Real Solutions For System Of Equations A Deep Dive

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Hey guys! Ever wondered how to figure out how many real solutions a system of equations has? It might sound intimidating, but it's actually super cool once you get the hang of it. We're going to dive into a problem that looks at the intersection of a parabola and a line. Specifically, we're going to tackle the system of equations:

y = x^2 + 4
y = 4x

Our mission? To determine how many real solutions exist for this system. Let's break it down step by step!

Understanding the Equations

Before we jump into solving, let's make sure we understand what these equations represent. The first equation, y = x^2 + 4, is a parabola. If you remember your quadratic equations, you'll know that the x^2 term means we're dealing with a U-shaped curve. The + 4 shifts the parabola upwards by 4 units on the y-axis. So, our parabola opens upwards and has its vertex (the lowest point) at (0, 4).

The second equation, y = 4x, is a straight line. This is a linear equation where the slope is 4, meaning for every 1 unit we move to the right on the x-axis, we move 4 units up on the y-axis. The line passes through the origin (0, 0) because there's no constant term added or subtracted.

So, we've got a parabola opening upwards and a straight line. The question of how many real solutions exist is essentially asking: how many times do these two graphs intersect? Each intersection point represents a solution (x, y) that satisfies both equations.

Solving the System of Equations

To find the solutions, we need to find the points where the parabola and the line meet. This means we need to find the x and y values that satisfy both equations simultaneously. A common way to do this is using the substitution method.

Since both equations are already solved for y, we can set them equal to each other:

x^2 + 4 = 4x

Now, we have a quadratic equation in terms of x. To solve it, let's rearrange the equation to get it into the standard quadratic form, which is ax^2 + bx + c = 0. Subtracting 4x from both sides gives us:

x^2 - 4x + 4 = 0

Great! Now we have a quadratic equation in standard form. There are a few ways to solve this. We could try factoring, using the quadratic formula, or even completing the square. In this case, the equation is easily factorable. We're looking for two numbers that multiply to 4 and add up to -4. Those numbers are -2 and -2. So, we can factor the quadratic equation as:

(x - 2)(x - 2) = 0

This simplifies to:

(x - 2)^2 = 0

Setting x - 2 equal to zero gives us:

x = 2

So, we have one solution for x, which is x = 2. This means the line and the parabola intersect at an x-coordinate of 2. To find the corresponding y-coordinate, we can plug this value of x back into either of our original equations. Let's use the simpler equation, y = 4x:

y = 4 * 2
y = 8

Therefore, the point of intersection is (2, 8).

Determining the Number of Real Solutions

We found one solution: (2, 8). This means the line and the parabola intersect at exactly one point. So, how many real solutions exist for this system of equations? The answer is one.

Visually, this means the line y = 4x is tangent to the parabola y = x^2 + 4. A tangent line touches the curve at only one point. If you were to graph these two equations, you'd see that the line just kisses the parabola at the point (2, 8).

Using the Discriminant

There's another cool way to determine the number of real solutions without fully solving the quadratic equation: the discriminant. Remember that the discriminant is the part of the quadratic formula under the square root: b^2 - 4ac. In our quadratic equation x^2 - 4x + 4 = 0, a = 1, b = -4, and c = 4. So, the discriminant is:

(-4)^2 - 4 * 1 * 4 = 16 - 16 = 0
  • If the discriminant is positive, there are two distinct real solutions.
  • If the discriminant is zero, there is exactly one real solution (a repeated root).
  • If the discriminant is negative, there are no real solutions (two complex solutions).

In our case, the discriminant is 0, confirming that there is exactly one real solution. This method is a handy shortcut when you just need to know the number of solutions and not the solutions themselves.

Why This Matters

Understanding how to solve systems of equations and determine the number of solutions isn't just an abstract math concept. It's incredibly useful in many real-world applications. For example, in physics, you might use systems of equations to model the trajectory of a projectile or the intersection of two paths. In economics, you could use them to find the equilibrium point where supply and demand curves intersect. Even in computer graphics, solving systems of equations is crucial for tasks like collision detection and rendering.

So, the next time you see a system of equations, remember that you're not just solving for x and y. You're uncovering relationships between different mathematical objects and gaining valuable insights into the world around you.

Exploring Different Scenarios

Now that we've solved this particular system of equations, let's think about how the number of solutions could change if we tweaked the equations a bit. This is where the beauty of mathematics really shines – exploring variations and seeing how they affect the outcome.

Case 1: No Real Solutions

Imagine we changed the equation of the line so that it never intersects the parabola. What would that look like? Let's consider the system:

y = x^2 + 4
y = 2x

Setting the equations equal gives us:

x^2 + 4 = 2x

Rearranging into standard quadratic form:

x^2 - 2x + 4 = 0

Now, let's calculate the discriminant:

b^2 - 4ac = (-2)^2 - 4 * 1 * 4 = 4 - 16 = -12

Since the discriminant is negative, there are no real solutions. This means the line y = 2x never intersects the parabola y = x^2 + 4. Visually, the line would pass below the parabola without ever touching it.

Case 2: Two Real Solutions

What if we changed the line so it intersects the parabola at two points? Let's look at this system:

y = x^2 + 4
y = 6x - 1

Setting the equations equal:

x^2 + 4 = 6x - 1

Rearranging:

x^2 - 6x + 5 = 0

Calculating the discriminant:

b^2 - 4ac = (-6)^2 - 4 * 1 * 5 = 36 - 20 = 16

A positive discriminant means two real solutions. Factoring the quadratic equation gives us:

(x - 1)(x - 5) = 0

So, x = 1 and x = 5. Plugging these into y = 6x - 1, we get the points (1, 5) and (5, 29). The line y = 6x - 1 intersects the parabola y = x^2 + 4 at two distinct points.

Key Takeaways

  • The number of real solutions to a system of equations tells us how many times the graphs of the equations intersect.
  • For a system involving a parabola and a line, there can be zero, one, or two real solutions.
  • The discriminant (b^2 - 4ac) of the resulting quadratic equation is a powerful tool for determining the number of solutions without solving the equation completely.
  • A positive discriminant means two real solutions.
  • A discriminant of zero means one real solution.
  • A negative discriminant means no real solutions.

By exploring these different scenarios, we gain a deeper understanding of how changing the equations affects the solutions. Math isn't just about finding the right answer; it's about understanding the relationships and patterns that emerge.

Final Thoughts

So, there you have it! We've successfully navigated the world of systems of equations, parabolas, and lines. We've seen how to solve for real solutions and how to use the discriminant as a shortcut. Remember, math is like a puzzle – each piece fits together to create a beautiful and interconnected picture. Keep exploring, keep questioning, and most importantly, keep having fun with it!