Problem 11 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$3 a=48$$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 16\).

Step 1: Understand the Problem

We need to solve the equation \(3a = 48\) for the variable \(a\). This means we want to find the value of \(a\) such that when it is multiplied by 3, the result is 48.

Step 2: Identify the Multiplication Property of Equality

The multiplication property of equality states that if you multiply or divide both sides of an equation by the same non-zero number, the equality remains true. In this case, we will use division, which is the inverse of multiplication, to isolate \(a\).

Step 3: Isolate the Variable 'a'

Divide both sides of the equation by the coefficient of \(a\), which is 3. This will give us \(a\) alone on one side of the equation:\[\frac{3a}{3} = \frac{48}{3}\] This simplifies to:\[ a = 16 \]

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(a = 16\) back into the original equation. Calculate 3 times 16 to check it equals 48:\[ 3 \times 16 = 48 \]Since this is true, \(a = 16\) is indeed the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, crucial for finding the unknown value that satisfies a given mathematical statement. When faced with an equation like \( 3a = 48 \), our goal is to determine the value of \( a \) that makes the equation true. This is achieved through systematic steps, ensuring that any operation performed on one side of the equation is mirrored on the other side to maintain balance. In our example, the operations used to solve for \( a \) involve division, specifically using the multiplication property of equality. This property helps maintain the integrity of the equation while isolating the variable. Through understanding and employing these steps, you can solve equations confidently and accurately.

Prealgebra Concepts

Prealgebra serves as a crucial precursor to more advanced algebraic studies, focusing on foundational skills like arithmetic operations, variables, and simple equations. Understanding core prealgebra concepts is vital for anyone beginning their journey into algebra.

Here are a few key concepts that are often used:

• Arithmetic Operations: Include addition, subtraction, multiplication, and division. These operations can be applied to numbers and variables alike.
• Variables: Symbols (often letters) that represent unknown values. In our equation \( 3a = 48 \), \( a \) is the variable.
• Inverse Operations: Used to solve equations. Multiplication and division or addition and subtraction are considered inverse operations, as they reverse each other's effects.

Emphasizing these topics helps to build a solid foundation in mathematics, making more complex topics easier to grasp later on.

Mathematical Reasoning

Mathematical reasoning is the logical thinking process that is key to solving problems effectively. It involves understanding the problem, selecting appropriate methods to solve it, and then verifying the results.

When solving an equation like \( 3a = 48 \), mathematical reasoning helps break down the problem into manageable steps:

• Identify the operation tying the variable to the constant (in this case, multiplication).
• Apply an inverse operation (division here) to both sides to isolate the variable.
• Verify the solution by substitution, ensuring the equation balances.

This kind of logical approach is fundamental not just in mathematics, but in everyday problem solving and critical thinking. By developing strong mathematical reasoning skills, students can tackle a wide array of problems with confidence.