System Of Equations: One, Infinite, Or No Solutions?

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System of Equations: One, Infinite, or No Solutions?

Hey guys! Today, we're diving deep into the super interesting world of systems of equations. Specifically, we're going to tackle a problem where we need to figure out if a given system has no solutions, infinitely many solutions, or just one unique solution. It's like being a detective for math problems! We've got this system right here:

-x + 2y = -3
5x - 8y = 9

Our mission, should we choose to accept it, is to analyze these two linear equations and determine the nature of their solution set. Are they going to meet at a single point? Will they run parallel forever without ever touching? Or perhaps they're actually the same line, meaning they overlap at every single point? Let's put on our thinking caps and find out!

Understanding the Possibilities

Before we jump into solving our specific system, let's quickly chat about what it means for a system of linear equations to have no solutions, infinitely many solutions, or exactly one solution. This is crucial for understanding why we arrive at our conclusions. Think of these equations as lines on a graph. The solution to a system of equations is where those lines intersect.

  • Exactly One Solution: This happens when the two lines have different slopes. They will intersect at precisely one point, and that point (x, y) is our unique solution. It’s like two roads crossing at an intersection – there's only one spot where they meet.
  • No Solutions: This scenario occurs when the two lines are parallel but distinct. Parallel lines have the same slope but different y-intercepts. They will never intersect, no matter how far you extend them. Think of two train tracks running side-by-side – they'll never meet.
  • Infinitely Many Solutions: This is the case when the two equations actually represent the same line. This means they have the same slope AND the same y-intercept. Every single point on that line is a solution because the lines are identical and overlap everywhere. It's like having two identical maps of the same city; they show you the exact same places.

Knowing these possibilities helps us interpret our results. We can often predict the outcome by just looking at the coefficients of our equations, but the safest bet is always to try and solve the system. We'll explore different methods to do this, and you guys can pick your favorite!

Method 1: The Substitution Method

The substitution method is a fantastic way to solve systems of equations, and it's particularly useful when you can easily isolate one variable in one of the equations. Let's look at our system again:

  1. -x + 2y = -3
  2. 5x - 8y = 9

Our goal here is to get one variable by itself in one equation and then substitute that expression into the other equation. This will reduce our system to a single equation with a single variable, which is way easier to solve!

Looking at equation (1), -x + 2y = -3, it seems pretty straightforward to isolate x. Let's add x to both sides and then subtract 3 from both sides:

-x = -3 - 2y x = 3 + 2y

Awesome! We've now expressed x in terms of y. The next step is to substitute this expression for x into equation (2). Remember, equation (2) is 5x - 8y = 9. So, wherever we see x in that equation, we'll replace it with (3 + 2y):

5(3 + 2y) - 8y = 9

Now, we just need to solve this single equation for y. Let's distribute the 5:

15 + 10y - 8y = 9

Combine the y terms (10y - 8y):

15 + 2y = 9

To isolate 2y, let's subtract 15 from both sides:

2y = 9 - 15 2y = -6

And finally, divide by 2 to find y:

y = -6 / 2 y = -3

Fantastic! We found the value of y. Now, to find the value of x, we just need to plug this y = -3 back into the expression we found for x earlier, which was x = 3 + 2y:

x = 3 + 2(-3) x = 3 - 6 x = -3

So, using the substitution method, we found that x = -3 and y = -3. This means our system has exactly one solution, which is the point (-3, -3). We did it! But let's try another method to confirm our findings, because why not?

Method 2: The Elimination Method

The elimination method (sometimes called the addition method) is another powerful technique for solving systems of equations. The main idea here is to manipulate one or both equations (by multiplying them by constants) so that when you add or subtract the equations, one of the variables cancels out. This leaves you with a single equation with one variable, just like with substitution!

Let's take our system again:

  1. -x + 2y = -3
  2. 5x - 8y = 9

We want to make the coefficients of either x or y opposites so they cancel out when added. Let's focus on the x terms. We have -x in the first equation and 5x in the second. If we multiply the first equation by 5, the x term will become -5x, which is the opposite of 5x in the second equation!

Let's do that. Multiply the entire first equation by 5:

5 * (-x + 2y) = 5 * (-3) -5x + 10y = -15

Now, our system looks like this:

1'. -5x + 10y = -15 2. 5x - 8y = 9

Look at that! The x coefficients are -5 and 5. If we add equation (1') and equation (2) together, the x terms will be eliminated:

(-5x + 10y) + (5x - 8y) = -15 + 9

Combine the x terms: -5x + 5x = 0x = 0 (they cancel out!) Combine the y terms: 10y - 8y = 2y Combine the constants: -15 + 9 = -6

So, our combined equation is:

2y = -6

This is the exact same equation we got for y using the substitution method! How cool is that? Solving for y:

y = -6 / 2 y = -3

Now that we have y = -3, we can substitute this value back into either of the original equations to solve for x. Let's use the first original equation: -x + 2y = -3.

-x + 2(-3) = -3 -x - 6 = -3

Add 6 to both sides:

-x = -3 + 6 -x = 3

Multiply by -1 to solve for x:

x = -3

Once again, we arrive at the same solution: x = -3 and y = -3. This confirms that the system has exactly one solution, which is the point (-3, -3). It's super reassuring when different methods give you the same answer, right?

Method 3: Graphical Analysis (Conceptual)

While we won't actually draw the graph here, it's super helpful to think about what's happening graphically. Each of our original equations represents a line.

  1. -x + 2y = -3
  2. 5x - 8y = 9

To understand the graphical representation, we can think about the slopes and y-intercepts of these lines. Let's rewrite each equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

For equation (1): -x + 2y = -3 Add x to both sides: 2y = x - 3 Divide by 2: y = (1/2)x - 3/2

So, the first line has a slope (m1) of 1/2 and a y-intercept (b1) of -3/2.

For equation (2): 5x - 8y = 9 Subtract 5x from both sides: -8y = -5x + 9 Divide by -8: y = (-5/-8)x + (9/-8) y = (5/8)x - 9/8

So, the second line has a slope (m2) of 5/8 and a y-intercept (b2) of -9/8.

Now, let's compare the slopes and y-intercepts:

  • Slopes: m1 = 1/2 and m2 = 5/8. Since 1/2 is not equal to 5/8, the slopes are different.
  • Y-intercepts: b1 = -3/2 and b2 = -9/8. These are also different.

Because the slopes (m1 and m2) are different, the lines are not parallel and they are not the same line. This means they must intersect at exactly one point. This graphical perspective confirms our algebraic findings: the system has exactly one solution. It's like seeing the lines on a graph – if their slopes aren't the same, they're destined to cross somewhere!

Conclusion: One Solution Found!

Alright guys, we've rigorously analyzed our system of equations using substitution, elimination, and even a conceptual graphical approach. Each method has consistently shown us that the two lines represented by our equations intersect at a single, unique point.

Our system:

-x + 2y = -3
5x - 8y = 9

leads to the solution:

x = -3 y = -3

Therefore, this system has exactly one solution. It's a great feeling to break down a problem like this and be absolutely sure of the answer. Remember these methods, and you'll be able to tackle any system of equations that comes your way! Keep practicing, and happy solving!