Problem 17 - Elementary and Intermediate Algebra

Question

Solve each equation. Check the result. $$ 3(x+1)=15 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Distribute the 3

Start by distributing the 3 to both terms inside the parentheses. This gives us the equation \(3 \cdot x + 3 \cdot 1 = 15\). Simplifying this, we get \(3x + 3 = 15\).

Step 2: Subtract 3 from both sides

Next, we need to isolate the term with the variable. Subtract 3 from both sides of the equation: \(3x + 3 - 3 = 15 - 3\). This simplifies to \(3x = 12\).

Step 3: Divide both sides by 3

To solve for \(x\), divide both sides of the equation by 3: \(\frac{3x}{3} = \frac{12}{3}\). Simplifying this gives \(x = 4\).

Step 4: Check the solution

Verify the solution by substituting \(x = 4\) back into the original equation: \(3(4 + 1) = 15\). Calculate inside the parentheses first to get \(3 \times 5 = 15\), which is true. Therefore, \(x = 4\) is the correct solution.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply each term inside a set of parentheses by a number outside of it. In our example, the equation is presented as \(3(x+1) = 15\). Using the distributive property, you multiply 3 by each term inside the parentheses separately. This results in \(3 \cdot x + 3 \cdot 1 = 15\). Simplifying each multiplication, we get \(3x + 3 = 15\). Break it down further:

• The number 3 is distributed to the \(x\) term, leading to \(3x\).
• The number 3 is also distributed to the 1, leading to \(3\).

This method is practical as it helps to eliminate parentheses, making it easier to isolate the variable later on. Think of the distributive property as dealing cards to each term inside the parentheses!
This step sets the stage for finding the solution.

Isolating the Variable

Once you've successfully applied the distributive property, the next step is to isolate the variable. "Isolating the variable" means getting \(x\) by itself on one side of the equation. With our equation \(3x + 3 = 15\), you start this process by removing constants from the side with the variable.

• Subtract 3 from both sides to balance the equation, resulting in \(3x = 12\).
• Next, divide both sides by 3 to solve for \(x\). This gives \(\frac{3x}{3} = \frac{12}{3}\), simplifying to \(x = 4\).

Each operation you perform should keep the equation balanced. Think of it as a delicate scale, where both sides need to remain equal. Break down each step without rushing, ensuring every operation targets the ultimate goal: getting \(x\) alone.

Checking Solutions

Checking your solution is an important and often overlooked step in solving equations. It's your way of verifying that your solution makes the original equation true. After finding \(x = 4\), substitute it back into the original equation to confirm it's correct.

• Replace \(x\) with 4 in the original equation, leading to \(3(4 + 1) = 15\).
• Simplify the expression inside the parentheses: \(4 + 1 = 5\).
• Multiply: \(3 \times 5 = 15\).

The left side equals the right side, so the solution is correct. This step acts like a final check in a multi-step puzzle, ensuring that all pieces actually fit together. Performing this verification gives you confidence in your work and reassures you that no mistakes were made while solving the equation.