Problem 50 - Prealgebra

Question

Solve each equation. Check your solution. $$3(n-1)=1.5(n+2)$$

Step-by-Step Solution

Answer

The solution is \( n = 4 \), verified by substituting back into the original equation.

Step 1: Distribute and Simplify Both Sides

First, we distribute the coefficients on both sides of the equation. For the left side, distribute the 3 into the parentheses: \[ 3(n-1) = 3n - 3 \] On the right side, distribute the 1.5 into the parentheses: \[ 1.5(n+2) = 1.5n + 3 \] Now, the equation looks like: \[ 3n - 3 = 1.5n + 3 \]

Step 2: Move Variables to One Side

To isolate the variable term, we subtract 1.5n from both sides of the equation: \[ 3n - 1.5n - 3 = 3 \] This simplifies to: \[ 1.5n - 3 = 3 \]

Step 3: Isolate the Constant

Next, move the constant term to the right side by adding 3 to both sides of the equation: \[ 1.5n - 3 + 3 = 3 + 3 \] This simplifies to: \[ 1.5n = 6 \]

Step 4: Solve for the Variable

Now, divide both sides by 1.5 to solve for \( n \): \[ n = \frac{6}{1.5} \] Simplifying, you find: \[ n = 4 \]

Step 5: Verify the Solution

Substitute \( n = 4 \) back into the original equation to verify the solution: \[ 3(n-1) = 1.5(n+2) \] This becomes: \[ 3(4-1) = 1.5(4+2) \] Simplify each side: \[ 3 \times 3 = 1.5 \times 6 \] \[ 9 = 9 \] Since both sides are equal, \( n = 4 \) is the correct solution.

Key Concepts

Variable Isolation

To solve an equation where a variable is unknown, the goal is to isolate this variable on one side of the equation. The process of variable isolation involves rearranging the equation so that the variable you want to solve for stands alone. Here's how you can achieve that:

• Identify the term containing the variable.
• Move other terms to the opposite side of the equation using addition or subtraction.
• Perform inverse operations, such as division or multiplication, to fully isolate the variable.

For example, in the equation from our exercise, after distributing and simplifying, we worked towards isolating the variable \( n \) by moving all terms without \( n \) to the right side. These careful steps ensure that only the variable remains on one side, making the equation easier to solve.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify and solve equations. It states that a term multiplied by a bracketed sum can be distributed to each term inside the brackets. The formula is \( a(b + c) = ab + ac \).
In our original exercise, we applied this property to both sides of the equation:

• On the left, \( 3(n-1) \) becomes \( 3n - 3 \).
• On the right, \( 1.5(n+2) \) becomes \( 1.5n + 3 \).

Understanding and correctly applying the distributive property is crucial for breaking down more complex equations into simpler forms. It helps you to reorganize the equation and move on to variable isolation with ease.

Verifying Solutions

After solving an equation, it's critical to verify that the solution you found is correct. Verifying a solution involves substituting the value back into the original equation to ensure both sides are equal. This step serves as a safety check to confirm that no errors were made during calculations.
In our exercise:

• We found \( n = 4 \) and substituted it back to check: \( 3(n-1) = 1.5(n+2) \).
• Plugging in, the equation becomes \( 3(4-1) = 1.5(4+2) \), simplifying to \( 9 = 9 \).

Since both sides equal, it assures us that \( n = 4 \) is indeed the correct solution. Always take the time to verify your solutions to build confidence and accuracy in problem-solving.