(2a-1)x+3y-5=0 3x+(b-1)y-2=0

Jasonbradley

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Jun 16, 2024

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Solving a System of Linear Equations

This article will guide you through the process of solving a system of two linear equations with two unknowns, represented as:

(2a-1)x + 3y - 5 = 0 (3)x + (b-1)y - 2 = 0

Where a and b are constants.

Understanding the Problem

A system of linear equations represents two or more lines on a graph. The solution to the system is the point where these lines intersect, or the common point that satisfies both equations.

Methods of Solving

There are several methods for solving systems of linear equations, including:

1. Substitution:

   • Solve one equation for one variable in terms of the other.
   • Substitute this expression into the second equation.
   • Solve for the remaining variable.
   • Substitute the value back into either original equation to find the other variable.

2. Elimination:

   • Multiply one or both equations by constants so that the coefficients of one variable are opposites.
   • Add the equations together to eliminate one variable.
   • Solve for the remaining variable.
   • Substitute the value back into either original equation to find the other variable.

Step-by-Step Solution

Let's solve our system using the elimination method.

1. Multiply the first equation by (b-1) and the second equation by -3: (2a-1)(b-1)x + 3(b-1)y - 5(b-1) = 0 -9x - 3(b-1)y + 6 = 0

2. Add the two equations together:

   • [(2a-1)(b-1) - 9]x - 5(b-1) + 6 = 0

3. Solve for x:

   • x = [5(b-1) - 6] / [(2a-1)(b-1) - 9]

4. Substitute the value of x back into either original equation to solve for y:

   • If we use the first equation:
   • (2a-1)[(5(b-1) - 6) / [(2a-1)(b-1) - 9]] + 3y - 5 = 0
   • Simplify and solve for y.

Conclusion

By following these steps, you can solve the given system of linear equations for x and y, given the values of a and b. The solution represents the point where the two lines intersect, demonstrating the unique solution for the system. Remember to choose a solution method that best suits your problem and to carefully perform each step to arrive at the correct answer.