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Solving linear equations means determining the value of the variable that makes the equation true. In our exercise, the equation is \[-x + 3 = 4x\]. The main idea here is to find out what number \(x\) must be to satisfy both sides of the equation.
Every term on each side of the equation influences the balance, much like a seesaw.
To "solve" means making sure the see-saw is level by having equivalent values both on the left and right.When you solve a linear equation, you're essentially trying to isolate the variable (\(x\) in this case) on one side. If done correctly, the rest of the equation turns into a simple numerical value on the other side, which represents the solution. It's important to follow a systematic approach to ensure that the equation remains balanced throughout.

• Start by simplifying each side of the equation if necessary, removing any parentheses, and combining like terms.
• Carefully perform mathematical operations to move variables to one side of the equation.
• Solve for the variable once the equation is balanced with the variable on one side and constants on the other.

Understanding these foundational steps will allow you to solve any linear equation methodically.

Isolating the variable is a critical step in solving linear equations. Let's take a deeper look into this concept using our example equation: \(-x + 3 = 4x\). To isolate the variable means to get \(x\) by itself on one side of the equation. Accomplishing this involves using different mathematical operations to rearrange the terms.In this exercise, we started by adding \(x\) to both sides to move all variable terms to one side.
This step transforms our equation into \[3 = 5x\]. Notice how adding \(x\) cancelled out the \(-x\) on the left side, neatly shifting \(x\) to one side.Once the variables are isolated on one side, you then adjust the equation to find a single instance of the variable. Here, dividing both sides by 5 simplifies it further to \[x = \frac{3}{5}\]. The aim of isolating variables is to simplify the equation down to a form where the variable equals a constant.
When you successfully isolate the variable, you directly uncover its value.

Mathematical operations like addition, subtraction, multiplication, and division are the tools that allow us to manipulate and solve equations. Let's delve into how these are applied in our example.Consider the given equation \(-x + 3 = 4x\).
Starting with operation, we add \(x\) to both sides. This operation is essential for balancing the equation, ensuring that our manipulations do not alter the inherent equality.

• Addition: Used to combine or shift terms within an equation. For example, we added \(x\) to each side.
• Subtraction: Helps in removing terms from one side to another side.
• Multiplication/Division: These are critical when you need to "undo" a multiplication or division in an equation. In our exercise, we divided by 5 to solve for \(x\).

The goal of using these operations is to methodically change the equation into a simpler form (ideally, \(variable = constant\)).
Each operation should always maintain the balance of the equation—whatever you do to one side, you must do to the other. This guarantees that the solution remains authentic, keeping the equation's truth intact.