Problem 19 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of three times a number and eight is \(23 .\) Find the number.

Step-by-Step Solution

Answer

The number is 5.

Step 1: Identify the Problem

We need to find the unknown number in the given word problem. The word problem mentions the sum of three times a number and eight is 23.

Step 2: Define a Variable

Let the unknown number be represented by the variable, say, x.

Step 3: Set Up the Equation

Translate the word problem into a mathematical equation. According to the problem: 'three times a number' translates to '3x' and 'the sum of three times a number and eight' translates to '3x + 8'. So, the equation becomes: 3x + 8 = 23

Step 4: Isolate the Variable

To find the value of x, we need to isolate x. First, subtract 8 from both sides of the equation: 3x + 8 - 8 = 23 - 8 This simplifies to: 3x = 15

Step 5: Solve for the Variable

Now, divide both sides by 3 to solve for x: 3x / 3 = 15 / 3 This simplifies to: x = 5

Step 6: Verify the Solution

To ensure the solution is correct, substitute x back into the original expression: 3(5) + 8 = 15 + 8 = 23 Since the left side equals the right side, the solution is verified as correct.

Key Concepts

Equation Solving

Solving an equation is all about translating a word problem into a mathematical statement and finding the unknown value.
For example, let's look at a problem: 'The sum of three times a number and eight is 23'.
First, translate the words into an equation. You can express 'a number' as a variable, say, \( x \).
Then, 'three times a number' becomes \( 3x \).
'The sum of three times a number and eight' becomes \( 3x + 8 \).
So the equation is: \( 3x + 8 = 23 \).
Next, solve this equation to find the unknown value by performing mathematical operations like addition, subtraction, multiplication, or division as needed.

Variable Isolation

To solve an equation, it's crucial to isolate the variable. This means getting \( x \) by itself on one side of the equation.
In our example, we start with the equation: \( 3x + 8 = 23 \).
Follow these steps:

• Step 1: Subtract 8 from both sides of the equation to eliminate the constant term.

\[ 3x + 8 - 8 = 23 - 8 \]
This simplifies to: \( 3x = 15 \).

• Step 2: Divide both sides by 3 to solve for \( x \).

\[ \frac{3x}{3} = \frac{15}{3} \]
Which simplifies to: \( x = 5 \).
Now, the variable is isolated and we have the value of \( x \).

Verification of Solution

Finally, verifying your solution is essential to ensure it's correct.
To verify, substitute the found value back into the original equation.
In our case, we found \( x = 5 \).
Substitute \( x \) back into the equation: \( 3x + 8 = 23 \).
This becomes \( 3(5) + 8 \).
Simplify: \( 15 + 8 = 23 \).
Since both sides of the equation are equal, we can confirm that our solution is correct.
Verification helps catch potential errors, making sure the variable value satisfies the original equation's conditions.