(2m-1)x+3y-5=0 3x+(n-1)y-2=0

4 min read Jun 16, 2024

Solving Systems of Linear Equations: A Step-by-Step Guide

This article will guide you through the process of solving a system of two linear equations in two unknowns, specifically focusing on the system:

(2m-1)x + 3y - 5 = 0 3x + (n-1)y - 2 = 0

Where 'm' and 'n' are constants.

Understanding Systems of Linear Equations

A system of linear equations is a set of two or more equations that share the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. In our case, we need to find values for 'x' and 'y' that make both equations true.

Methods for Solving

There are several methods for solving systems of linear equations:

Substitution: Solve one equation for one variable, then substitute that expression into the other equation.
Elimination: Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations together.

We'll demonstrate both methods for our system.

Solving by Substitution

Solve one equation for one variable. Let's solve the first equation for 'x': (2m-1)x = 5 - 3y x = (5 - 3y) / (2m-1)
Substitute the expression for 'x' into the second equation. 3 * [(5 - 3y) / (2m-1)] + (n-1)y - 2 = 0
Simplify and solve for 'y'. (15 - 9y) / (2m-1) + (n-1)y - 2 = 0 (15 - 9y) + (n-1)(2m-1)y - 2(2m-1) = 0 y [ (n-1)(2m-1) - 9 ] = 4m - 17 y = (4m - 17) / [(n-1)(2m-1) - 9]
Substitute the value of 'y' back into either of the original equations to solve for 'x'.

Solving by Elimination

Multiply one or both equations by constants so that the coefficients of one variable are opposites. Let's eliminate 'x'. Multiply the first equation by 3 and the second equation by -(2m-1): 3[(2m-1)x + 3y - 5 = 0] -(2m-1)[3x + (n-1)y - 2 = 0]
Add the equations together. 6mx - 3x + 9y - 15 = 0 -6mx + (2m-1)(n-1)y + 2(2m-1) = 0 [ (2m-1)(n-1) + 9 ] y = 15 - 4m
Solve for 'y'. y = (15 - 4m) / [ (2m-1)(n-1) + 9 ]
Substitute the value of 'y' back into either of the original equations to solve for 'x'.

Conclusion

By using either substitution or elimination, you can solve for 'x' and 'y' in the given system of equations, depending on the values of 'm' and 'n'. Remember to be careful with the algebraic manipulations and to check your solutions by substituting them back into the original equations.