Problem 1 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$\frac{1}{4} x=2$$

Step-by-Step Solution

Answer

The solution is \( x = 8 \).

Step 1: Understand the Equation

You are given the equation \( \frac{1}{4} x = 2 \). This equation states that one-fourth of \( x \) is equal to 2. The goal is to find the value of \( x \).

Step 2: Use the Multiplication Property of Equality

According to the multiplication property of equality, you can multiply both sides of an equation by the same number without changing the equality. Here, multiply both sides by 4 to eliminate the fraction on the left side.

Step 3: Perform the Multiplication

Multiply both sides by 4: \[ 4 \cdot \frac{1}{4} x = 4 \cdot 2 \]

Step 4: Simplify Both Sides

On the left side, multiplying \( \frac{1}{4} \) by 4 cancels out the fraction, leaving \( x \). On the right side, 4 multiplied by 2 equals 8. Therefore: \[ x = 8 \]

Step 5: Verify Solution

Substitute \( x = 8 \) back into the original equation to verify the solution: \[ \frac{1}{4} \cdot 8 = 2 \] Simplifying gives \( 2 = 2 \), confirming the solution is correct.

Key Concepts

Solving Equations

When we talk about solving equations, we are seeking to find the value of the unknown variable that makes the equation true. In an equation like \( \frac{1}{4} x = 2 \), we want to determine what value of \( x \) will satisfy this condition. Solving equations involves a number of strategies and steps:

• First, understand what the equation represents. It's a statement that two expressions are equal. Here, it tells us that one-fourth of \( x \) amounts to 2.
• Next, we use algebraic principles to isolate the variable, which in this case is \( x \). The key is to perform the same operation on both sides without altering the equation's truth.
• By using operations like addition, subtraction, multiplication, and division, we can steadily work towards isolating the variable.

Once \( x \) is on its own on one side of the equation, you have found your solution. The beauty of solving equations is in the logic and balance—it’s a method of making complex relationships simple and understandable.

Prealgebra

Prealgebra sets the stage for algebra by introducing fundamental concepts and skills. It serves as a bridge between basic arithmetic and the more analytical realm of algebra.

• One of the central ideas in prealgebra is the concept of variables. Variables like \( x \) in our problem are symbols that represent numbers, allowing us to write expressions that can model real-world situations.
• Equations are another key focus. They are mathematical statements of equality where we apply operations to uncover the value of unknown quantities.
• Prealgebra also emphasizes the importance of properties of numbers, such as the multiplication property of equality, which is critical for manipulating and solving equations.

Understanding these foundational topics not only aids in solving simple equations but also prepares you for the more complex algebra problems you'll encounter later. It’s important to grasp these ideas deeply to build a strong mathematical foundation.

Fractions in Equations

Fractions often appear in equations, and understanding how to handle them is crucial. Fractions can be tricky, but they become manageable once you know how to work with them effectively.

• At their core, fractions represent division. In our example, \( \frac{1}{4} x = 2 \), the fraction \( \frac{1}{4} \) tells us that \( x \) is divided by 4.
• To eliminate the fraction, we use the multiplication property of equality. By multiplying both sides by 4, we effectively "cancel out" the division, leaving \( x \) by itself.
• This property is particularly helpful, as it not only simplifies the equation but also illustrates the balance that's inherent in equality. Whatever you do to one side of the equation, you must do to the other to preserve balance.

Solving equations involving fractions requires this balance to maintain the truth of the equation while simplifying it to reveal the unknown value. Such problems hone your skills in arithmetic operations with fractions and deepen your understanding of equality principles in mathematics.