It allows us to manipulate these expressions to solve equations. This is particularly useful in translating word problems into mathematical problems we can work with.
Keep in mind: translating a word problem into an algebraic expression is often the first step in problem-solving.

Systems of Equations

Systems of equations involve solving two or more equations that are interrelated. In our scenario, even though we have a single equation, it is built from relationship expressions:
1. The smaller number as \( x \).2. The larger number as \( x + 24 \).
The sum of these numbers leads us to the equation \( x + (x + 24) = 58 \). This single equation represents a small system that we can solve to find the two unknowns.
When dealing with systems of equations in more complex situations, you'll often find more than one equation needed. You solve them using various methods, such as:

• Substitution
• Elimination
• Graphical methods

For our straightforward problem, working through the algebraic rearrangement suffices to find the solution.

Problem Solving

Problem-solving in mathematics involves a logical approach to finding solutions to given problems. It often begins with understanding the problem and then expressing it in a mathematical form.
Here are the steps we followed in our example:

• Identify what's given and what needs to be found - one number exceeds another by 24, and their total is 58.
• Translate these conditions into equations using algebraic expressions.
• Use arithmetic to manipulate the equation - merging like terms and isolating the variable to solve for \( x \).
• Finally, substitute back to find both numbers - substituting to verify our answers makes sure they're correct.

Effective problem-solving often involves checking your work and ensuring every step is logical and leads to the desired outcome. This structured approach helps in tackling more complex mathematical problems seamlessly.