Problem 12 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$2 a=28$$

Step-by-Step Solution

Answer

The value of \(a\) is 14.

Step 1: Identify the Equation Structure

The given equation is \(2a = 28\). It contains a variable \(a\) multiplied by 2 on the left side of the equation.

Step 2: Apply the Multiplication Property of Equality

To isolate \(a\), we can divide both sides of the equation by 2. This is the multiplication property of equality, as it involves canceling the multiplication of 2 by dividing by 2. So, divide both sides by 2: \( \frac{2a}{2} = \frac{28}{2} \).

Step 3: Simplify Both Sides of the Equation

When you divide \(2a\) by 2, \(a\) is left. Divide 28 by 2 to get 14. So, the equation simplifies to \(a = 14\).

Key Concepts

Solving Equations

Solving equations is a fundamental part of mathematics, especially when learning algebra. It involves finding the value of a variable that makes the equation true. In the equation \(2a = 28\), the variable is \(a\), and our goal is to solve for this unknown.
To solve an equation, you perform operations that will simplify and transform the equation until the variable is isolated and you know its value:

• Identify what is being done to the variable.
• Perform the inverse operation to both sides of the equation to balance it out.

In our example, the variable \(a\) is multiplied by 2. To isolate \(a\), you need to do the opposite, which means dividing. By applying these steps systematically, you will solve the equation.

Prealgebra Concepts

Prealgebra lays the groundwork for understanding algebra and higher-level mathematics. It includes basic concepts such as arithmetic operations, fractions, and simple equations. Recognizing properties like the multiplication property of equality is part of this foundational stage.
The multiplication property of equality states that if you multiply or divide both sides of an equation by the same nonzero number, the two sides remain equal. This is because both operations are performed equally on each side:

• The equation maintains balance, much like a scale.
• Operations do not change the equality but transform the appearance of the equation.

These principles help students move confidently into variable manipulation and algebraic expressions.

Variable Isolation

Variable isolation is the process of getting the variable you are solving for by itself on one side of the equation. This often involves using inverse operations to "undo" mathematical operations applied to the variable. In the example equation \(2a = 28\):

• The variable \(a\) is being multiplied by 2.
• To isolate \(a\), divide both sides by 2.
• The left side simplifies \(\frac{2a}{2}\) to \(a\), effectively eliminating the coefficient.

Once the variable stands alone, you have found its value. In our case, dividing 28 by 2 gives us \(a = 14\).
Understanding and mastering variable isolation is a crucial skill, as it opens the door to solving more complex algebraic equations.