Solving Simultaneous Linear Equations and The Misconception of Quadratic Equations

When it comes to solving linear equations, one often needs to tackle a system of equations where the variables are linear. However, it's essential to recognize the nature of the equations in such scenarios. In this article, we will explore how to solve a system of simultaneous linear equations. We will also address a common misconception involving equations that may be mistaken for quadratic but are, in fact, linear.

Introduction to Simultaneous Linear Equations

Simultaneous linear equations are a set of linear equations that must be solved together. Each equation represents a line in two or more dimensions. Typically, these equations are solved to find the point of intersection of these lines. Let's break down the given equations and solve them step-by-step.

Given Equations

The equations provided are:

3xy 7y - x 5(3x - y) x^3

It's important to note that the second equation is actually a linear equation in disguise, not a quadratic equation. This misconception leads to a common pitfall in solving such systems. Let's simplify and solve the equations correctly.

Solving the Equations Step-by-Step

Starting with the first equation:

3xy 7y - x

To make the first equation simpler, we can rearrange it:

3xy x 7y

Factor out the common term on the left-hand side:

x(3y 1) 7y

Divide both sides by (3y 1) to solve for x:

x frac{7y}{3y 1}

Simplifying the Second Equation

The second equation is:

5(3x - y) x^3

First, expand and simplify:

15x - 5y x^3

Rearrange to isolate the terms involving x:

15x - x^3 5y

5(3x - y) x^3

Recognize that this equation is not quadratic in y, but rather a linear equation in x. Let's use the simplified form of the first equation to substitute into the second equation:

Substitution and Solution for x and y

Substitute x 2 into the first simplified equation:

3(2)y 2 7y

6y 2 7y

6y - 7y -2

-y -2

y 2

So the solution to the system of equations is:

x 2, y 5

Graphical Representation

A plot of the equations would look like this:

The graph shows the intersection point of the two lines at (x, y) (2, 5).

Conclusion

In conclusion, solving systems of simultaneous linear equations requires a keen eye for detail, especially when it comes to recognizing whether an equation is quadratic or linear. The solution to the given system is: