Problem 26 - Algebra 1

Question

Solve the equation. $$n-5=-9$$

Step-by-Step Solution

Answer

The solution to the equation \(n - 5 = -9\) is \(n = -4\).

Step 1: Isolate the variable

In the given equation \(n - 5 = -9\), add 5 to both sides to isolate the variable 'n'. That gives \(n = -9 + 5\)

Step 2: Solve for 'n'

Solving the addition in the previous step, \(n = -4\). Hence, the solution to the equation \(n - 5 = -9\) is \(n = -4\).

Key Concepts

Understanding Variables

In the world of algebra, variables are fundamental. A variable is essentially a symbol, often a letter, used to represent an unknown number. In the equation we are solving, which is \(n - 5 = -9\), the variable is \(n\). This means \(n\) stands for some number that we need to discover.

Variables are placeholders for values that might change, or those that aren't yet known. This is powerful in mathematics, as it allows us to form general expressions or equations that solve problems for numerous scenarios, not just one specific instance.

The goal in solving an equation is to determine what value the variable should take to make the equation true. This involves performing operations like addition or subtraction to both sides of the equation while maintaining balance.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. To solve an equation like \(n - 5 = -9\), we need to isolate the variable \(n\).

• First, identify what operations have been performed on the variable. In this example, 5 has been subtracted from \(n\).
• The next step is to do the opposite operation to both sides of the equation to bring \(n\) alone on one side. Since we have \(n - 5 = -9\), adding 5 balances the equation because it counteracts the subtraction.
• Executing this operation yields: \(n = -9 + 5\).

At this point, we have simplified the equation to a basic arithmetic problem, and you're one step closer to finding \(n\). It’s vital to ensure that every step maintains the equality of both sides of the equation, making the solution valid.

Mastering Algebraic Manipulation

Algebraic manipulation is at the heart of solving equations. This skill involves strategically adding, subtracting, multiplying, or dividing both sides of an equation to keep it balanced while progressing towards isolating the variable.

In our example problem \(n - 5 = -9\), the equation requires adding 5 to both sides. This is a direct application of the algebraic rule to 'do unto one side what you do to the other'.

• This principle ensures that we treat the equation equally and move steadily towards an answer. By adding 5, we effectively cancel out the \(-5\) on the left and thus isolate \(n\).
• This brings us to the simple expression: \(n = -4\). The manipulation results in simplifying the equation, which makes solving accessible and straightforward.

Algebraic manipulation doesn't just make solving equations possible; it empowers you to handle even complex expressions systematically without losing balance on either side of the equation.