Problem 4 - College Algebra with Modeling and Visualization

Question

What property is used to solve 15x = 5?

Step-by-Step Solution

Answer

The Division Property of Equality is used.

Step 1: Identify Equation Type

The given equation is 15\(x = 5\). This is a simple linear equation where the variable \(x\) is multiplied by a constant.

Step 2: Recognize the Property Needed

To solve for \(x\), we recognize the need to isolate \(x\). This requires us to divide both sides of the equation by 15.

Step 3: Apply the Division Property of Equality

The Division Property of Equality states that if you divide both sides of an equation by the same non-zero value, the sides remain equal. Here, divide both sides by 15: \(\frac{15x}{15} = \frac{5}{15}\).

Step 4: Simplify the Equation

After dividing, simplify the equation: \(x = \frac{5}{15}\). Further simplification gives \(x = \frac{1}{3}\), which is the value of \(x\).

Key Concepts

Division Property of Equality

When solving equations, one important method is the Division Property of Equality. This property allows us to maintain equality in equations while isolating the variable we want to solve for.
In the given example, 15\(x = 5\), we want to solve for \(x\). The Division Property of Equality tells us that we can divide both sides of the equation by the same non-zero number to keep the sides equal. Here, we divide both sides by 15.

• Resulting equation becomes: \(\frac{15x}{15} = \frac{5}{15}\).
• This simplifies to: \(x = \frac{5}{15}\).

Using this property ensures that the equality of the equation is preserved, while also bringing us closer to finding \(x\). Just remember: always divide both sides by the same number, and never divide by zero!

Solving Linear Equations

Solving linear equations involves finding the unknown value that makes the equation true. In our original problem, we dealt with the linear equation 15\(x = 5\).
Here's the process to solve it:

• Identify the goal: Our goal is to find the value of \(x\) that makes both sides of the equation equal.
• Choose an operation: In this case, division is the operation needed to isolate \(x\) because \(x\) is currently multiplied by 15.
• Apply Division: Use the Division Property of Equality, which allows us to divide both sides of the equation to maintain balance and isolate \(x\).

After simplifying, we find \(x = \frac{1}{3}\), completing the solution. By breaking down the problem into these simple steps, we can solve many types of linear equations by isolating the variable.

Simplification of Fractions

Simplification of fractions is an essential math skill, especially when solving equations. It helps us rewrite fractions in the simplest form, making them easier to work with.
In the exercise, after applying the Division Property of Equality, we reached the equation \(x = \frac{5}{15}\). To simplify:

• Find the greatest common factor of the numerator and denominator. For 5 and 15, the greatest common factor is 5.
• Divide both the numerator and the denominator by this factor: \( \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \).

The value \(x = \frac{1}{3}\) shows the simplest form of the fraction. Simplifying fractions is crucial because it provides the cleanest and most understandable form of the solution. A simplified fraction is less cluttered and easier to interpret, especially when it represents the solution to an equation.