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Chapter 1: Problem 25

Solve each equation, if possible. $$ 3+2 n=4 n+7 $$

Short Answer

Expert verified
The solution is \( n = -2 \).

Step by step solution

01

Isolate Variable Terms

First, move the variable terms on one side of the equation and the constant terms on the other side. Subtract 2n from both sides: \[ 3 + 2n - 2n = 4n + 7 - 2n \] This simplifies to: \[ 3 = 2n + 7 \]
02

Isolate Constant Terms

Next, isolate the constant term by subtracting 7 from both sides: \[ 3 - 7 = 2n + 7 - 7 \] This simplifies to: \[ -4 = 2n \]
03

Solve for the Variable

Divide both sides by the coefficient of the variable, which is 2: \[ \frac{-4}{2} = \frac{2n}{2} \] This simplifies to: \[ n = -2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving linear equations, one major goal is to isolate the variable. Think of it like peeling an onion, layer by layer. The aim is to have the variable (e.g., \( n \)) alone on one side of the equation.
Let's look at the example: \( 3 + 2n = 4n + 7 \). To isolate \( n \), we first eliminate terms with \( n \) on one side.
Subtracting \( 2n \) from both sides does the trick: \[ 3 + 2n - 2n = 4n + 7 - 2n \] This simplifies to: \[ 3 = 2n + 7 \]
Now \( n \) is only on the right side.
Constants
Constants in equations are numbers that do not change. They are the fixed values. In our example, \( 3 \) and \( 7 \) are constants.
After we isolated \( n \) on one side of the equation, we need to move the constants to the other side.
We do this by subtracting \( 7 \) from both sides: \[ 3 - 7 = 2n + 7 - 7 \] Which simplifies to: \[ -4 = 2n \]
Notice how we carefully balance the equation? Whatever we do to one side, we do to the other.
Simplifying Equations
Simplifying an equation means making it as straightforward as possible. We combine like terms and reduce everything to simple forms.
In the step where we moved constants, we ended up with: \[ -4 = 2n \]
This equation is simplified because it has only one term with \( n \) and one constant. No extra terms or fractions.
The simpler the equation, the easier it is to solve.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying an equation using basic algebra rules. Our goal is to get the variable by itself.
From the simplified equation \( -4 = 2n \), we divide both sides by 2 (the coefficient of \( n \)): \[ \frac{-4}{2} = \frac{2n}{2} \] This simplifies to: \[ n = -2 \]
By performing these operations, we ensure our solution is correct and balanced.

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