Problem 8 - Elementary Algebra

Question

Solve each equation. $$3(n+4)=7$$

Step-by-Step Solution

Answer

The solution is \(n = -\frac{5}{3}\).

Step 1: Distribute the 3

Start by applying the distributive property to the term \(3(n + 4)\). This simplifies to \(3n + 12\). The equation becomes \(3n + 12 = 7\).

Step 2: Subtract 12 from Both Sides

To isolate the term with \(n\), subtract 12 from both sides of the equation: \(3n + 12 - 12 = 7 - 12\). This simplifies to \(3n = -5\).

Step 3: Divide by 3

Now, divide both sides by 3 to solve for \(n\): \(\frac{3n}{3} = \frac{-5}{3}\). This simplifies to \(n = -\frac{5}{3}\).

Key Concepts

Distributive Property

The distributive property is an essential concept when solving equations that involve multiplication distributed over addition or subtraction.
For example, in the expression \(3(n + 4)\), the number 3 needs to be distributed across both \(n\) and 4.
This transforms the expression into \(3n + 12\). Each term within the parentheses gets multiplied by 3. Here is how it works:

• Multiply 3 by \(n\) to get \(3n\).
• Multiply 3 by 4 to get 12.

Distributing helps to simplify expressions, allowing us to work with them more easily. It's like breaking down a task into smaller, manageable pieces.

Isolation of Variables

Isolation of variables is a critical step in solving equations. The goal is to get the variable on one side of the equation and the numbers on the other.
In our problem, after applying the distributive property, we have \(3n + 12 = 7\). The next goal is to have \(n\) alone. This usually involves:

• Adding or subtracting terms on both sides to remove the constant term from one side. For this example, we subtract 12 from both sides to get \(3n = -5\).

This approach allows us to break down the equation, moving closer to identifying the value of \(n\). It’s a systematic process of untangling the equation, so the mystery number (the variable) can stand out on its own.

Equation Simplification

Equation simplification involves making an equation easier to solve.
It often means reducing more complex expressions into simpler ones. In our case, after isolating the variable, we simplified \(3n = -5\) by dividing each side by 3.
This gives us \(n = -\frac{5}{3}\), which is the simplest form of the equation.
Here's the process:

• To simplify \(3n = -5\), divide both sides by 3. This keeps the equation balanced.
• This operation yields the final solution where \(n = -\frac{5}{3}\).

Simplification is like tidying up an equation. This makes it easier to see the solution clearly and ensure that there are no calculations left unanalyzed.