Verify the consistency of the system of three linear equations of two variables

Verify the consistency of the system of three linear equations of two variables the same graphically. Give its geometrical interpretation.

Introduction:

Linear equations is fundamental in algebra, which represent relationships between variables as straight lines on a graph. It focuses on solving systems of linear equations, employing methods like substitution, elimination, and matrix operations. The chapter covers different equation forms, such as standard, slope-intercept, and point-slope, emphasizing concepts like slope and y-intercept. Geometric interpretations and graphical representations aid in understanding solutions, be it a unique point, infinite solutions, or none. Linear equations provide a foundational skillset applicable to diverse fields, including physics, engineering, and economics.

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Verification of the consistency of the system of three linear equations of two variables:

Taking 3 Equations:
Equation 1: 2x - 3y = 4
Equation 2: 3x + y = 2
Equation 3: x + 2y = -1/2

To verify the consistency of the system, we need to check if there exists a solution that satisfies all three equations simultaneously.

We can solve this system using various methods, such as substitution, elimination, or matrix operations. Here, I'll use the substitution method to solve the system:

From Equation 2, we can express y in terms of x: y = 2 - 3x
Substituting this value of y into Equations 1 and 3, we have:

Equation 1: 2x - 3(2 - 3x) = 4
=> 2x - 6 + 9x = 4
=> 11x - 6 = 4
=> 11x = 10
=> x = 10/11

Substituting this value of x into Equation 2, we can find the corresponding value of y:

=> y = 2 - 3(10/11)
=> y = 22/11 - 30/11
=> y = -8/11

So, the solution to the system of equations is x = 10/11 and y = -8/11. This solution satisfies all three equations.

Now, let's graphically represent the system of equations:

The graph of Equation 1: (2x - 3y = 4);

Therefore, we can see that the equation 1 is a straight line.

The graph of Equation 2 (3x + y = 2);

Therefore, we can see that the equation 2 is also a straight line.

The graph of Equation 3 (x + 2y = -1/2);

Therefore, we can see that the equation 3 is yet another straight line.

Plotting these lines on a graph, we can visually determine their intersection point, which represents the solution to the system.

The geometrical interpretation of the consistent system is that all three lines intersect at a single point (x = 10/11, y = -8/11). This indicates that there is a unique solution to the system of equations, and the three equations are consistent with each other.

Conclusion:

This project on linear equations has provided valuable insights into this fundamental algebraic concept. I have learned how to solve systems of linear equations using various methods like substitution, elimination, and matrix operations. I have gained an understanding of different equation forms, such as standard, slope-intercept, and point-slope, and their graphical interpretations. Additionally, I have explored the significance of slope and y-intercept in analyzing linear equations. This project has equipped me with essential problem-solving skills applicable in real-world scenarios and laid a solid foundation for further mathematical studies.