Solving Systems Of Equations By Graphing A Step-by-Step Guide

Let's dive into how Sylvie tackled a system of equations using a graphical approach. If you're just starting out with systems of equations, or if you want to brush up on your skills, this is the perfect place to be. We'll break down the problem step by step, making sure you understand the key concepts and how to apply them. Sylvie's problem involves two equations: y = (2/3)x + 1 and y = (-2/3)x - 1. Our main goal here is to figure out which graph shows the solution to this system. So, buckle up, guys, because we're about to embark on a mathematical journey!

Understanding Systems of Equations

Before we jump into the specifics of Sylvie's equations, let's take a moment to understand what a system of equations really is. Simply put, a system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. In simpler terms, it's the point (or points) where the lines or curves represented by the equations intersect. Imagine it like this: each equation draws a line on a graph, and the point where these lines cross is the solution we're after. There are several ways to solve systems of equations, including substitution, elimination, and, as Sylvie did, graphing.

When we talk about solving a system of equations graphically, we're essentially looking for the point of intersection of the lines represented by the equations. Each equation in the system can be graphed as a line on a coordinate plane. The solution to the system is the point where these lines intersect because this point satisfies both equations. If the lines never intersect, it means there's no solution to the system. If the lines are the same, it means there are infinitely many solutions because every point on the line satisfies both equations. Understanding this visual representation can make solving systems of equations much more intuitive. So, keep this in mind as we move forward with Sylvie's problem – we're essentially searching for that elusive intersection point.

Graphing is a fantastic method because it gives you a visual representation of what's happening with the equations. You can see the lines and their relationship to each other. This visual cue can often make the concept easier to grasp, especially for those who are more visual learners. Moreover, graphing can quickly reveal if a system has one solution, no solution, or infinitely many solutions. It's like having a map that guides you directly to the answer! So, let's keep this visual approach in mind as we tackle Sylvie's equations and find the solution by graphing.

Analyzing Sylvie's Equations

Now, let's zoom in on the specific equations Sylvie is working with: y = (2/3)x + 1 and y = (-2/3)x - 1. The first thing you might notice is that these equations are in slope-intercept form, which is written as y = mx + b. This form is super helpful because it immediately tells us two things about the line: the slope (m) and the y-intercept (b). The slope tells us how steep the line is and in what direction it's going, while the y-intercept tells us where the line crosses the y-axis.

For the first equation, y = (2/3)x + 1, the slope (m) is 2/3, and the y-intercept (b) is 1. This means the line rises 2 units for every 3 units it runs to the right, and it crosses the y-axis at the point (0, 1). For the second equation, y = (-2/3)x - 1, the slope (m) is -2/3, and the y-intercept (b) is -1. This line falls 2 units for every 3 units it runs to the right, and it crosses the y-axis at the point (0, -1). Notice how the slopes are different – one is positive, and the other is negative. This tells us that the lines will intersect at some point, which means there is a unique solution to the system.

Another crucial observation here is that the slopes of the two lines are 2/3 and -2/3. These slopes are not only different but also negative reciprocals of each other (if you multiply them, you get -1). When two lines have slopes that are negative reciprocals, it means they are perpendicular, or they intersect at a right angle (90 degrees). This is a key piece of information that will help us visualize the graph and identify the solution. So, when you're looking at a system of equations, always pay close attention to the slopes and y-intercepts – they can give you a ton of insight into the behavior of the lines and the nature of the solution.

Graphing the Equations

Okay, so we've analyzed the equations and understand their slopes and y-intercepts. Now it's time to bring these equations to life on a graph! To graph the equation y = (2/3)x + 1, we start by plotting the y-intercept, which is (0, 1). From this point, we use the slope (2/3) to find another point on the line. The slope tells us to go up 2 units and right 3 units. So, starting from (0, 1), we move up 2 units to (3, 3). Now we can draw a straight line through these two points. Congrats, guys, we've graphed our first equation!

Next up is the equation y = (-2/3)x - 1. We'll follow the same process here. The y-intercept is (0, -1), so we'll start by plotting that point. The slope is -2/3, which means we go down 2 units and right 3 units. Starting from (0, -1), we move down 2 units and right 3 units to the point (3, -3). Now, we draw a straight line through (0, -1) and (3, -3). And boom, we've got our second line graphed!

Now for the magic moment: the point where the two lines intersect is the solution to our system of equations. If you've graphed the lines accurately, you'll notice that they intersect at the point (-1.5, 0). This point satisfies both equations, meaning if you plug in x = -1.5 into both equations, you'll get y = 0. Graphing the equations gives us a clear visual confirmation of the solution. It's like seeing the answer right in front of our eyes! Remember, this intersection point is the golden ticket – it’s the solution to our system.

Identifying the Solution

Alright, we've graphed the equations, and we've pinpointed the intersection. But let's make absolutely sure we know what the solution represents. Remember, the solution to a system of equations is the set of values for x and y that make both equations true. In our case, the intersection point is (-1.5, 0). This means that when x is -1.5, y is 0 for both equations.

To double-check our answer, let's plug these values back into the original equations. For the first equation, y = (2/3)x + 1, we substitute x = -1.5 and y = 0: 0 = (2/3)(-1.5) + 1. Simplifying the right side, we get 0 = -1 + 1, which is true. For the second equation, y = (-2/3)x - 1, we substitute x = -1.5 and y = 0: 0 = (-2/3)(-1.5) - 1. Simplifying the right side, we get 0 = 1 - 1, which is also true. So, our point (-1.5, 0) indeed satisfies both equations, confirming that it is the solution to the system.

Identifying the solution isn't just about finding the intersection point on the graph; it's about understanding what that point represents and verifying that it makes both equations true. This step is crucial because it ensures that we haven't made any mistakes in our graphing or calculations. So, always take a moment to check your answer – it's the ultimate seal of approval!

Conclusion

So, there you have it, guys! Sylvie successfully found the solution to her system of equations by graphing. By understanding the concept of systems of equations, analyzing the equations in slope-intercept form, carefully graphing the lines, and identifying the intersection point, she nailed it. Remember, the key takeaways here are the importance of the slope and y-intercept, the visual representation provided by graphing, and the significance of the intersection point as the solution.

Graphing systems of equations might seem intimidating at first, but with a little practice and a clear understanding of the steps involved, you can become a pro at solving these problems too. Whether you're dealing with linear equations, quadratic equations, or more complex systems, the fundamental principles remain the same. Keep practicing, keep exploring, and most importantly, keep having fun with math! Remember, each equation tells a story, and graphing is like bringing that story to life. So, go out there and unleash your inner mathematician!

By breaking down the problem into manageable steps, we've seen how Sylvie's approach can be applied to various systems of equations. Whether it's finding the intersection point or understanding the nature of the lines, graphing provides a powerful visual tool to solve these problems. Keep honing your skills, and you'll be graphing solutions like a champ in no time! And always remember, math is not just about finding the right answer; it's about the journey of discovery and understanding the underlying concepts. So, keep exploring, keep questioning, and keep learning!

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Which graph correctly shows the solution to the system of equations y = (2/3)x + 1 and y = (-2/3)x - 1?

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Solving Systems of Equations by Graphing A Step-by-Step Guide