Graphing Systems Of Equations Finding Solutions

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Hey guys! Today, we're diving into the world of graphing systems of equations. It might sound a bit intimidating, but trust me, it's totally manageable, and once you get the hang of it, it's actually pretty cool. We're going to tackle the following system:

3x+6y=123x+2y=8\begin{array}{l} 3x + 6y = 12 \\ 3x + 2y = 8 \end{array}

And our mission? To figure out the solution to this system. We'll explore different possibilities – infinitely many solutions, one unique solution (maybe (2,1)?), or something else entirely. So, let's jump right in and break it down step by step!

Step-by-Step Guide to Graphing and Solving

1. Preparing the Equations: Slope-Intercept Form

Before we can even think about graphing, we need to get our equations into a friendlier format – the slope-intercept form. This form is like the VIP pass to the graphing party because it tells us the slope and y-intercept of each line, making them super easy to plot. The slope-intercept form looks like this:

y=mx+by = mx + b

Where:

  • m is the slope (the steepness of the line)
  • b is the y-intercept (where the line crosses the y-axis)

Let's transform our equations. First up:

3x+6y=123x + 6y = 12

Our goal is to isolate y. Here's how we do it:

  1. Subtract 3x from both sides:

    6y=−3x+126y = -3x + 12

  2. Divide both sides by 6:

    y=−36x+126y = \frac{-3}{6}x + \frac{12}{6}

  3. Simplify:

    y=−12x+2y = -\frac{1}{2}x + 2

Bam! Our first equation is ready. We can see that the slope (m) is -1/2, and the y-intercept (b) is 2. That means this line slopes downwards slightly, and it crosses the y-axis at the point (0, 2).

Now, let's tackle the second equation:

3x+2y=83x + 2y = 8

Same drill:

  1. Subtract 3x from both sides:

    2y=−3x+82y = -3x + 8

  2. Divide both sides by 2:

    y=−32x+82y = \frac{-3}{2}x + \frac{8}{2}

  3. Simplify:

    y=−32x+4y = -\frac{3}{2}x + 4

Alright! Equation two is also in slope-intercept form. The slope here is -3/2, which is steeper than our first line, and the y-intercept is 4. This line also slopes downwards, but more sharply, and it crosses the y-axis at (0, 4).

2. Graphing the Lines: Visualizing the Equations

Okay, we've got our equations prepped and ready. Now comes the fun part – graphing the lines! You'll need a coordinate plane (either on paper or using a graphing tool). Remember, the coordinate plane has two axes: the x-axis (horizontal) and the y-axis (vertical).

Let's start with the first equation:

y=−12x+2y = -\frac{1}{2}x + 2

  1. Plot the y-intercept: We know the y-intercept is 2, so put a point on the y-axis at (0, 2). This is where our line starts its journey.

  2. Use the slope to find another point: The slope is -1/2. Remember, slope is rise over run. So, -1/2 means we go down 1 unit and right 2 units from our y-intercept. From (0, 2), go down 1 and right 2 – you'll land at the point (2, 1).

  3. Draw the line: Now, grab a ruler (or use the line tool on your graphing software) and connect those two points. Extend the line across the coordinate plane. That's the graph of our first equation!

Now, let's graph the second equation:

y=−32x+4y = -\frac{3}{2}x + 4

  1. Plot the y-intercept: The y-intercept is 4, so put a point on the y-axis at (0, 4).

  2. Use the slope to find another point: The slope is -3/2. This means we go down 3 units and right 2 units from our y-intercept. From (0, 4), go down 3 and right 2 – you'll land at the point (2, 1).

  3. Draw the line: Connect the y-intercept and the new point with a line, extending it across the plane. This is the graph of our second equation.

3. Finding the Solution: Where the Lines Meet

The whole point of graphing these lines is to find the solution to the system of equations. So, what exactly is the solution? It's the point (or points) where the two lines intersect. Think of it as the place where the two equations agree – the (x, y) values that make both equations true at the same time.

Take a good look at your graph. Do the lines intersect? If they do, that intersection point is your solution. If the lines are parallel (they run side-by-side and never cross), then there's no solution. And if the lines overlap completely (they're the same line), then there are infinitely many solutions.

In our case, if you've graphed the equations accurately, you'll notice that the lines intersect at a single point. And guess what? That point is (2, 1)! This is our one unique solution.

4. Verifying the Solution: Making Sure It Works

We've found our potential solution, but it's always a good idea to double-check and make sure it actually works. To verify, we'll plug the x and y values of our solution (2 and 1, respectively) back into our original equations and see if they hold true.

Let's start with the first equation:

3x+6y=123x + 6y = 12

Substitute x = 2 and y = 1:

3(2)+6(1)=123(2) + 6(1) = 12

6+6=126 + 6 = 12

12=1212 = 12

Yep, the first equation checks out!

Now, let's try the second equation:

3x+2y=83x + 2y = 8

Substitute x = 2 and y = 1:

3(2)+2(1)=83(2) + 2(1) = 8

6+2=86 + 2 = 8

8=88 = 8

Awesome! The second equation also holds true. This confirms that our solution, (2, 1), is indeed the correct one.

The Solution: One Unique Answer

After all that graphing, analyzing, and verifying, we've arrived at our answer. The solution to the system of equations is:

B. There is one unique solution, (2, 1).

We found the point where the two lines intersect, and we confirmed that this point satisfies both equations. That's the power of graphing systems of equations!

Exploring Other Solution Types: A Quick Look

We found a system with one unique solution, but it's worth knowing that systems of equations can behave in other ways too. Let's briefly touch on the other possibilities:

1. Infinitely Many Solutions

Imagine you graph two equations, and they turn out to be the exact same line. They overlap perfectly. In this case, every single point on the line is a solution to the system. There are countless points, hence infinitely many solutions. This happens when the equations are essentially multiples of each other.

For example, consider this system:

x+y=22x+2y=4\begin{array}{l} x + y = 2 \\ 2x + 2y = 4 \end{array}

If you divide the second equation by 2, you'll get the first equation. These are the same line in disguise!

2. No Solution

Now, picture two lines that are perfectly parallel. They have the same slope but different y-intercepts. They'll run alongside each other forever, never touching. This means there's no point of intersection, and therefore, no solution to the system.

Here's an example:

y=x+1y=x+3\begin{array}{l} y = x + 1 \\ y = x + 3 \end{array}

Both lines have a slope of 1, but they have different y-intercepts (1 and 3). They're parallel and will never meet.

Tips and Tricks for Graphing Success

Graphing systems of equations can be a breeze if you keep these tips in mind:

  • Always rewrite equations in slope-intercept form: It makes graphing so much easier!
  • Use a ruler or straightedge: Neat lines lead to accurate solutions.
  • Plot at least two points per line: This ensures you've drawn the line correctly.
  • Check your solution: Plug the (x, y) values back into the original equations to verify.
  • Pay attention to slopes and y-intercepts: They'll tell you a lot about how the lines will behave.
  • Consider using graphing tools: Online calculators and software can be a lifesaver, especially for complex equations.

Real-World Applications: Why This Matters

You might be thinking,