Solving Systems Of Equations Rafik's Graphical Method

by ADMIN 54 views

Hey guys! Ever wondered how systems of equations work and what it means when they have just one solution? Let's dive into this topic using Rafik's approach to solving the system: 2x + y = 4 and x - y = -1. We'll break it down using the graphical method, making it super easy to understand. So, buckle up and let's get started!

Introduction to Systems of Equations

Before we jump into solving, let's quickly recap what systems of equations are. A system of equations is simply a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all equations in the system true simultaneously. This might sound a bit complicated, but trust me, it's not! Think of it as finding a sweet spot where all conditions are met.

The number of solutions a system can have varies. It can have one solution, no solution, or infinitely many solutions. When we say a system has one solution, it means there's only one unique set of values for the variables that satisfies all equations. This is what we're focusing on today with Rafik's example.

Rafik's System of Equations: 2x + y = 4 and x - y = -1

Okay, let's look at the specific system Rafik is dealing with:

  1. 2x + y = 4
  2. x - y = -1

Our goal is to find the values of x and y that satisfy both of these equations. There are several methods to solve such systems, including substitution, elimination, and the graphical method. Today, we're going to focus on the graphical method because it gives us a visual understanding of what's happening. This method is especially useful for understanding why some systems have one solution, while others might have none or infinitely many.

The Graphical Method: A Visual Approach

The graphical method involves plotting the equations on a coordinate plane and finding the point of intersection. Each equation represents a line, and the point where the lines intersect is the solution to the system. If the lines intersect at exactly one point, the system has one unique solution. If the lines are parallel and never intersect, the system has no solution. And if the lines overlap each other, the system has infinitely many solutions.

Step-by-Step Guide to Graphing the Equations

To graph the equations, we first need to rewrite them in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This form makes it easier to plot the lines on the graph.

Equation 1: 2x + y = 4

Let's rewrite the first equation: 2x + y = 4. To isolate y, we subtract 2x from both sides:

y = -2x + 4

Now, we can easily 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), and for every 1 unit we move to the right along the x-axis, we move 2 units down along the y-axis.

Equation 2: x - y = -1

Next, let's rewrite the second equation: x - y = -1. To isolate y, we first subtract x from both sides:

-y = -x - 1

Now, multiply both sides by -1 to get y by itself:

y = x + 1

Here, the slope (m) is 1, and the y-intercept (b) is 1. So, this line crosses the y-axis at the point (0, 1), and for every 1 unit we move to the right along the x-axis, we move 1 unit up along the y-axis.

Plotting the Lines

Now that we have both equations in slope-intercept form, we can plot them on a graph. Grab a piece of graph paper or use an online graphing tool. First, plot the y-intercept for each line and then use the slope to find additional points. For example:

  • For y = -2x + 4, plot the point (0, 4). Then, using the slope of -2, move 1 unit to the right and 2 units down to find another point (1, 2). Draw a line through these points.
  • For y = x + 1, plot the point (0, 1). Then, using the slope of 1, move 1 unit to the right and 1 unit up to find another point (1, 2). Draw a line through these points.

Finding the Point of Intersection

Once you've plotted both lines, you'll notice that they intersect at a single point. This point of intersection is the solution to the system of equations. By looking at the graph, we can see that the lines intersect at the point (1, 2). This means that x = 1 and y = 2 is the unique solution to the system.

Verifying the Solution

To make sure we've got the right answer, let's plug x = 1 and y = 2 back into the original equations:

Equation 1: 2x + y = 4

2(1) + 2 = 4

2 + 2 = 4

4 = 4 (This is true!)

Equation 2: x - y = -1

1 - 2 = -1

-1 = -1 (This is also true!)

Since both equations hold true with x = 1 and y = 2, we've verified that this is indeed the solution to the system. Great job, Rafik, for setting up this problem!

Why One Solution? Understanding Intersecting Lines

The reason this system has exactly one solution is that the lines represented by the equations intersect at only one point. Think about it: two straight lines in a plane can either intersect at one point, be parallel (never intersect), or be the same line (overlap completely). When the lines have different slopes, they're guaranteed to intersect at one point.

In our case, the slopes of the lines are -2 and 1, which are different. This ensures that the lines will intersect at a single point, giving us one unique solution. If the slopes were the same but the y-intercepts were different, the lines would be parallel, and there would be no solution. If both the slopes and y-intercepts were the same, the lines would overlap, and there would be infinitely many solutions.

Real-World Applications

Systems of equations with one solution pop up all over the place in real-world scenarios. Here are a couple of examples:

  1. Mixing Solutions: Imagine you're mixing two different chemical solutions to get a specific concentration. Each solution has a different concentration of the chemical, and you need to figure out how much of each solution to mix. This can be modeled as a system of equations, where the amount of each solution is the variable, and the desired concentration and total volume are the constants. The single solution tells you the exact amounts to mix.
  2. Supply and Demand: In economics, the price and quantity of a product are determined by the supply and demand curves. The point where these curves intersect represents the equilibrium price and quantity—the price at which the quantity supplied equals the quantity demanded. This is a classic example of a system of equations with one solution.

Common Mistakes and How to Avoid Them

When solving systems of equations graphically, there are a few common mistakes that students often make. Let's go over them so you can avoid them:

  1. Inaccurate Graphing: The most common mistake is not graphing the lines accurately. This can happen if you miscalculate the slope or y-intercept or if you don't plot the points carefully. Always double-check your calculations and use a ruler to draw straight lines.
  2. Misreading the Intersection: Even if your lines are graphed correctly, it's easy to misread the coordinates of the point of intersection. Make sure you carefully read the x and y values of the point where the lines cross.
  3. Not Rewriting Equations: Trying to graph equations without first rewriting them in slope-intercept form can be tricky. It's much easier to identify the slope and y-intercept when the equation is in the form y = mx + b.
  4. Forgetting to Verify: Always verify your solution by plugging the values of x and y back into the original equations. This will catch any errors you might have made along the way.

Practice Problems

To really nail down your understanding, let's try a couple of practice problems:

Practice Problem 1

Solve the following system of equations graphically:

  1. y = 2x - 1
  2. y = -x + 5

Practice Problem 2

Solve the following system of equations graphically:

  1. x + y = 3
  2. 2x - y = 0

Try graphing these systems on your own. Remember to rewrite the equations in slope-intercept form first, if necessary. Plot the lines carefully and find the point of intersection. Then, verify your solution by plugging the values back into the original equations.

Conclusion: Mastering Systems of Equations

So, guys, we've explored how to solve systems of equations with one solution using the graphical method, inspired by Rafik's approach. We learned that a system with one solution means the lines intersect at a single point, and we saw how to find that point by graphing the equations. We also discussed real-world applications and common mistakes to avoid.

Remember, understanding systems of equations is a crucial skill in math and has applications in various fields. Keep practicing, and you'll become a pro at solving these problems! Whether it's mixing solutions, analyzing supply and demand, or any other scenario, you'll be ready to tackle it with confidence. Keep up the great work, and happy solving!