Hey guys! Today, we're diving into the exciting world of linear equations and how to solve them by graphing. It might sound a bit intimidating at first, but trust me, it's super cool and actually pretty straightforward once you get the hang of it. We'll be tackling a specific system of equations, breaking down each step, and making sure you understand exactly what's going on. So, grab your graph paper (or your favorite digital graphing tool), and let's get started!
Understanding the Problem
Before we jump into graphing, let's take a closer look at the system of equations we're going to solve. We have:
x + y = -3
6x + 6y = -18
Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy both of these equations simultaneously. In other words, we're looking for the point where the lines represented by these equations intersect on a graph.
The beauty of systems of linear equations lies in their ability to model real-world situations. Imagine you're trying to figure out the cost of buying a certain number of apples and bananas, or perhaps you're planning a road trip and need to calculate travel time based on speed and distance. These kinds of scenarios can often be represented and solved using systems of equations.
Now, you might be wondering, why graphing? Well, graphing provides a visual representation of the equations, making it easier to understand the relationship between the variables and the solution. It's like seeing the answer right before your eyes! Plus, it's a fantastic way to check your work if you've solved the system algebraically.
Converting to Slope-Intercept Form
Okay, so we've got our equations, and we know we want to graph them. But to graph them easily, we need to get them into a special form called slope-intercept form. This form is super helpful because it tells us two crucial things about the line: its slope and its y-intercept. The slope-intercept form looks like this:
y = mx + b
Where:
- m is the slope of the line (how steep it is).
- b is the y-intercept (where the line crosses the y-axis).
Let's start with our first equation:
x + y = -3
To get y by itself, we need to subtract x from both sides of the equation. This gives us:
y = -x - 3
Awesome! Now it's in slope-intercept form. We can see that the slope (m) is -1 (since there's an implied -1 in front of the x), and the y-intercept (b) is -3. That means this line crosses the y-axis at the point (0, -3), and for every 1 unit we move to the right on the graph, we move 1 unit down.
Now, let's tackle the second equation:
6x + 6y = -18
This one's a little trickier, but we can handle it! First, we subtract 6x from both sides:
6y = -6x - 18
Then, to get y all by itself, we divide both sides by 6:
y = -x - 3
Wait a minute... This looks familiar! It's the exact same equation as our first one! This means that both equations represent the same line. We'll explore what that means for our solution in a bit.
Converting equations to slope-intercept form is a fundamental skill in algebra. It not only makes graphing easier but also helps you quickly understand the characteristics of a line. By identifying the slope and y-intercept, you can visualize the line's direction and position on the coordinate plane. This skill is essential for solving various mathematical problems and understanding real-world applications that involve linear relationships.
Graphing the Equations
Alright, we've got our equations in slope-intercept form, so it's time to put our graphing skills to the test! Remember, we have:
y = -x - 3
This equation represents a line with a slope of -1 and a y-intercept of -3. To graph it, we can start by plotting the y-intercept, which is the point (0, -3). Then, we can use the slope to find another point on the line. Since the slope is -1, we can think of it as -1/1. This means we move 1 unit down and 1 unit to the right from our y-intercept. This gives us the point (1, -4). We can plot this point as well.
Now, we just need to draw a straight line through these two points, and voilà! We've graphed our first equation. But hey, remember that our second equation is the same line? That's right! So, when we graph the second equation, we'll end up drawing the exact same line. This is a crucial observation, and it tells us a lot about the solution to our system.
Graphing linear equations is a powerful tool for visualizing their behavior and understanding their relationships. By plotting points and drawing lines, we can gain insights into the solutions of systems of equations and the properties of linear functions. This visual approach can make abstract concepts more concrete and help us solve problems more effectively.
Interpreting the Graph and Finding the Solution
So, we've graphed our equations, and we've noticed something interesting: both equations represent the same line. What does this mean for the solution to our system? Well, remember that the solution to a system of equations is the point (or points) where the lines intersect.
In this case, since both equations represent the same line, they intersect at every single point on the line! This means there are infinitely many solutions to this system of equations. Any point that lies on the line y = -x - 3 will satisfy both equations.
This is a special case called a dependent system. A dependent system is a system of equations where the equations are essentially multiples of each other, resulting in the same line when graphed. This contrasts with independent systems, where the equations represent distinct lines that intersect at a single point, and inconsistent systems, where the equations represent parallel lines that never intersect.
Understanding the different types of systems is essential for solving linear equations effectively. By recognizing dependent systems, you can avoid unnecessary calculations and quickly determine that there are infinitely many solutions. This knowledge will save you time and effort in solving mathematical problems and applying linear equations to real-world scenarios.
Expressing the Solution
Since we have infinitely many solutions, we can't list them all individually. Instead, we express the solution in a general form. We can say that the solution set consists of all points (x, y) that satisfy the equation y = -x - 3. We can also express this solution set using set-builder notation:
{(x, y) | y = -x - 3}
This notation means