Problem 8 - Fundamentals of Mathematics

Question

If five more than three times a number is thirty-two, what is the number?

Step-by-Step Solution

Answer

The number is 9.

Step 1: Understand the Problem Statement

We need to find a number such that when it is multiplied by three and then increased by five, the result is thirty-two.

Step 2: Formulate the Equation

Let the unknown number be represented by the variable \( x \). According to the problem, three times this number plus five equals thirty-two. We can write this as the equation: \[ 3x + 5 = 32 \]

Step 3: Solve for the Variable

Subtract five from both sides of the equation to isolate the term with the variable: \[ 3x + 5 - 5 = 32 - 5 \] This simplifies to: \[ 3x = 27 \]

Step 4: Isolate the Variable

Divide both sides by three to solve for \( x \): \[ x = \frac{27}{3} \] This simplifies to: \[ x = 9 \]

Step 5: Verify the Solution

Substitute \( x = 9 \) back into the original equation to ensure it satisfies the problem. Calculate \( 3(9) + 5 \) which equals \( 27 + 5 = 32 \). The solution is correct.

Key Concepts

Equations

In algebra, equations are mathematical sentences that show that two expressions are equal. They are a central concept in math that helps us describe patterns and relationships.
An equation often consists of:

• Variables (like \( x \) or \( y \)), which represent unknown values that we want to find.
• Constants, which are fixed values (like numbers 3 or 5).
• Operations (like addition, subtraction, multiplication, or division).

Consider the equation we derived from the problem: \( 3x + 5 = 32 \).
- Here, \( 3x \) means three times the unknown number \( x \).- The \(+ 5\) adds five to the product, and- the \(= 32\) indicates that these values together equal thirty-two.Understanding how to set up equations from word problems allows us to translate real-life scenarios into a mathematical format that can be solved systematically.

Variable Isolation

Variable isolation refers to the process of rearranging an equation to get the variable on one side by itself. This is a crucial step in solving equations as it allows us to find the unknown value.
In the example equation \( 3x + 5 = 32 \),

• Start by isolating the term with the variable. We subtract 5 from both sides to remove the constant: \( 3x = 27 \).


• Next, to completely isolate \( x \), divide both sides by 3: \( x = \frac{27}{3} \).

This step-by-step process reveals the value of \( x \).
The main goal is to apply inverse operations to "cancel out" elements on the side of the equation with the variable until the variable is isolated.

Substitution Method

The substitution method allows us to verify our solution by substituting the value we found back into the original equation.
Once we've solved for \( x \) and found that \( x = 9 \), it is essential to check our work:

• Substitute \( 9 \) back into the original equation \( 3x + 5 = 32 \).


• Calculate: \( 3(9) + 5 \) which simplifies to \( 27 + 5 \).


• Verify: This equals \( 32 \), confirming that our solution is correct.

The substitution method ensures the solution is accurate and aligns with the problem's condition. This method not only checks correctness but also helps build confidence in solving equations.