Graphing Systems Of Equations A Step-by-Step Solution

Hey guys! Today, we're diving into the exciting world of graphing systems of equations and pinpointing their solutions. It might sound a bit intimidating, but trust me, it's like solving a puzzle, and we'll break it down step by step. We'll be tackling the following system:

$$\left\{ \begin{array}{l} y = \frac{1}{3}x - 3 \\ 2x + y = 4 \end{array} \right.$$

So, grab your graph paper (or your favorite digital graphing tool), and let's get started!

Understanding Systems of Equations

Before we jump into graphing, let's quickly recap what a system of equations actually is. Simply put, it's a set of two or more equations that we're looking to solve simultaneously. That means we want to find the values for the variables (in this case, x and y) that make all the equations in the system true at the same time. Think of it like finding the common ground where all the equations agree.

Each equation in our system represents a line on a graph. The solution to the system is the point (or points) where these lines intersect. This intersection point is the x, y coordinate that satisfies both equations. There are a few possibilities when it comes to solutions:

• One Solution: The lines intersect at a single point. This is the most common scenario.
• No Solution: The lines are parallel and never intersect. They have the same slope but different y-intercepts.
• Infinitely Many Solutions: The lines are actually the same line! They overlap completely, meaning every point on the line is a solution.

So, our mission is to graph these lines and see what kind of solution we're dealing with.

Graphing the First Equation: y = (1/3)x - 3

Let's start with the first equation: y = (1/3)x - 3. This equation is in slope-intercept form (y = mx + b), which makes graphing a breeze! Remember that m represents the slope and b represents the y-intercept.

In our equation, the slope (m) is 1/3, and the y-intercept (b) is -3. This means the line crosses the y-axis at the point (0, -3). The slope of 1/3 tells us that for every 3 units we move to the right on the graph, we move 1 unit up.

To graph this line, we can follow these steps:

1. Plot the y-intercept: Plot the point (0, -3) on the graph. This is our starting point.
2. Use the slope to find another point: From the y-intercept, move 3 units to the right and 1 unit up. This will give you a second point on the line. You can repeat this process to find even more points if you like.
3. Draw the line: Connect the points with a straight line. Extend the line across the graph.

Now we have our first line graphed! It's a visual representation of all the possible x and y values that satisfy the equation y = (1/3)x - 3.

Graphing the Second Equation: 2x + y = 4

Next up, we have the equation 2x + y = 4. This equation isn't in slope-intercept form yet, but no worries! We can easily convert it.

To get it into y = mx + b form, we need to isolate y on one side of the equation. Let's subtract 2x from both sides:

y = -2x + 4

Awesome! Now it's in slope-intercept form. We can see that the slope (m) is -2, and the y-intercept (b) is 4. This means the line crosses the y-axis at the point (0, 4). The slope of -2 (or -2/1) tells us that for every 1 unit we move to the right on the graph, we move 2 units down.

Let's graph this line using the same steps as before:

1. Plot the y-intercept: Plot the point (0, 4) on the graph.
2. Use the slope to find another point: From the y-intercept, move 1 unit to the right and 2 units down. This gives us another point on the line.
3. Draw the line: Connect the points with a straight line. Extend the line across the graph.

Great job! We now have both lines graphed on the same coordinate plane.

Finding the Solution: Where the Lines Intersect

This is where the magic happens! The solution to the system of equations is the point where the two lines intersect. Take a good look at your graph. Do you see where the lines cross each other?

If you've graphed the lines accurately, you should see that they intersect at the point (3, -2). This means that x = 3 and y = -2 is the solution to our system of equations.

To double-check our answer, we can substitute these values back into both original equations and see if they hold true:

• Equation 1: y = (1/3)x - 3 -2 = (1/3)(3) - 3 -2 = 1 - 3 -2 = -2 (This is true!)
• Equation 2: 2x + y = 4 2(3) + (-2) = 4 6 - 2 = 4 4 = 4 (This is also true!)

Since the values x = 3 and y = -2 satisfy both equations, we've confirmed that (3, -2) is indeed the solution to the system.

Alternative Solution Methods: A Sneak Peek

Graphing is a fantastic way to visualize the solution to a system of equations, but it's not always the most precise method, especially if the intersection point has fractional coordinates. Luckily, there are other powerful algebraic techniques we can use, such as:

• Substitution: Solving one equation for one variable and substituting that expression into the other equation.
• Elimination (or Addition): Manipulating the equations to eliminate one variable, allowing us to solve for the other.

We won't go into detail about these methods today, but it's good to know they exist as alternative approaches. They're especially useful when dealing with more complex systems or when you need a highly accurate solution.

Wrapping Up: Systems Solved!

And there you have it! We've successfully graphed a system of equations and found its solution. Remember, the key is to graph each line accurately and then identify the point of intersection. This point represents the x and y values that make both equations true.

Graphing systems of equations is a fundamental skill in algebra, and it's a great way to build your understanding of how equations and their solutions work. So keep practicing, and you'll become a system-solving pro in no time!

The solution to the system is (3, -2).