Problem 65 - Algebra and Trigonometry

Question

In Exercises 63–68, find the solution set for each system by graphing both of the system’s equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{l} {x=y^{2}-3} \\ {x=y^{2}-3 y} \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions of the system would be the exact coordinate points where the graphs of the two equations intersect. To find these points, a graphical method or software might be needed.

Step 1: Graph the First Equation

The first equation is \(x=y^{2}-3\). Here, for each value of y, x is calculated as the square of y minus 3. Calculating x for a range of y values and plotting on the graph will give a parabola.

Step 2: Graph the Second Equation

Now focusing on the second equation \(x=y^{2}-3y\). Similarly, for each y, calculate x as the square of y minus thrice of y. Plotting these values on the same graph as Step 1, a second parabola is formed.

Step 3: Find Intersection Points

The points where the two parabolas intersect are the solutions to the system of quadratic equations. By visually comparing the two graphs, you can find where they intersect.

Step 4: Check the Solution

Substitute the solutions back into the original equations to verify. For any solution, when substituted into the first and second equation, you should end up with the same value for x in both equations.

Key Concepts

Quadratic Equations

Quadratic equations take the form of an equation where the highest exponent of the variable is squared, like in \[x = y^2 - 3\] or \[x = y^2 - 3y\]. These are second-degree equations and can have various forms: standard, vertex, or factored form. They are fundamental in graphing parabolas. Each equation represents a curve called a parabola on a coordinate grid. In this specific system of equations exercise, both equations are quadratic in the variable y.

• A quadratic equation results in a U-shaped curve when graphed.
• It's crucial to know the general properties of quadratics to understand how they behave on graphs.

The system in this exercise involves finding common solutions for both quadratic equations by looking at where their graphs intersect.

Graphing

Graphing is the process of drawing a mathematical function on a set of axes. For quadratic functions, we graph them to visualize the shape and make it easier to see where they might intersect with other equations. To graph a quadratic, you start by calculating points for various values of y. For each y value, you substitute it into the equation to find x, and plot the point \((x, y)\).

• It's like drawing a connect-the-dots with your calculated points.
• These points will form a bow-shaped curve when connected.

Graphing makes it easier to solve systems of equations because you can visualize where the curves intersect.

Parabolas

Parabolas are the curved shapes you get when you graph a quadratic equation. They open upwards or downwards depending on the sign of the leading coefficient (the number in front of \(y^2\)). These curves are symmetrical, and their highest or lowest point is called the vertex. In our exercise, both equations \(x = y^2 - 3\) and \(x = y^2 - 3y\) graph into parabolas.

• If you imagine slicing a cone, the shape of the cut would be a parabola.
• The points where these parabolas touch or cross each other are very important in systems of equations.

Understanding parabolas helps in identifying which parts of the graphs will intersect.

Intersection Points

Intersection points are where two graphs cross each other. These points are solutions to a system of equations because they satisfy all equations in the system. In the current exercise, the intersections of our two parabolas in the graphs represent the x and y values where both quadratic equations are true.

• Finding these points visually on a graph gives you the solution quickly.
• Always technically check each solution by plugging back into both equations to ensure they work.

In practice, intersection points tell you the \(x\) and \(y\) pairs where both equations intersect and hence, both hold true simultaneously.