Solving Simultaneous Equations: A Detailed Guide with Step-by-Step Solutions

Simultaneous equations are a fundamental concept in algebra, and understanding how to solve them is crucial for many applications in mathematics and real-world problems. This guide will walk you through a detailed step-by-step process to find the values of x and y in the simultaneous equations 2x - 3y 0 and 3x 4y 3.

Introduction to Simultaneous Equations

Simultaneous equations involve two or more equations with multiple variables. The goal is to find values of these variables that satisfy all the given equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables, x and y.

The Given Equations

Let's consider the following simultaneous equations:

1. 2x - 3y 0

2. 3x 4y 3

Solving the System of Equations

We will use the substitution method to solve this system of equations.

Step 1: Express x in terms of y from the first equation

From the first equation:

2x - 3y  0

Solving for x:

2x  3yx  (3y)/2

Step 2: Substitute this expression for x into the second equation

Substitute x (3y)/2 into the second equation:

3x   4y  33((3y)/2)   4y  3(9y)/2   4y  3

Combine the terms:

(9y)/2   (8y)/2  3(17y)/2  3

Solving for y:

17y  6y  6/17

Step 3: Substitute the value of y back into the expression for x

Substitute y 6/17 back into the expression for x:

x  (3y)/2x  (3 cdot (6/17))/2x  (18/17)/2x  9/17

Thus, the solution to the system of equations is:

x  9/17y  6/17

Verification of the Solution

To verify the solution, substitute the values of x and y back into the original equations:

1. For the first equation, 2x - 3y 0: 2(9/17) - 3(6/17) 0 18/17 - 18/17 0 0 0 (True)

2. For the second equation, 3x 4y 3: 3(9/17) 4(6/17) 3 27/17 24/17 3 51/17 3 (True)

Conclusion

We have successfully solved the given system of simultaneous equations using the substitution method. The values of x and y are:

x  9/17y  6/17

This method is effective and can be applied to a wide variety of simultaneous equations. Understanding the steps involved will help you tackle more complex problems in the future.

Keywords: simultaneous equations, linear equations, substitution method