Solving Simultaneous Equations: A Clear Guide

Introduction to Simultaneous Equations

Simultaneous equations are systems of linear equations that can be solved to find the values of variables that satisfy all equations in the system. They are a fundamental concept in algebra and have numerous applications in real-world scenarios, such as economics, engineering, and physics.

Understanding the Problem

The problem at hand involves solving the following system of equations:

Equation 1: Y x - 3 Equation 2: 3x 4y 2

There are two methods to solve this system: graphical and algebraic. We will explore both for a comprehensive understanding.

Graphical Method

The graphical method involves plotting both equations on a coordinate plane and finding the point of intersection.

Plot the equation Y x - 3. This is a straight line with a slope of 1 and a y-intercept at -3. Plot the equation 3x 4y 2 by finding the x and y intercepts. For the y-intercept, set x 0: 4y 2 implies y 0.5. For the x-intercept, set y 0: 3x 2 implies x 2/3. Find the point of intersection of both lines. This will be the solution to the system of equations.

While the graphical method is intuitive and easy to understand, it may not always yield precise solutions, especially when dealing with equations that do not intersect at integer points.

Algebraic Method

The algebraic method involves manipulating the equations to isolate the variables and solve the system. Let's go through the steps:

Express Y from the first equation: Y x - 3. Multiply the first equation by 3: 3Y 3x - 9. Subtract the modified first equation from the second equation: 3x 4y - (3x - 9) 2 - (-9). Simplify this to get 7y -7, so y -1. Substitute y -1 into the first equation: -1 x - 3. Solving this gives x 2. Verify the solution by substituting x 2 and y -1 back into both equations.

Substitute x 2 and y -1 into Equation 1: -1 2 - 3 is true. Substitute into Equation 2: 3(2) 4(-1) 6 - 4 2 is also true.

Conclusion

By either graphical or algebraic means, we have determined that the solution to the system of equations is x 2 and y -1.

Simultaneous equations are a powerful tool in solving real-world problems and form the basis for more advanced algebraic concepts. Understanding how to solve them is crucial for success in mathematics and various scientific fields.