Problem 9 - Elementary Algebra

Question

Solve each equation. $$6(n+7)=8$$

Step-by-Step Solution

Answer

The solution is \(n = \frac{-17}{3}\).

Step 1: Distribute the 6

First, distribute the 6 in the equation \(6(n+7)=8\). This means multiplying 6 with both \(n\) and 7, which results in \(6n + 42 = 8\).

Step 2: Isolate the Variable Term

Next, we want to isolate the term with the variable \(n\). Subtract 42 from both sides of the equation \(6n + 42 = 8\). This simplifies to \(6n = 8 - 42\).

Step 3: Simplify the Right Side

Simplify the right side of the equation: \(8 - 42 = -34\). So the equation now is \(6n = -34\).

Step 4: Solve for n

To solve for \(n\), divide both sides of the equation \(6n = -34\) by 6. This gives \(n = \frac{-34}{6}\).

Step 5: Simplify the Fraction

Simplify \(\frac{-34}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. The simplified form is \(n = \frac{-17}{3}\).

Key Concepts

Distributive Property

The distributive property is an essential algebraic rule used to simplify expressions and solve equations. It states that for any numbers or variables, \(a(b + c) = ab + ac\). This property allows you to distribute a single term across terms inside a parenthesis.
For example, in the equation \(6(n + 7) = 8\), by applying the distributive property, we multiply 6 with both \(n\) and 7, resulting in \(6n + 42 = 8\). This step is crucial in turning a more complex expression into a simpler algebraic form.
Using this property effectively prepares the equation for further manipulation to isolate the variable, making the equation easier to solve.

Solving Linear Equations

Solving linear equations involves a series of operations to find the value of the unknown variable. The core idea is to isolate the variable on one side of the equation. Let’s go through this process step by step with the equation \(6n + 42 = 8\):

• Isolate the Variable Term: Subtract 42 from both sides so you can isolate \(6n\). This gives \(6n = 8 - 42\).


• Simplify: Calculate \(8 - 42\), which is \(-34\). Now the equation is \(6n = -34\).


• Solve for the Variable: Divide both sides by 6 to solve for \(n\), leading to \(n = \frac{-34}{6}\).

Linear equations are fundamental in algebra because they represent some of the simplest equations students analyze, fostering a clear understanding of algebraic manipulations.

Fractions in Algebra

Fractions often appear in algebraic solutions, especially when solving equations. They are simply division problems. In our equation, once we isolate \(n\), we had \(n = \frac{-34}{6}\).
To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 34 and 6 is 2. Dividing both by 2, we get \(n = \frac{-17}{3}\).
Working with fractions requires careful simplification:

• Find the GCD: Determine the largest number that can divide both the numerator and denominator.


• Divide Both Parts: Simplify the fraction by dividing both the numerator and denominator by the GCD.

Understanding fractions in algebra enables you to simplify and solve equations neatly, which is crucial for tackling more complex algebraic challenges.