Problem 11 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{c}{9}=4$$

Step-by-Step Solution

Answer

The solution is \( c = 36 \).

Step 1: Understand the Equation

We are given the equation \( \frac{c}{9} = 4 \). This is a simple linear equation where \( c \) is divided by 9, and the result is 4. Our goal is to solve for \( c \).

Step 2: Isolate the Variable

To isolate \( c \), we need to eliminate the division by 9. We do this by multiplying both sides of the equation by 9.

Step 3: Multiply Both Sides

Multiply both sides by 9: \[ 9 \times \frac{c}{9} = 9 \times 4 \]This simplifies to:\[ c = 36 \]

Step 4: Check the Solution

To ensure the solution is correct, substitute \( c = 36 \) back into the original equation: \[ \frac{36}{9} = 4 \]Simplify the left side:\[ 4 = 4 \]The left side equals the right side, confirming that \( c = 36 \) is correct.

Key Concepts

Checking Solutions for Accuracy

After finding a solution to a linear equation, it's important to check if it's correct. This step ensures you have not made any mistakes during the process of solving the equation.
To check your solution, simply substitute the found value back into the original equation and verify if both sides are equal.

• For example, in the equation \( \frac{c}{9} = 4 \), once we solve it and find \( c = 36 \), we substitute 36 back into the original equation.
• The equation becomes \( \frac{36}{9} = 4 \). Simplifying the left side yields \( 4 = 4 \), which confirms our solution is correct.

Checking solutions is an essential step as it verifies your answer is accurate. This practice builds confidence and helps prevent small errors from slipping through.

Isolating Variables in Equations

Isolating the variable is a key step in solving equations. The goal is to have the variable alone on one side of the equation, effectively "solving" for it.
The process begins by reversing operations that are applied to the variable:

• If a variable is divided by a number, like \( \frac{c}{9} = 4 \), you can eliminate the division by multiplying both sides by the same number (in this case, 9).
• Similarly, if a variable is added to a number, you subtract that number from both sides to isolate the variable.

The logic here is localizing the variable to understand its specific value in the context of the equation. This process doesn't change the equation's balance because the same arithmetic operation is applied to both sides.

Using the Multiplication Property of Equality

When solving equations, one tool that we frequently use is the multiplication property of equality. This property states that if you multiply both sides of an equation by the same nonzero number, the equation remains balanced.
Here's how it works:

• Consider the equation \( \frac{c}{9} = 4 \). To solve for \( c \), we need to get rid of the division by 9. This is done by multiplying both sides of the equation by 9.
• The left side \( 9 \times \frac{c}{9} \) simplifies to \( c \) because \( 9 \) and \( \frac{1}{9} \) cancel each other out.
• The right side becomes \( 9 \times 4 = 36 \).

The multiplication property of equality is crucial because it allows us to manipulate equations without altering their true meaning. This concept helps us maintain equality while making solving equations possible.