(2a-1)x-3y=5 3x+(b-2)y=3

4 min read Jun 16, 2024

Solving a System of Linear Equations: A Step-by-Step Guide

This article will explore how to solve the system of linear equations:

(2a-1)x - 3y = 5 3x + (b-2)y = 3

We'll delve into the process of solving for x and y using the elimination method, with a focus on understanding the steps involved.

Understanding the System of Equations

This system of equations represents two lines in the xy-plane. Solving the system means finding the point of intersection of these lines, if they intersect. This point represents the solution where both equations are simultaneously satisfied.

Solving using Elimination Method

The elimination method works by manipulating the equations to eliminate one of the variables. Here's a detailed breakdown:

Choose a variable to eliminate: In this case, let's eliminate x.
Multiply equations to match coefficients: We need to find a common multiple for the coefficients of x in both equations.
- Multiply the first equation by 3: 3(2a-1)x - 9y = 15
- Multiply the second equation by (2a-1): 3(2a-1)x + (b-2)(2a-1)y = 3(2a-1)
Subtract the equations: Subtracting the second equation from the first will eliminate x:

- [9 + (b-2)(2a-1)]y = 15 - 3(2a-1)
Solve for y: Simplify the equation and solve for y:

y = [3(2a-1) - 15] / [9 + (b-2)(2a-1)]
Substitute y to find x: Substitute the value of y back into either of the original equations. Let's use the first equation:

(2a-1)x - 3[3(2a-1) - 15] / [9 + (b-2)(2a-1)] = 5
Solve for x: Simplify and solve the equation for x.

Important Notes:

No unique solution: If the coefficients of x and y are proportional in both equations, the lines are parallel and the system has no unique solution. This will result in a 0/0 situation when solving for y.
Infinite solutions: If both equations are equivalent, they represent the same line and have infinitely many solutions. This will result in a 0/0 situation when solving for y.

Conclusion

By carefully following the elimination method, we can successfully solve a system of linear equations for x and y. The process involves manipulating the equations to eliminate one variable, then substituting the value to find the other. It's important to be mindful of the cases where the system has no unique solution or infinite solutions.