Solving Linear Equations: A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving into the fascinating world of linear equations. Specifically, we're going to tackle a system of two linear equations and explore how to find solutions (if they exist!). Linear equations pop up everywhere, from simple everyday problems to complex scientific models. So, understanding how to solve them is a super valuable skill to have. We'll break down each step, making it easy to follow along, even if you're just starting out with algebra. So, grab your pencils and let's get started on this mathematical journey together! The system of equations we'll be focusing on is:

$$\begin{cases} 6x - 9y = 12 \\ 3x - 4.5y = -6 \end{cases}$$

Our goal is to find values for x and y that satisfy both equations simultaneously. We'll explore different methods to achieve this, highlighting the nuances and potential pitfalls along the way. Whether you're a student prepping for an exam or simply curious about math, this guide is designed to help you grasp the core concepts and build confidence in your problem-solving abilities.

Analyzing the Equations

Before we jump into solving, let's take a closer look at the equations. Analyzing the equations first can often give us clues about the nature of the solution. By observing the coefficients, we can sometimes quickly determine if the equations are independent, inconsistent, or dependent. This preliminary analysis saves time and helps in choosing the most efficient method for solving the system. For example, if we notice that one equation is a multiple of the other, it indicates that the equations are dependent and might have infinitely many solutions or no solution at all if inconsistent.

Firstly, observe the equations closely. We have:

$$\begin{cases} 6x - 9y = 12 \\ 3x - 4.5y = -6 \end{cases}$$

Notice anything interesting? The coefficients in the second equation look suspiciously related to those in the first equation. Let's investigate further. If we divide the first equation by 2, we get:

$$3x - 4.5y = 6$$

Comparing this to the second equation, which is:

$$3x - 4.5y = -6$$

We see something very important: The left-hand sides of both equations are identical, but the right-hand sides are different. This tells us that the two equations are inconsistent. Inconsistent equations mean that there is no solution that satisfies both equations simultaneously. Think of it like this: the equations are making contradictory claims about the relationship between x and y. This observation is crucial because it saves us from wasting time trying to find a solution that doesn't exist!

Why No Solution Exists: A Deeper Dive

To understand why these inconsistent equations have no solution, let’s think about what each equation represents graphically. Each linear equation represents a straight line on a coordinate plane. The solution to a system of two linear equations is the point where the two lines intersect. If the lines are parallel, they never intersect, indicating that there is no solution. In our case, the two equations represent parallel lines. Because the left-hand sides are the same ($3x - 4.5y$), but the right-hand sides are different (6 and -6), the lines have the same slope but different y-intercepts. This guarantees that they are parallel and never meet.

Another way to think about it is to try to manipulate the equations to see if we can arrive at a contradiction. Suppose we multiply the second equation by -1. This gives us:

$$-3x + 4.5y = 6$$

Now, if we add this modified equation to the first equation (divided by 2, which is $3x - 4.5y = 6$), we get:

$$(3x - 4.5y) + (-3x + 4.5y) = 6 + 6$$

Simplifying, we have:

$$0 = 12$$

This is clearly a contradiction! Zero cannot equal twelve. This contradiction confirms that the system of equations has no solution. This detailed explanation should give you a solid understanding of why these equations are inconsistent and have no solution.

Methods That Would Fail (and Why)

It's instructive to consider what would happen if we tried to solve this system using standard methods like substitution or elimination. This helps illustrate why recognizing inconsistency early on is beneficial. Let's briefly explore these methods and see where they lead us.

Substitution Method

In the substitution method, we solve one equation for one variable and then substitute that expression into the other equation. Let's try solving the second equation for x:

$$3x - 4.5y = -6$$

$$3x = 4.5y - 6$$

$$x = 1.5y - 2$$

Now, substitute this expression for x into the first equation:

$$6(1.5y - 2) - 9y = 12$$

$$9y - 12 - 9y = 12$$

$$-12 = 12$$

Again, we arrive at a contradiction! This shows that the substitution method fails because it leads to an impossible statement, confirming that there is no solution.

Elimination Method

In the elimination method, we multiply one or both equations by constants so that the coefficients of one of the variables are opposites. Then, we add the equations to eliminate that variable. To eliminate x, we can multiply the second equation by -2:

$$-2(3x - 4.5y) = -2(-6)$$

$$-6x + 9y = 12$$

Now, add this modified equation to the first equation:

$$(6x - 9y) + (-6x + 9y) = 12 + 12$$

$$0 = 24$$

Once again, we get a contradiction! The elimination method also fails, leading us to the same conclusion: the system has no solution. These examples demonstrate that even if we blindly apply standard methods, we will eventually encounter a contradiction, highlighting the importance of analyzing the equations first.

Graphical Interpretation

As we mentioned earlier, each linear equation in two variables represents a straight line when graphed on a coordinate plane. The solution to a system of two linear equations corresponds to the point(s) where the two lines intersect. When the lines are parallel, they never intersect, indicating that there is no solution to the system.

In our case, the equations:

$$\begin{cases} 6x - 9y = 12 \\ 3x - 4.5y = -6 \end{cases}$$

Represent two parallel lines. To visualize this, we can rewrite each equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

For the first equation:

$$6x - 9y = 12$$

$$-9y = -6x + 12$$

$$y = \frac{2}{3}x - \frac{4}{3}$$

For the second equation:

$$3x - 4.5y = -6$$

$$-4.5y = -3x - 6$$

$$y = \frac{2}{3}x + \frac{4}{3}$$

Notice that both equations have the same slope ($\frac{2}{3}$), but different y-intercepts (-$\frac{4}{3}$ and $\frac{4}{3}$). This confirms that the lines are parallel and will never intersect, visually demonstrating why the system has no solution.

Conclusion

So, there you have it, guys! The system of equations:

$$\begin{cases} 6x - 9y = 12 \\ 3x - 4.5y = -6 \end{cases}$$

has no solution. We determined this by carefully analyzing the equations and recognizing that they are inconsistent. We also explored how standard solution methods would fail and examined the graphical interpretation, which shows that the equations represent parallel lines. Remember, always take a moment to analyze the equations before diving into solving. It can save you time and prevent frustration! Understanding these concepts will not only help you solve linear equations but also build a strong foundation for more advanced mathematical topics. Keep practicing, and you'll become a math whiz in no time! If you have any questions, feel free to ask. Happy solving!