Problem 47 - Algebra A Combined Function

Question

Solve. $$ n+4=3.6 $$

Step-by-Step Solution

Answer

\( n = -0.4 \)

Step 1: Isolate the variable

To solve for the variable \(n\), we need to get \(n\) by itself on one side of the equation. The equation is given as \(n + 4 = 3.6\). To isolate \(n\), subtract 4 from both sides of the equation.

Step 2: Perform the arithmetic

Subtract 4 from both sides: \[ n + 4 - 4 = 3.6 - 4 \] This simplifies to: \[ n = -0.4 \] The arithmetic involves subtracting 4 from 3.6, which gives \(-0.4\).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in mathematics that involves finding the value of the unknown variable that makes the equation true. An equation is like a balance scale where both sides represent equal amounts, and the goal is to find the value that keeps this balance intact. In our example, we have a simple equation:

• \(n + 4 = 3.6\)

Here, the equation involves a variable \(n\) whose value we need to determine. To solve an equation, we perform a series of operations to isolate the variable on one side of the equation. This helps us find the exact value of the unknown variable. Remember, whatever we do to one side of the equation, we must do to the other to maintain equality.
Understanding the basics of solving equations is crucial as it forms the foundation for more complex mathematical operations and problem-solving scenarios.

Isolation of Variables

Isolating the variable means manipulating the equation to have the unknown variable \(n\) by itself on one side, generally the left side. This allows us to see what \(n\) equals directly. In the equation \(n + 4 = 3.6\), isolating \(n\) involves getting rid of the \(+4\) that accompanies it.
We achieve this by performing an operation that cancels out the +4. Specifically, subtraction is used here. By subtracting 4 from both sides of the equation, we can move towards expressing \(n\) solely on one side:

• \(n + 4 - 4 = 3.6 - 4\)

This simplifies to \(n = -0.4\). The principle of keeping equations balanced is critical. Every step in removing other components around the variable must be mirrored on the equation's opposite side.
Mastering the isolation of variables is an essential skill for tackling various algebraic equations and translates into other areas of algebra, such as solving systems of equations.

Arithmetic Operations

Arithmetic operations involve basic mathematical procedures including addition, subtraction, multiplication, and division. These operations are pivotal in the manipulation of equations. In our example, we predominantly use subtraction.
The equation \(n + 4 = 3.6\) requires us to eliminate the 4 connected to \(n\). This is achieved by subtracting 4 from both sides of the equation:

• \(3.6 - 4 = -0.4\)

Subtraction modifies the equation by reducing the positive 4 on the left to zero, effectively isolating \(n\). On the right, it changes the total from 3.6 to -0.4. It is these precise manipulations that allow us to simplify the equation and discover the variable's value.
Arithmetic operations provide the practical calculations necessary for solving equations. They empower you to shift expressions around an equation and arrive at simple solutions.