How to Solve a Pure Equation

Equations are an essential part of mathematics, serving as a tool for solving problems and understanding relationships between variables. A pure equation, also known as a simplified equation, is an equation that contains only one variable. Solving pure equations requires a systematic approach to isolate the variable and determine its value. In this article, we will explore step-by-step instructions on how to solve a pure equation.

Step 1: Understand the Equation
The first step in solving a pure equation is to ensure that you fully understand what the equation is asking for and what it represents. Read the equation carefully, identify the variable, and determine the desired value or condition.

Step 2: Simplify the Equation
Check if the equation is already in its simplest form or if any simplification can be made. Remove any unnecessary terms or constants to bring the equation to its most basic structure. Remember, the goal is to have only one variable on one side of the equation.

Step 3: Apply Inverse Operations
The next step is to apply inverse operations to isolate the variable. Inverse operations are mathematical operations that undo one another. For example, to perform the inverse operation of addition, you subtract, and vice versa. The four basic inverse operations are addition/subtraction, multiplication/division, exponentiation, and taking square roots.

Step 4: Perform the Same Operation on Both Sides
When applying inverse operations, it is crucial to maintain the equation’s balance by performing the same operation on both sides. This ensures that the equation remains true. Remember, you want to isolate the variable and determine its unique value.

Step 5: Simplify Further if Necessary
After applying inverse operations, simplify the equation further if needed. Combine like terms, cancel out any common factors, or rearrange terms to bring the equation to its simplest form.

Step 6: Check the Solution
The final step is to check if the solution obtained satisfies the original equation. Substitute the found value back into the original equation and verify if both sides are equal. If they are, then the solution is valid; otherwise, reassess the steps taken to find any errors.

Example:
Let’s solve the pure equation: 3x + 7 = 22

Step 1: Understand the equation: This equation asks us to find the value of x.

Step 2: Simplify the equation: The equation is already in its simplest form.

Step 3: Apply inverse operations: To isolate x, we need to subtract 7 from both sides of the equation.

3x + 7 – 7 = 22 – 7
3x = 15

Step 4: Perform the same operation on both sides: Divide both sides of the equation by 3.

(3x) / 3 = 15 / 3
x = 5

Step 5: Simplify further if necessary: The equation is already simplified.

Step 6: Check the solution: Substitute x = 5 back into the original equation.

3(5) + 7 = 22
15 + 7 = 22
22 = 22

Since both sides of the equation are equal, the solution x = 5 is valid.

In conclusion, solving a pure equation involves a systematic approach of understanding the equation, applying inverse operations, and simplifying the equation step by step. By following these steps, you can successfully isolate the variable and determine its value. Pure equations serve as a fundamental concept in mathematics and play a crucial role in problem-solving and understanding the relationships between variables.

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