Problem 26 - Elementary Algebra

Question

In the following exercises, solve each number word problem. One number is five more than the other. Their sum is 33 . Find the numbers.

Step-by-Step Solution

Answer

The numbers are 14 and 19.

Step 1: Define the Variables

Let the first number be denoted by \(x\). The second number is five more than the first, so it can be denoted by \(x + 5\).

Step 2: Set Up the Equation

The sum of the two numbers is given as 33. Therefore, we can write the equation: \[ x + (x + 5) = 33 \]

Step 3: Combine Like Terms

Combine the terms involving \(x\) on the left side of the equation. This gives: \[ 2x + 5 = 33 \]

Step 4: Solve for \(x\)

Subtract 5 from both sides to isolate the terms with \(x\): \[ 2x = 28 \] Now divide both sides of the equation by 2: \[ x = 14 \]

Step 5: Find the Second Number

Since the second number is \(x + 5\), we substitute \(x = 14\) into this expression: \[ 14 + 5 = 19 \]

Step 6: Verify the Answer

Verify the solution by checking the sum: \[ 14 + 19 = 33 \] This confirms that our solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In this context, we aim to find the value of the unknown variable, which makes the equation true. For the given problem, we start by defining the equation based on the conditions provided. The sum of the two numbers is expressed as: \[ x + (x + 5) = 33 \]The goal is to solve for \(x\) by performing operations that maintain equality and simplify the equation step-by-step. This involves operations like addition, subtraction, multiplication, or division. It’s essential to balance these operations on both sides of the equation to isolate the variable. By carefully following each step, we find the solution.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions and equations. In our exercise, after defining the variables, our equation is:\[ x + (x + 5) = 33 \]To simplify this, we need to combine the terms involving \(x\). Both terms on the left-hand side of the equation have \(x\) as a common factor. By combining them, we get:\[ 2x + 5 = 33 \]This makes the equation easier to solve. Combining like terms helps reduce complexity and makes it easier to perform subsequent operations to solve the equation. Always look for terms with the same variable and power, and combine their coefficients.

Defining Variables

Defining variables is a crucial initial step when solving word problems. It involves choosing symbols to represent unknown quantities in the problem. For our exercise, we need to find two numbers where one number is five more than the other, and their sum is 33. We start by letting one of the numbers be denoted by \( x \). Accordingly, the other number, which is five more than the first, is represented as \( x + 5 \). Using variables helps us translate the word problem into mathematical equations. It is helpful to clearly define what each variable represents to avoid confusion and ensure accurate solutions. This representation allows us to apply algebraic methods to find the solution.

Isolating Terms

Isolating terms is a process where we aim to get the variable by itself on one side of the equation. This helps us find the value of the variable. After combining like terms, our modified equation is:\[ 2x + 5 = 33 \]To isolate \( 2x \), we first subtract 5 from both sides:\[ 2x = 28 \]Next, we divide both sides by 2 to find \( x \):\[ x = 14 \]This process of isolating terms is essential because it simplifies the equation down to a point where the variable's value can be easily identified. The steps must be carefully followed to ensure the integrity and accuracy of the solution. By isolating terms, we make the equation easier to solve and closer to finding the final answer.