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 32Our 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