Problem 3 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$\frac{1}{2} x=-3$$

Step-by-Step Solution

Answer

The solution is \( x = -6 \).

Step 1: Identify the Equation

The equation given is \( \frac{1}{2} x = -3 \). Our goal is to solve for \( x \) by isolating it on one side of the equation.

Step 2: Understand the Multiplication Property of Equality

The multiplication property of equality states that if we multiply both sides of an equation by the same nonzero number, the sides remain equal. This property can be used to eliminate a fraction by multiplying by its reciprocal.

Step 3: Multiply Both Sides by 2

To eliminate the fraction \( \frac{1}{2} \), we multiply both sides of the equation by 2. This helps isolate \( x \). So, \( 2 \times \frac{1}{2} x = 2 \times (-3) \) gives us \( x = -6 \).

Step 4: Verify the Solution

Substitute \( x = -6 \) back into the original equation to verify:\( \frac{1}{2} \times (-6) = -3 \). Since both sides are equal, the solution is correct.

Key Concepts

Solving Equations

Solving equations is a foundational skill in mathematics that involves finding what values make the equation true. These values are often called 'solutions.' In prealgebra, solving equations usually means isolating a variable — commonly denoted as \(x\) — to one side to find its value.
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Equations are like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. In our exercise, we started with \(\frac{1}{2}x = -3\). To solve for \(x\), the equation tells us how much \(x\) is worth. By isolating \(x\), we can see this value clearly.
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In solving \(\frac{1}{2}x = -3\), we use the multiplication property of equality. This allows us to multiply both sides by the same number to get rid of fractions or other coefficients, making our solution easier to find.

Prealgebra Concepts

Prealgebra introduces students to basic arithmetic and algebraic concepts, preparing them for more advanced topics. It is crucial as it lays the groundwork for understanding how numbers and variables interact.

In the prealgebra stage, we learn about fundamental properties like:

• Multiplication and division of numbers
• Understanding variables and expressions
• Using properties like addition and multiplication to simplify equations

In our example, we see the multiplication property of equality in action. By multiplying both sides of \(\frac{1}{2}x = -3\) by 2, we apply a typical prealgebra strategy to solve for \(x\). This simplification technique is vital for solving more complicated equations you will encounter in algebra and beyond.

Working with Fractions

Fractions can initially seem intimidating, but they're just another way of representing numbers. Understanding how to work with fractions is essential for solving equations that contain them.

In our exercise, \(\frac{1}{2}x = -3\) involves a fraction. We overcome this by multiplying both sides by the reciprocal of \(\frac{1}{2}\), which is 2. This effectively eliminates the fraction on the side with \(x\), simplifying the problem.

Here's a quick look at fractions and their reciprocals:

• Multiplying a fraction by its reciprocal (e.g., \(\frac{a}{b} \times \frac{b}{a}\)) results in 1. This helps in simplifying equations.
• Reciprocals are helpful when you need to 'clear' a fraction from one side of an equation.

Through these steps, working with fractions becomes manageable, turning them from obstacles into helpful tools for solving equations. Remember, practice is key — the more you work with fractions, the more intuitive they'll become.