Infinite Solutions Unveiled Solving Y=3x+2 And 3y=9x+6 Graphically
Hey guys! Today, we're diving into a super interesting math problem that involves solving a system of equations. Specifically, we're looking at the system:
y = 3x + 23y = 9x + 6
We're going to tackle this problem using a graphical approach – either with graph paper or some cool graphing technology. And the big question we want to answer is: what's the solution to this system? Let's get started!
Understanding Systems of Equations
Before we jump into the specifics, let's quickly recap what a system of equations is all about. Basically, a system of equations is just a set of two or more equations that we're trying to solve simultaneously. We're looking for values of the variables (in this case, x and y) that satisfy all the equations in the system at the same time. Think of it like finding the sweet spot where all the equations agree!
There are a few ways we can solve systems of equations. We could use algebraic methods like substitution or elimination, but for this problem, we're focusing on the graphical approach. Graphing is a fantastic way to visualize what's going on and see the solutions (or lack thereof) right before our eyes.
The Graphical Approach: A Quick Overview
When we graph equations, we're essentially plotting all the points that satisfy each equation. For linear equations (which is what we have here), the graph will be a straight line. The solution to a system of equations, graphically, is the point (or points) where the lines intersect. Why? Because the intersection point represents the (x, y) values that make both equations true.
So, if the lines intersect at one point, we have one unique solution. If the lines don't intersect at all (they're parallel), we have no solutions. And if the lines overlap completely (they're the same line), we have infinitely many solutions. This last scenario is the one we're going to discover in this problem!
Let's Get Graphing
Okay, let's grab our graph paper (or fire up our graphing calculator/software) and plot these equations. The first equation, y = 3x + 2, is in slope-intercept form (y = mx + b), which makes it super easy to graph. We know the slope (m) is 3, and the y-intercept (b) is 2. This means we can start by plotting the point (0, 2) on the y-axis, and then use the slope to find other points on the line (rise 3, run 1).
Now, let's look at the second equation: 3y = 9x + 6. It's not quite in slope-intercept form yet, but we can easily get it there by dividing every term by 3. This gives us:
y = 3x + 2
Wait a minute… that's the same equation as the first one! This is a huge clue. It tells us that when we graph these two equations, we're actually graphing the same line twice.
Visualizing the Overlap
If you graph these equations, you'll see that they perfectly overlap. Every single point on the line y = 3x + 2 is also a point on the line 3y = 9x + 6. This means there are infinitely many points that satisfy both equations simultaneously. Think about it: any x value you pick, you can plug it into the equation y = 3x + 2 to get a corresponding y value, and that (x, y) pair will work for both equations.
The Solution: Infinite Solutions
So, what's the solution to this system? The answer is B. Infinite solutions.
Why Infinite Solutions?
The key takeaway here is that the two equations are actually dependent. They represent the same line in disguise. When you have dependent equations in a system, it means the equations are essentially multiples of each other. One equation doesn't provide any unique information that the other one doesn't already have. This leads to an infinite number of solutions because every point on the line satisfies both equations.
Understanding Different Solution Types
To solidify our understanding, let's quickly touch on the other possible solution types for systems of linear equations:
- One Unique Solution: This happens when the lines intersect at a single point. The slopes of the lines are different.
- No Solutions: This occurs when the lines are parallel (they have the same slope but different y-intercepts). They never intersect.
- Infinite Solutions: This is what we saw in our problem. The lines are the same (dependent equations), and they overlap completely.
Dependent and Independent Equations: It's also useful to understand the terms dependent and independent equations. We already know that dependent equations are essentially the same line. Independent equations, on the other hand, are distinct lines. They either intersect at one point (one solution) or are parallel (no solutions).
Let's Dive Deeper into Graphing Techniques
Okay, guys, now that we've solved our problem and nailed down the concept of infinite solutions, let's take a moment to explore some different graphing techniques. Mastering these techniques will make solving systems of equations graphically a breeze!
Method 1: Slope-Intercept Form (y = mx + b)
We touched on this earlier, but let's really dig in. Slope-intercept form is your best friend when it comes to graphing linear equations. Remember, y = mx + b, where:
mis the slope (the
