Solving Algebraic Equations: A Step-by-Step Guide with Practical Examples

Solving Algebraic Equations: A Step-by-Step Guide with Practical Examples

Algebra is a fundamental branch of mathematics that involves solving equations to find unknown variables. In this article, we will walk through the process of solving a specific set of equations and understanding the logic behind each step. This guide includes practical examples and a detailed explanation of the techniques involved.

Understanding the Problem

We are given the following equations:

3x 7y 78 x 3y 32

Our goal is to solve for the value of 2x y. This guide will take you through the step-by-step process to achieve this.

Step-by-Step Solution

Step 1: Solving for x

The second equation is most straightforward to solve:

x 3y 32

Isolate x by subtracting 3y from both sides:

x 32 - 3y

Step 2: Substitution

Substitute the expression for x into the first equation:

3(32 - 3y) 7y 78

Expand and simplify:

96 - 9y 7y 78

96 - 2y 78

Subtract 96 from both sides:

-2y 78 - 96

-2y -18

Divide by -2 to solve for y:

y 9

Step 3: Solving for x

Now substitute y 9 back into the equation for x:

x 32 - 3(9)

x 32 - 27

x 5

Step 4: Calculating 2x y

Finally, substitute the values of x and y into the expression 2x y:

2x y 2(5) 9

2(5) 9 10 9

2x y 19

Verification

Let's verify our solution with the given equations:

Verification for the First Equation (3x 7y 78)

Substitute x 5 and y 9:

3(5) 7(9) 15 63 78

This checks out.

Verification for the Second Equation (x 3y 32)

Substitute x 5 and y 9:

5 3(9) 5 27 32

This also checks out.

Conclusion

Through this step-by-step guide, we have successfully solved the given equations and found that the value of 2x y is 19. This method, known as substitution, is a powerful tool for solving algebraic equations.

Algebra and equation solving are essential skills in many fields, including mathematics, engineering, and science. Regular practice and a solid understanding of the underlying concepts are key to success in algebra.

Resources for Further Practice:

Online algebra solvers and practice problems Math textbooks and workbooks on algebra YouTube tutorials and educational videos on algebra