Understanding and Solving Algebraic Equations: A Comprehensive Guide

Algebra is the fundamental branch of mathematics that uses letters and symbols to represent numbers and quantities in formulae and equations. Solving algebraic equations is a critical skill that is widely used in various fields, from engineering to finance. This article will guide you through solving a linear equation step by step, ensuring you understand each concept before moving to the next. We will use examples to illustrate each method, providing a clear and detailed explanation.

Solving the Equation 97x 3 100

To solve the equation 97x 3 100 for the variable x, follow these steps:

Isolate the variable term: Start by moving the constant term (3) to the right side of the equation. This is achieved by subtracting 3 from both sides of the equation. 97x 3 - 3 100 - 3

This simplifies to:

97x 97

Isolate the variable: Divide both sides of the equation by the coefficient of x (97), which is 97, to get the value of x. 97x / 97 97 / 97

This simplifies to:

x 1

Verify the solution: Substitute the value of x back into the original equation to ensure it is correct. 97(1) 3 100

This simplifies to:

97 3 100

100 100

Therefore, x 1 is indeed the solution to the equation 97x 3 100.

Further Examples

Let's explore a few more examples to further illustrate the concept of solving equations.

Example: Solve 5x 10

To solve 5x 10:

Isolate the variable: Divide both sides by 5. 5x / 5 10 / 5

This simplifies to:

x 2

This is a straightforward example, but it reinforces the concept of isolating the variable.

Example: Solve 5x - x 100

To solve the equation 5x - x 100:

Combine like terms: Simplify the left side of the equation by combining like terms. 4x 100

No further steps are necessary as we have already simplified the equation.

Isolate the variable: Divide both sides by 4. 4x / 4 100 / 4

This simplifies to:

x 25

Thus, the solution to the equation is x 25.

Example: Solve 97x 3 100

As previously discussed, solving 97x 3 100 involves:

Isolate the variable term: Subtract 3 from both sides. 97x 97

Then, isolate the variable by dividing both sides by 97.

x 1

Finally, verify the solution by substituting x 1 back into the original equation.

The value x 1 satisfies the equation, demonstrating the method's accuracy.

Conclusion

Solving equations is a fundamental skill in mathematics, applicable across various disciplines. The examples provided in this article illustrate the process step by step, emphasizing the importance of maintaining balance on both sides of the equation. Whether you are solving a simple linear equation or a more complex algebraic expression, the key principles remain the same: isolate the variable and verify the solution.

Keywords

algebraic equations, solving equations, linear equations