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A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane. These equations are the simplest form of algebraic expressions and typically involve variables raised only to the power of one. Linear equations can take different forms, such as the standard form \(Ax + By = C\) and the slope-intercept form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

The exercise above involves a linear equation, \(5x - 4y = 20\), in two variables, \(x\) and \(y\). The goal is to express one variable in terms of the other. Understanding linear equations is crucial because they serve as the foundation for more complex algebraic operations. They allow us to model and solve real-world problems like budgeting, planning projects, and even predicting trends.

Solving for a variable involves isolating the variable on one side of the equation to find its value based on given information. The exercise has us solve for \(y\) in the equation \(5x - 4y = 20\). To find \(y\), we first need to isolate it, which means moving all terms with \(y\) to one side and everything else to the other.

### Step 1: Isolate the y-Term You begin by subtracting any term not involving \(y\) from both sides of the equation. Initially:```5x - 4y = 20```Subtract \(5x\) from both sides:```-4y = 20 - 5x```
### Step 2: Solve for y Now divide each term by \(-4\) to clear out any coefficient that isn't 1:```y = \frac{20 - 5x}{-4}```Simplify the equation:- First, \(\frac{20}{-4} = -5\)- Second, \(\frac{-5x}{-4} = \frac{5}{4}x\)Putting it all together, we get:\y = \frac{5}{4}x - 5This step-by-step isolation and division make it clear how the solution arrives at the expression for \(y\).

Equation manipulation involves using algebraic operations to rearrange and simplify equations. Mastering these techniques is essential for solving equations, especially when dealing with multiple variables or complex expressions.

In the example \(5x - 4y = 20\), manipulation techniques allowed us to express \(y\) in terms of \(x\). Let's break down the key manipulation moves:

• Adding/Subtracting: To isolate variables, terms can be moved across the equals sign by performing inverse operations, like subtracting \(5x\) to contain \(y\) on one side.
• Dividing/Multiplying: To solve for \(y\), dividing each term in the equation by \(-4\) ensures that the final result has \(y\) standing alone.
• Simplification: Reducing fractions or expressions, like changing \(-\frac{20}{4}\) to \(-5\), makes it clearer and easier to interpret.

Understanding these basics conveys the power and flexibility of equation manipulation, equipping you to handle a variety of algebraic situations with confidence.