Solving System Of Equations: A Graphical Approach

Let's dive into understanding how to solve a system of equations graphically, specifically focusing on the system:

$$\begin{array}{l} x^2+y^2=2 \\ y=2 x^2-3 \end{array}$$

We'll explore what these equations represent and how their graphs interact to give us the solutions. Buckle up, guys, it's gonna be a fun ride!

Understanding the Equations

First, let's break down each equation to understand what kind of shapes they represent on a graph. This will give us a visual intuition before we even start plotting points or using graphing tools.

Equation 1: x² + y² = 2

Hey, what's up? The first equation, x² + y² = 2, should ring a bell! It's in the form of a circle equation: x² + y² = r², where r is the radius of the circle. In our case, r² = 2, so the radius r = √2. This means we have a circle centered at the origin (0,0) with a radius of approximately 1.414. So, envision a circle neatly centered at the origin, not too big, not too small.

Equation 2: y = 2x² - 3

The second equation, y = 2x² - 3, represents a parabola. Specifically, it's a parabola that opens upwards. The general form of a parabola is y = ax² + bx + c. Here, a = 2, b = 0, and c = -3. The coefficient 'a' determines how wide or narrow the parabola is (a larger 'a' means a narrower parabola), and since a = 2, it's a bit narrower than the standard y = x² parabola. The constant term c = -3 shifts the entire parabola vertically downwards by 3 units. This means the vertex (the lowest point) of the parabola is at (0, -3).

Graphical Interpretation

Now that we know what each equation represents, let's think about what it means to solve this system of equations graphically. When we solve a system of equations, we're looking for the points where the graphs of the equations intersect. These intersection points represent the (x, y) pairs that satisfy both equations simultaneously. For our system, we’re looking for where the circle and the parabola meet.

Imagine plotting both the circle and the parabola on the same graph. The circle is centered at the origin, and the parabola opens upwards with its vertex at (0, -3). The solutions to the system are the points where these two curves intersect. The number of intersection points will tell us how many real solutions the system has.

Possible Scenarios

Think about the different ways a circle and a parabola can interact. Here are a few possibilities:

1. No Intersection: The circle might be positioned such that the parabola never touches it. In this case, the system has no real solutions.
2. One Intersection: The parabola could just graze the circle at one point. This means the system has one real solution.
3. Two Intersections: The parabola could cut through the circle at two distinct points, indicating two real solutions.
4. Three Intersections: While less common with a simple parabola, it's theoretically possible if the curves are positioned just right.
5. Four Intersections: The parabola could intersect the circle at four distinct points, indicating four real solutions.

Without plotting the exact graphs (which we can do using graphing software or by hand), it’s hard to say definitively how many intersection points there are. However, visualizing the shapes helps us understand the possibilities.

Solving Graphically (The Actual Process)

To solve this graphically, you would typically follow these steps:

1. Plot the Circle: Draw the circle x² + y² = 2 on a coordinate plane. Remember, it's centered at (0,0) with a radius of √2.
2. Plot the Parabola: Draw the parabola y = 2x² - 3 on the same coordinate plane. The vertex is at (0, -3), and it opens upwards.
3. Identify Intersections: Look for the points where the circle and parabola intersect. These are the solutions to the system.
4. Estimate Coordinates: Estimate the (x, y) coordinates of each intersection point. These are the approximate solutions to the system.

Using a graphing calculator or software like Desmos can make this process much easier and more accurate. Just input the two equations, and the software will plot them and show you the intersection points.

Analyzing the System

Given the equations x² + y² = 2 and y = 2x² - 3, we can substitute the second equation into the first to find the intersection points algebraically and confirm our graphical intuition. Replacing y in the circle equation with 2x² - 3 gives us:

x² + (2x² - 3)² = 2

Expanding and simplifying:

x² + (4x⁴ - 12x² + 9) = 2 4x⁴ - 11x² + 7 = 0

Let z = x², then we have a quadratic equation in terms of z:

4z² - 11z + 7 = 0

We can solve this quadratic equation using the quadratic formula or by factoring. Factoring, we get:

(4z - 7)(z - 1) = 0

So, z = 7/4 or z = 1. Since z = x², we have:

x² = 7/4 or x² = 1

Taking the square root of both sides:

x = ±√(7/4) = ±√7 / 2 or x = ±1

Now, we find the corresponding y values using the equation y = 2x² - 3:

For x = ±√7 / 2:

y = 2(7/4) - 3 = 7/2 - 3 = 1/2

For x = ±1:

y = 2(1) - 3 = -1

Thus, the solutions are (√7 / 2, 1/2), (-√7 / 2, 1/2), (1, -1), and (-1, -1). These are the four points where the circle and parabola intersect.

Conclusion

So, guys, to wrap it up, by understanding the equations and visualizing their graphs, we can determine the nature of the solutions. In this case, the system of equations x² + y² = 2 and y = 2x² - 3 has four real solutions, which correspond to the four intersection points between the circle and the parabola. This graphical approach provides a powerful way to understand and solve systems of equations. And, with tools like graphing calculators and software, it becomes even easier to find those intersection points accurately.

Remember, visualizing the equations and understanding their graphical representations is key to mastering these problems! Keep practicing, and you'll become a pro at solving systems of equations.