Graphing Systems Of Equations: Step-by-Step Solutions

Understanding Systems of Equations

Let's dive into the world of systems of equations. In the realm of mathematics, a system of equations is essentially a set of two or more equations containing the same variables. The solution to a system of equations represents the point(s) where the graphs of the equations intersect. Graphing is a powerful visual method for solving these systems, allowing us to see the relationships between the equations and pinpoint their common solutions. In this comprehensive guide, we will explore the process of graphing a given system of equations and determining its solution. We will focus specifically on the system: $\left\{\begin{array}{l}y=\frac{1}{3} x-1 \\ x-3 y=-3\end{array}\right.$, breaking down each step to ensure a clear understanding. Before we jump into the specifics of our example, let's establish a foundation by understanding the different types of solutions a system of equations can have. There are three primary scenarios: a unique solution (where the lines intersect at one point), no solution (where the lines are parallel and never intersect), and infinitely many solutions (where the lines are the same). Recognizing these possibilities is crucial for interpreting the results of our graphing process. Now, let's turn our attention back to the system at hand, $\left\{\begin{array}{l}y=\frac{1}{3} x-1 \\ x-3 y=-3\end{array}\right.$. Our journey will begin with a detailed examination of each equation individually, understanding their slopes and y-intercepts, and then meticulously plotting these lines on a coordinate plane.

Graphing the First Equation: $y = \frac{1}{3}x - 1$

Let's begin by tackling the first equation: $y = \frac{1}{3}x - 1$. This equation is presented in slope-intercept form, which is a fantastic way to easily visualize and graph a linear equation. The slope-intercept form is generally written as y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. Now, let's dissect our equation. By comparing it to the slope-intercept form, we can readily identify that the slope (m) is $\frac{1}{3}$ and the y-intercept (b) is -1. The y-intercept, -1, provides us with our first point on the graph: (0, -1). We can plot this point directly on the coordinate plane. The slope, $\frac{1}{3}$, tells us how to move from one point on the line to another. Remember, the slope is rise over run. A slope of $\frac{1}{3}$ means that for every 1 unit we move vertically (rise), we move 3 units horizontally (run). Starting from our y-intercept (0, -1), we can move up 1 unit and then right 3 units to find another point on the line. This new point would be (3, 0). We can repeat this process to find additional points, or we can simply draw a straight line through the two points we've already plotted. Accuracy is key here! Use a ruler or straightedge to ensure your line is precise. A slightly skewed line can lead to an incorrect solution when we consider the second equation. Now that we've successfully graphed the first equation, let's move on to the second equation and repeat the process. This methodical approach will help us clearly visualize the system and identify its solution. Remember, the goal is to find the point where both lines intersect, as that point represents the solution that satisfies both equations simultaneously.

Graphing the Second Equation: $x - 3y = -3$

Now, let's shift our focus to the second equation: $x - 3y = -3$. Unlike the first equation, this one isn't presented in slope-intercept form. To make graphing easier, we need to rearrange it into the familiar y = mx + b format. This involves isolating 'y' on one side of the equation. Here's how we can do it:

1. Subtract 'x' from both sides: $-3y = -x - 3$
2. Divide both sides by -3: $y = \frac{1}{3}x + 1$

Now we have the equation in slope-intercept form! We can see that the slope (m) is $\frac{1}{3}$ and the y-intercept (b) is 1. This means we have a y-intercept at the point (0, 1). Let's plot this point on our coordinate plane. Just like before, we can use the slope to find another point on the line. A slope of $\frac{1}{3}$ indicates that we move up 1 unit for every 3 units we move to the right. Starting from the y-intercept (0, 1), we move up 1 unit and right 3 units, landing us at the point (3, 2). We now have two points, allowing us to draw a straight line through them. Again, precision is paramount. Use a ruler to create an accurate line representing the equation $y = \frac{1}{3}x + 1$. As we graph this second line, we begin to observe its relationship with the first line we graphed, $y = \frac{1}{3}x - 1$. A keen eye will notice something significant about their slopes. Both lines share the same slope, $\frac{1}{3}$. This observation is a crucial clue in determining the type of solution our system of equations has. Remember the three possibilities we discussed earlier: one solution, no solution, or infinitely many solutions. The fact that the slopes are identical points us towards a specific scenario. Let's delve deeper into the implications of having the same slope as we move on to analyzing the solution.

Analyzing the Solution

With both equations graphed on the coordinate plane, we can now analyze their relationship and determine the solution to the system. Remember, the solution represents the point(s) where the lines intersect. If the lines intersect at one point, we have a unique solution. If they are parallel and never intersect, there's no solution. And if they overlap completely, we have infinitely many solutions. Looking at our graph, what do we observe? The two lines, $y = \frac{1}{3}x - 1$ and $y = \frac{1}{3}x + 1$, appear to be parallel. But let's not rely solely on visual inspection. We already noted that both lines have the same slope, $\frac{1}{3}$. This confirms that they are indeed parallel. Parallel lines, by definition, never intersect. They run alongside each other, maintaining a constant distance, but never meeting. Therefore, in this particular system of equations, there is no solution. This means there is no single point (x, y) that satisfies both equations simultaneously. The lines simply do not share any common points. To solidify our understanding, let's briefly revisit the concept of slope-intercept form. The slope (m) dictates the steepness and direction of the line, while the y-intercept (b) indicates where the line crosses the y-axis. In our case, the lines have the same steepness (same slope) but different y-intercepts. This is the hallmark of parallel lines. Now, imagine if the lines had the same slope and the same y-intercept. They would essentially be the same line, overlapping completely. This scenario would lead to infinitely many solutions, as every point on the line would satisfy both equations. Understanding these nuances is essential for confidently solving systems of equations graphically. To summarize, by graphing the system $\left\{\begin{array}{l}y=\frac{1}{3} x-1 \\ x-3 y=-3\end{array}\right.$, we have visually and analytically determined that there is no solution because the lines are parallel.

Conclusion

In conclusion, we've successfully navigated the process of graphing a system of equations and identifying its solution. We started with the system $\left\{\begin{array}{l}y=\frac{1}{3} x-1 \\ x-3 y=-3\end{array}\right.$, graphed each equation individually by converting them into slope-intercept form, and then analyzed the resulting graph. Our key finding was that the two lines are parallel, indicating that there is no solution to this system of equations. This exercise highlights the power of graphical methods in understanding and solving systems of equations. By visualizing the equations as lines on a coordinate plane, we can readily observe their relationships and determine whether they intersect, are parallel, or overlap. This visual approach provides a valuable complement to algebraic methods, enhancing our overall problem-solving abilities. Remember, understanding the slope-intercept form (y = mx + b) is crucial for efficient graphing. The slope (m) tells us the steepness and direction of the line, while the y-intercept (b) gives us a starting point on the y-axis. By mastering these concepts, you can confidently tackle a wide range of systems of equations. Graphing systems of equations is a fundamental skill in algebra and has numerous applications in various fields, including science, engineering, and economics. It allows us to model real-world scenarios involving multiple variables and constraints, providing valuable insights and solutions. So, keep practicing, keep graphing, and keep exploring the fascinating world of mathematics! For further learning and resources on systems of equations, you can visit trusted websites like Khan Academy.