Solving the Linear System: x y 1 and y 3x - 3

Understanding and solving linear systems is a fundamental aspect of algebra and is widely applied in various fields such as engineering, physics, and economics. This article will demonstrate how to solve the linear system x y 1 and y 3x - 3 step-by-step, providing a clear and detailed explanation of the process.

Introduction to Linear Systems

A linear system is a set of linear equations involving several variables. Solving such a system involves finding the values of the variables that satisfy all the equations simultaneously. In this case, we have the following system of equations:

x y 1 y 3x - 3

Step-by-Step Solution

Step 1: Express y in terms of x

The second equation provides a direct expression for y in terms of x. Therefore, we can easily substitute this into the first equation. However, since the first equation is already in a simpler form, we proceed directly to substitution:

y 1 - x

Step 2: Substitute the expression for y into the first equation

Next, substitute the expression for y from the first equation into the second equation:

1 - x  3x - 3

Step 3: Simplify the resulting equation

Moving all terms involving x to one side and constants to the other side of the equation, we get:

1 3 3x x

4 4x

Divide both sides by 4 to solve for x:

x  1

Step 4: Find the value of y

Substitute the value of x back into the expression for y obtained from the first equation:

y 1 - x

y 1 - 1

y 0

Verification

Finally, verify the solution by substituting the values of x and y into the original equations:

Equation 1: x y 1

1 0 1 checkmark

Equation 2: y 3x - 3

0 3(1) - 3 checkmark

Conclusion

The solution to the linear system x y 1 and y 3x - 3 is x 1 and y 0. This method of solving linear systems is a versatile and elegant approach that can be applied to a wide range of problems in mathematics and its applications.

Related Topics and Further Exploration

For those interested in delving deeper into linear systems and their applications, the following topics may be of interest:

Simultaneous Equations: Understanding how to solve systems with multiple equations. Linear Algebra: Expanding the concept of linear systems to more complex scenarios involving matrices. Applications in Real World: Exploring the practical applications of linear systems in fields such as economics, engineering, and physics.