91Ó°ÊÓ

Linear equations are mathematical statements of equality involving variables raised to the first power.
They are among the most fundamental types of algebraic equations you will encounter. A general linear equation in one variable looks like this: \( ax + b = c \), where \( a, b, \) and \( c \) are constants, and \( x \) is the variable.
In the context of our exercise, \( \frac{y-8}{3} = 4 \) is a linear equation because \( y \) is a single variable and appears to the first power only.

Linear equations can be solved using simple operations such as addition, subtraction, multiplication, and division.

• These operations help us isolate the variable.
• Our goal is to have the variable by itself on one side of the equation.

Understanding these basic principles makes working with linear equations much more straightforward.

Algebraic manipulation is the process of rearranging and simplifying equations and expressions to solve for variables. In our exercise, this involves taking the linear equation \( \frac{y-8}{3} = 4 \) and performing operations to solve for \( y \).
Here's a breakdown of the steps involved:

• Eliminating fractions: To do this, multiply both sides of the equation by the denominator, which will eliminate the fraction. For our equation, multiplying by 3 simplifies it to \( y - 8 = 12 \).
• Isolating the variable: Next, add or subtract quantities as needed to both sides to get the variable by itself. In our case, we add 8 to both sides, resulting in \( y = 20 \).

Mastering these algebraic manipulations enhances your ability to solve not only linear equations but also more complex algebraic expressions.

The step-by-step approach to solving equations helps in breaking down the problem into manageable parts. For our exercise, this approach was applied as follows:

Step 1: Understanding the Equation

Identify that \( \frac{y-8}{3} = 4 \) is a linear equation, noting the fractional form and where the variable appears.

Step 2: Eliminate the Fraction

By multiplying both sides by 3, you remove the fraction, simplifying the equation to \( y - 8 = 12 \).
This step clears the path to easily isolating the variable in the next step.

Step 3: Isolate the Variable

Add 8 to both sides of the equation to solve for \( y \).
This leaves you with \( y = 20 \), clearly showcasing that \( y \) is solved.

This systematic method is essential for building confidence in handling more complex equations in the future.