Problem 36 - Algebra 1: Concepts and Skills

Question

Solve the equation by distributing the fraction first. $$ \frac{2}{3}(9 n-6)=4(n+1) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Fraction

Start by distributing the fraction \(\frac{2}{3}\) through the parentheses on the left side of the equation. This calculation gives us \(\frac{2}{3} * 9n - \frac{2}{3} * 6 = 6n - 4\).

Step 2: Simplify the Equation

Next, distribute the 4 on the right side of the equation across the parentheses, which provides us with \(4n + 4\). Thus we get the new equation \(6n - 4 = 4n + 4\).

Step 3: Isolate the Variable

Subtract 4n from both sides to start isolating the variable n. This gives us \(2n - 4 = 4 \). Then, add 4 to both sides to completely isolate the term with n in it: \(2n = 8\).

Step 4: Solve for n

Finally, divide both sides by 2 to solve for n. Hence, the solution to the equation is \(n = 4\).

Key Concepts

Distributing Fractions

When dealing with algebraic expressions, distributing fractions across terms inside parentheses is an essential skill to master. It involves multiplying the fraction by each term inside the parentheses. For instance, let's take the fraction \frac{2}{3} and distribute it over the terms in the parentheses \frac{2}{3}(9n - 6)\. This means we multiply \frac{2}{3}\ by each term separately: \(9n\) and \(-6\). As a result, we get \(6n - 4\), since \(\frac{2}{3} \times 9n \) simplifies to \(6n\) and \(\frac{2}{3} \times -6\) simplifies to \(-4\).

To make this process easier, keep these tips in mind:

• Use the numerator of the fraction to multiply each term inside the parentheses.
• Treat the denominator as a division to be applied after multiplication.
• Always simplify the products if possible.

Simplifying Equations

Simplifying equations is a critical step in solving them. It involves combining like terms and reducing expressions to their simplest form. In our exercise, once we've distributed the fractions, we end up with an equation that can be further simplified. For example, after distribution, we have the equation \(6n - 4 = 4n + 4\). To simplify, we identify and combine like terms, which in algebra are terms that have the exact variable raised to the same power.

Some simplifying guidelines include:

• Combine like terms on each side of the equation first.
• Look for opportunities to factor expressions or cancel out terms.
• Rearrange terms so that all variables are on one side and constants on the other.

By constantly practicing these steps, the process of simplifying equations will become an intuitive part of your algebra toolkit.

Isolating Variables

To solve an equation for a variable means to isolate the variable on one side of the equation. The goal is to get the variable by itself, with a coefficient of 1. In the process of solving \(6n - 4 = 4n + 4\), we aim to isolate \(n\). We do this by getting rid of terms that do not contain \(n\) from the side of the equation where \(n\) is present.

Here's how to isolate a variable effectively:

• Subtract or add terms to both sides to move all terms with the variable to one side and constants to the other.
• Perform the same operation (addition, subtraction, multiplication, or division) on both sides to maintain balance.
• Simplify after each operation to keep track of changes you’ve made.
• Continue until the variable is by itself and the equation is balanced.

By following this process, you will end up with an equation in the form of \(n =\) some number, which is the solution! The satisfaction of clearing the clutter and isolating that variable cannot be overstated—it's one of the joys of algebra!