Solving Simultaneous Equations Using Linear Algebra: A Step-by-Step Guide

Simultaneous equations are a fundamental concept in algebra, often encountered in various fields such as engineering, physics, and economics. In this article, we will demonstrate how to solve the following pair of simultaneous equations using linear algebra methods:

Equations:

2y 2x 12 4x 2y 14

Let's break down the solution into a series of detailed steps and explain the algebraic methods used.

Step 1: Simplify the Equations

Start by simplifying each equation so that they are in a standard form. The given equations can be simplified as follows:

Equation 1:

2y 2x 12

Dividing by 2:

2y 2x 12 becomes y x 6

Equation 2:

4x 2y 14

Dividing by 2:

4x 2y 14 becomes 2x y 7

Step 2: Isolate One Variable

Let's isolate 'y' in both equations:

Equation 1:

y x 6

Rearranging, we get:

y 6 - x

Equation 2:

2x y 7

Rearranging, we get:

y 7 - 2x

Step 3: Substitute and Solve for x

Now, we can set the two expressions for 'y' equal to each other:

6 - x 7 - 2x

Move all x terms to one side:

6 - 7 -2x x

-1 -x

x 1

Step 4: Substitute x Back into One of the Equations

Now that we have the value of x, let's substitute it back into one of the equations to find y. We'll use the simplified Equation 1:

y x 6

Substituting x 1:

y 1 6

y 6 - 1

y 5

Step 5: Check the Solution

To ensure the solution is correct, substitute both x and y values back into the original equations:

2y 2x 12:

2(5) 2(1) 10 2 12 (True)

4x 2y 14:

4(1) 2(5) 4 10 14 (True)

Conclusion

The solution to the system of simultaneous equations is x 1 and y 5. Therefore, the value of xy is:

xy 25

Additional Resources

For a deeper understanding of solving simultaneous equations and the methods used, refer to the following resources:

Math Is Fun - Systems of Linear Equations Khan Academy - Systems of Equations S.O.S. Math - Systems of Equations

Understanding these methods not only helps in solving complex problems but also reinforces fundamental algebraic skills.