Problem 10 - Prealgebra

Question

Solve each equation. Check your solution. $$3(2 g+4)=6(g+2)$$

Step-by-Step Solution

Answer

The solution is all real numbers.

Step 1: Expand Both Sides

Let's begin by expanding both sides of the equation. The left side is expanded as follows:\[ 3(2g + 4) = 6g + 12 \]For the right side:\[ 6(g + 2) = 6g + 12 \]Now the equation is:\[ 6g + 12 = 6g + 12 \]

Step 2: Simplify the Equation

Notice that both sides of the equation are identical:\[ 6g + 12 = 6g + 12 \]This means the equation is true for all values of \( g \). Thus, the solution is all real numbers.

Step 3: Verify the Solution

We can check our solution by choosing a specific value for \( g \) and substituting it back into the original equation. For example, let\( g = 0 \):Left side: \[ 3(2*0 + 4) = 3(4) = 12 \]Right side: \[ 6(0 + 2) = 6(2) = 12 \]Since both sides are equal, \( g = 0 \) satisfies the equation, confirming our solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are statements of equality that involve variables and constants combined with mathematical operations. The primary goal of solving an algebraic equation is to find the value or set of values of the variables that make the equation true. In this exercise, we are given the equation \(3(2g + 4) = 6(g + 2)\). This involves expanding and simplifying both sides to reveal any commonalities or contradictions.
An algebraic equation can have different types of solutions depending on its structure:

• A unique solution if it results in one specific value for the variable.
• No solution when the equation results in a contradiction.
• Infinite solutions, like in this case, when both sides simplify to the same expression, indicating the equation is always true.

To solve, you often start by expanding expressions and simplifying wherever possible to reveal a clearer form of the equation. Here, we expanded and realized both sides were identical, leading to infinite solutions.

Verification of Solutions

Verification is a critical step in equation solving. It ensures that the solution works for the original equation. It's like double-checking your work to prevent any errors. In this example, after simplifying, both sides of the equation were identical, meaning the equation holds true for any value of \( g \).
However, to fully verify, you can choose specific numbers for the variable and substitute them back into the original equation to check for consistency. Here, choosing \( g = 0 \) showed both sides of the equation equaled 12. This confirms that our interpretation of the equation having infinite solutions is accurate. Verification not only confirms the solution but also deepens our understanding of algebraic manipulation.

Mathematical Properties

Understanding mathematical properties is crucial in solving algebraic equations. Here, distributive and equality properties played significant roles.

• Distributive Property: This property allows us to expand expressions like \(3(2g + 4)\) to \(6g + 12\). It guides the initial steps of solving equations by distributing multiplication over addition inside parentheses.
• Properties of Equality: Once both sides simplify to the same expression, we utilize equality properties, acknowledging that the equation holds for all \( g \). This recognition comes from the principle that if \( a = a \), then any real number substitution keeps the equation true.

Recognizing and applying these properties allows us to transition smoothly through the steps, ensuring correctness and revealing the solution type (e.g., all real numbers in this example). Mastery over these concepts is foundational to efficiently solving and understanding algebraic equations.