Problem 13 - Prealgebra

Question

Solve each equation. Check your solution. $$6 n-18=4(n+2.1)$$

Step-by-Step Solution

Answer

The solution is \(n = 13.2\).

Step 1: Distribute the 4 on the Right

To solve the equation \(6n - 18 = 4(n + 2.1)\), first distribute the 4 to both terms inside the parentheses on the right side. This gives you: \(6n - 18 = 4n + 8.4\).

Step 2: Move Variables to One Side

Subtract \(4n\) from both sides of the equation to get the terms with \(n\) on one side: \(6n - 4n - 18 = 8.4\), simplifying to \(2n - 18 = 8.4\).

Step 3: Isolate the Variable Term

Add 18 to both sides to isolate the \(n\) term: \(2n - 18 + 18 = 8.4 + 18\). This simplifies to \(2n = 26.4\).

Step 4: Solve for n

Divide both sides by 2 to solve for \(n\): \(n = \frac{26.4}{2}\), resulting in \(n = 13.2\).

Step 5: Check the Solution

Substitute \(n = 13.2\) back into the original equation to check if it holds true: \(6(13.2) - 18 = 4(13.2 + 2.1)\). Calculate each side separately: \(6(13.2) - 18 = 79.2 - 18 = 61.2\) and \(4(13.2 + 2.1) = 4(15.3) = 61.2\). Both sides equal, confirming the solution is correct.

Key Concepts

Equation Solving

Equation solving is a process of finding the value of a variable that makes an equation true. At its core, it involves ensuring that both sides of the equation balance or hold the same value.
For example, in the equation \(6n - 18 = 4(n + 2.1)\), our goal is to determine the value of \(n\) that makes both sides equal.
To solve such equations, one performs a series of algebraic manipulations: simplifying expressions, rearranging terms, and undoing operations through inverse operations, like adding to undo subtraction or dividing to undo multiplication.

Distributive Property

The distributive property is a fundamental algebraic property used often while solving equations. It states that multiplication distributed over addition acts as: \(a(b + c) = ab + ac\).
This property is crucial when dealing with terms inside parentheses. Especially when you have an equation like \(6n - 18 = 4(n + 2.1)\).

• Here, you distribute the 4 across each term inside the parentheses: distribute it to \(n\) and then to 2.1.
• This simplifies the equation to \(4n + 8.4\).

Applying this step correctly is essential as it helps in clearing out the parentheses, making it simpler to isolate and solve for the variable.

Isolate the Variable

Isolating the variable means manipulating the equation in such a way that one side contains only the variable you are solving for.
In our example, once we applied the distributive property resulting in \(6n - 18 = 4n + 8.4\), the next steps are crucial:

• First, get all \(n\) terms on one side by subtracting \(4n\) from both sides, simplifying to \(2n - 18 = 8.4\).
• Then, use addition to move the constant term (-18) to the other side by adding 18, resulting in \(2n = 26.4\).
• Lastly, divide by the coefficient of \(n\), which is 2, to finally solve for \(n\), getting \(n = 13.2\).

All these steps help in simplifying the equation to a point where you can clearly see the variable isolated.

Check the Solution

Checking the solution is an essential step in solving equations. It confirms whether the value obtained for the variable satisfies the original equation.
After finding \(n = 13.2\), substitute it back into the original equation: \(6(13.2) - 18 = 4(13.2 + 2.1)\).

• Calculate each side separately to ensure they are equal: the left side calculates to \(61.2\) and so does the right side.
• The fact that both sides equal confirms our solution \(n = 13.2\) is indeed correct.

This step is vital, as it validates the entire solution process and ensures no errors were made during the algebraic manipulations.