Problem 1 - Prealgebra

Question

Solve each equation. Check your solution. $$3(g-3)=6$$

Step-by-Step Solution

Answer

The solution is \(g = 5\). This satisfies the equation when checked.

Step 1: Distribute the Constant

Multiply the constant on the outside of the parentheses by each term inside the parentheses. This gives us \(3 \cdot g - 3 \cdot 3\) which simplifies to \(3g - 9\). Thus, the equation becomes \(3g - 9 = 6\).

Step 2: Isolate the Variable Term

Add 9 to both sides of the equation to move the constant term to the other side. This gives us \(3g - 9 + 9 = 6 + 9\), resulting in \(3g = 15\).

Step 3: Solve for the Variable

Divide both sides of the equation by 3 to solve for \(g\). This results in \(\frac{3g}{3} = \frac{15}{3}\), simplifying to \(g = 5\).

Step 4: Check the Solution

Substitute \(g = 5\) back into the original equation and verify if both sides are equal. The original equation was \(3(g-3) = 6\). Substitute \(g\) to get \(3(5-3) = 6\) which simplifies to \(3 \cdot 2 = 6\), verifying that both sides equal 6, confirming \(g = 5\) is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify equations. It is used when a single term is multiplied by terms inside parentheses. By distributing this single term, you break it down into separate components that are easier to work with.For instance, in the equation \(3(g-3)=6\), you apply the distributive property to eliminate the parentheses:

• Multiply 3 by \(g\) to get \(3g\).
• Then multiply 3 by -3 to get -9.

This process transforms the equation into \(3g - 9 = 6\), making it easier to solve. The distributive property ensures that every term inside the parenthesis is accounted for, setting up the next steps for simplifying the equation.

Isolating Variables in Equations

Isolating the variable means getting the variable (in this case, \(g\)) all alone on one side of the equation. This process allows us to determine its value directly. We achieve this by performing operations that simplify one side of the equation, bringing us closer to the solution.After applying the distributive property, the equation simplified to \(3g - 9 = 6\). Our next step is to eliminate any constants next to the variable:

• Add 9 to both sides of the equation to cancel out the -9, resulting in \(3g = 15\).
• Next, divide both sides by 3 to isolate \(g\), giving us \(g = 5\).

By keeping the variable on one side, we've made the equation simple enough that the solution becomes clear. This step-by-step approach helps to prevent mistakes and ensures that every operation we perform maintains the balance of the equation.

Checking Solutions for Accuracy

Once you've found a solution, it's crucial to verify your work by checking the solution against the original equation. This step ensures that no errors were made during the solving process and confirms the correctness of your result.To check the solution \(g = 5\), substitute it back into the original equation \(3(g-3)=6\):

• Replace \(g\) with 5, leading to \(3(5-3)=6\).
• Simplify the expression inside the parentheses: \(5-3\) equals 2.
• Multiply 3 by 2, resulting in 6.

Since both sides of the equation are equal (\(6=6\)), our solution \(g = 5\) is correct. This method of checking solutions reinforces your understanding and boosts confidence in your algebraic skills.