Problem 67 - Beginning and Intermediate Algebra

Question

Write each word statement as an equation. Use \(x\) as the variable. Then find the solution of the equation from the set \(\\{2,4,6,8,10\\} .\) \text { The sum of a number and } 8 \text { is } 18 \text { . }

Step-by-Step Solution

Answer

The solution is \(x = 10\).

Step 1 - Define the variable

Let the unknown number be represented by the variable \(x\).

Step 2 - Write the word statement as an equation

The statement says, 'The sum of a number and 8 is 18'. This can be written as: \[ x + 8 = 18 \]

Step 3 - Solve the equation

Solve for \(x\) by isolating it on one side of the equation: \[ x + 8 = 18 \] Subtract 8 from both sides: \[ x = 18 - 8 \] \[ x = 10 \]

Step 4 - Verify the solution

Check if the solution \(x = 10\) is part of the given set \(\{2, 4, 6, 8, 10\}\). Since 10 is in the set, \(x = 10\) is the correct solution.

Key Concepts

Algebraic Expressions

Algebraic expressions form the foundation of algebra. They consist of variables, numbers, and operations. In this problem, the expression we've worked with is the sum of a variable and a constant. Specifically, the equation given was: \[ x + 8 = 18 \] Here:

• \textbf{Variable:} `\(x\)`, representing the unknown number.
• \textbf{Constant:} `\(8\)`, a fixed number added to the variable.
• \textbf{Expression:} `\(x + 8\)`, combining both elements with an addition operation.

Algebraic expressions allow us to model real-world situations mathematically, making it easier to manipulate and solve them.

Isolating the Variable

Isolating the variable means getting the variable alone on one side of the equation. This process helps in finding the value of the variable.

In our example, we start with the equation: \[ x + 8 = 18 \] To isolate `\(x\)`, follow these steps:

• \textbf{Step 1:} Recognize that we need `\(x\)` by itself on one side.
• \textbf{Step 2:} Subtract `\(8\)` from both sides to keep the equation balanced.

The subtraction gives us: \[ x = 18 - 8 \]
Simplifying further: \[ x = 10 \] At this point, `\(x\)` is isolated, meaning we have successfully found the value of `\(x\)`.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true.

Let's revisit our equation: \[ x + 8 = 18 \]
As detailed in previous sections, we isolated `\(x\)` to find: \[ x = 10 \]
Finally, we check the solution: Our goal was to see if 10 is a part of the set `\(\{2, 4, 6, 8, 10\}\)`. Since 10 is indeed in the set, we confirm: \[ x = 10 \]
This confirms that our solution is correct. When solving other equations, you can follow a similar process:

• Translate any word problems into algebraic expressions.
• Isolate the variable.
• Solve and verify the solution.