Problem 13 - Algebra 2

Question

Solve each equation or formula for the specified variable. $$ 4 y-2 n=9, \text { for } y $$

Step-by-Step Solution

Answer

\( y = \frac{9 + 2n}{4} \)

Step 1: Isolate the term with the variable

To solve for the variable \( y \), we need to isolate the term containing \( y \). Begin by adding \( 2n \) to both sides of the equation: \[ 4y = 9 + 2n \].

Step 2: Solve for the variable

Now that we have \( 4y = 9 + 2n \), we need to solve for \( y \). Divide both sides of the equation by 4 to isolate \( y \): \[ y = \frac{9 + 2n}{4} \].

Key Concepts

Isolating Variables

When solving an equation, the first step is often to isolate the variable you're solving for. Isolating a variable means to get that variable alone on one side of the equation. This is done so that you can see exactly what the variable equals. It's a basic step whenever you're dealing with equations.
It's like peeling away layers until you find your variable.

For the given equation \( 4y - 2n = 9 \), we want to isolate \( y \). This means removing anything that isn't \( y \) from the side of the equation that \( y \) is on.

• Start by undoing addition or subtraction.
• Next, undo any multiplication or division.

By applying these steps to the equation, you add \( 2n \) to both sides, making the equation \( 4y = 9 + 2n \). Now, \( y \) is more approachable, with fewer distractions.

Linear Equations

Linear equations are one of the simplest types of equations you'll encounter in math, represented in the form \( ax + b = c \). They are called 'linear' because they graph as straight lines on a coordinate plane. Each variable in a linear equation is raised only to the first power.
If you have more than one variable, like in the equation \( 4y - 2n = 9 \), this is still a linear equation because both \( y \) and \( n \) are raised to the first power.

Solving linear equations generally involves a series of arithmetic steps to isolate the variable you are interested in. The goal is to manipulate the equation using operations like addition, subtraction, multiplication, and division.

• Each operation should help get closer to having the variable by itself.
• This logical flow makes linear equations a perfect introduction to understanding more complex algebraic ideas.

Once you understand how to work with linear equations, a lot of algebra becomes much more approachable.

Step-by-Step Solutions

Step-by-step solutions are vital in math as they break down complex problems into manageable parts. This process aids in understanding and reduces errors.
With a step-by-step approach, you see exactly how to handle each part of the equation. You learn not only to solve equations but also why each step matters.

Applying a step-by-step solution to our equation \( 4y - 2n = 9 \):

• First, identify the goal — isolating \( y \).
• Next, add \( 2n \) to both sides. This yields \( 4y = 9 + 2n \).
• Finally, divide everything by 4. This isolates \( y \) and gives you \( y = \frac{9 + 2n}{4} \).

This sequential approach helps clarify each stage, ensuring you understand how each operation leads to the next. By decomposing an equation into steps, it becomes more approachable and less intimidating.