Problem 33 - Algebra 1: Concepts and Skills

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: Identify the equation

We are given the equation \(n-5 = -9\). The goal is to solve for \(n\). This means, we need to isolate \(n\) on one side of the equation.

Step 2: Set \(n\) on one side of the equation

To do this, we add \(5\) to both sides of the equation to cancel the \(-5\) on the left side of the equation. This gives us \(n-5+5=-9+5\).

Step 3: Simplify both sides

When we simplify the equation, we obtain: \(n=-4\).

Key Concepts

Algebraic Equations

An algebraic equation is a mathematical statement that shows the relationship between two expressions that are set equal to each other. These equations are foundational in algebra and are vital for solving various mathematical problems. They commonly include variables (such as n), constants (like the number 5 in our exercise), and operational symbols (plus, minus, multiplication, etc.).

In the given exercise, n - 5 = -9, n is the variable we want to find, and the equation tells us that when 5 is subtracted from n, the result is -9. To solve algebraic equations, one must manipulate the equation using arithmetic operations while maintaining the equality so as to find the value of the unknown variable.

Isolating Variables

Isolating the variable, often the aim in solving algebraic equations, involves manipulating the equation to get the unknown variable by itself on one side of the equation. This isolation is critical for determining the value of the variable. In our exercise, we want to isolate n.

To do this, we perform the same operations on both sides of the equation to keep it balanced. Adding 5 to each side effectively cancels the -5 next to the n, following the principle of maintaining equilibrium. This step is crucial and is usually the first action taken to solve simple algebraic equations where the variable is part of a basic arithmetic operation.

Equation Simplification

Simplifying an equation makes it easier to solve by combining like terms and reducing expressions to their simplest form. The process typically involves basic arithmetic operations: addition, subtraction, multiplication, and division. When simplifying, it's essential to perform the same operation on both sides of the equation to keep it balanced.

In our exercise, the simplification step involves adding the numbers on each side of the equation, which gives us n = -4. This simple form is what we aim for when solving equations—having the variable on one side and the simplest expression, often a single number, on the other side.