Solving Linear Equations: An In-Depth Guide with Step-by-Step Solution Techniques

Linear equations are a fundamental part of algebra and are used extensively in various fields such as physics, engineering, and economics. Understanding how to solve these equations is crucial for tackling more complex mathematical problems. In this article, we will walk through the process of solving a specific linear equation, using clear and detailed steps. The equation in question is 4b 6 2 - b - 4.

Step-by-Step Solution of 4b 6 2 - b - 4

To solve the linear equation 4b 6 2 - b - 4, we will follow a structured approach:

Step 01 - Simplify the Right Side of the Equation

The right side of the equation 2 - b - 4 simplifies to 2 - 4 - b -2 - b.

So, the equation becomes:

4b   6  -2 - b

Step 02 - Add b to Both Sides to Combine Like Terms

Adding b to both sides of the equation, we get:

4b   b   6  -2

This simplifies to:

5b   6  -2

Step 03 - Isolate the b Variable by Subtracting 6 from Both Sides

Subtracting 6 from both sides, we obtain:

5b  -2 - 6

This simplifies to:

5b  -8

Step 04 - Solve for b by Dividing Both Sides by 5

Dividing both sides by 5, we get:

b  -frac{8}{5}

Thus, the solution to the equation is:

b  -frac{8}{5}

Verification of the Solution

To verify our solution, we can substitute b -frac{8}{5} back into the original equation and check if it holds true:

4 cdot -frac{8}{5}   6 approx 2 - left(-frac{8}{5}right) - 4

Calculating the left side:

-frac{32}{5}   6  -frac{32}{5}   frac{30}{5}  -frac{2}{5}

Calculating the right side:

2   frac{8}{5} - 4  -frac{2}{5}

Since both sides are equal, the solution b -frac{8}{5} is correct.

Rules and Techniques for Solving Linear Equations

When solving linear equations, it's important to follow certain rules and techniques that ensure accuracy:

Rule 1: Bring Constants and Variables to Opposite Sides

Move all terms containing the variables to one side of the equation and all constants to the other side.

Rule 2: Change the Sign of Terms Transposed to the Other Side of the Equation

Whenever a term is moved from one side to the other, change its sign.

By applying these rules and following the steps above, we can solve any linear equation systematically.

Linear equations provide a strong foundation for understanding more advanced mathematical concepts. Mastering the techniques for solving these equations can significantly improve your problem-solving skills in various fields.