Problem 16 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of a number and nine is 17 . Find the number.

Step-by-Step Solution

Answer

The number is 8.

Step 1: Understand the problem

The problem states that the sum of a number and nine is 17. We need to find this unknown number.

Step 2: Set up the equation

Let's denote the unknown number by the variable \( x \). According to the problem, the sum of this number and nine is equal to 17. Therefore, we can write the equation as follows: \[ x + 9 = 17 \]

Step 3: Isolate the variable

To find the value of \( x \), we need to isolate it on one side of the equation. We can do this by subtracting 9 from both sides: \[ x + 9 - 9 = 17 - 9 \]

Step 4: Simplify the equation

Simplify both sides of the equation: \[ x = 8 \]

Step 5: Verify the solution

Plug the found value back into the original statement to verify: \[ 8 + 9 = 17 \] Since the left side equals the right side, our solution is correct.

Key Concepts

algebra

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, symbols represent quantities without fixed values, known as variables. These variables allow us to write general formulas and equations to solve problems. When working with algebra, you’ll often see variables like x or y used to represent unknown values.
In the given exercise, we used algebra to describe the relationship between an unknown number and the numbers provided in the problem. This approach makes it easier to systematically find the solution.

equation solving

Equation solving is a core aspect of algebra with the main goal of finding the value of the unknown variable(s) in the equation. An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation typically involves isolating the variable on one side to determine its value.
In our example, we created the equation \( x + 9 = 17 \) to express the problem statement. By manipulating this equation, we eventually isolated the variable x, leading us to the solution.

word problems

Word problems present mathematical scenarios through descriptive text. The goal is to translate these descriptions into mathematical equations that can then be solved. Word problems can encompass many real-life situations, making them very practical.
To solve word problems, follow these steps:

• Understand the problem's context and requirements.
• Translate the words into a mathematical equation.
• Solve the equation.
• Verify the solution within the context of the problem.

This exercise was a word problem where we needed to find a number based on its sum with nine equalling 17. We translated the word problem into the equation \( x + 9 = 17 \) and solved it.

variable isolation

Variable isolation is the process of rearranging an equation to get the unknown variable by itself on one side. This process typically involves using arithmetic operations to 'move' numbers and simplify the equation.
To isolate a variable:

• Identify the operations applied to the variable.
• Undo these operations with inverse operations (e.g., subtract if added, divide if multiplied).
• Apply these steps to both sides of the equation to maintain equality.

In our example, to isolate x in \( x + 9 = 17 \), we subtracted 9 from both sides, resulting in \( x = 8 \). Successfully isolating the variable is crucial for finding the correct solution.