Exploring the Intersection of Two Equations: y x - 5 and y 3 - 2x^2

This article explores the process of solving the system of equations y x - 5 and y 3 - 2x^2. We will start by examining the algebraic solution and proceed to a graphical approach using the R software's ggplot2 and cowplot packages. By the end, you will understand the solutions both mathematically and visually.

Algebraic Solution

Let's begin by solving the system y x - 5 and y 3 - 2x^2. We can set the two equations equal to each other to find the points of intersection:

y x - 5

y 3 - 2x^2

Setting y y, we get:

x - 5 3 - 2x^2

Which simplifies to:

2x^2 x - 8 0

This is a quadratic equation, and solving for x requires the quadratic formula or factoring:

x -4 or x 1.5

Substituting these x-values back into either equation to find y, we get:

x -4:

y -4 - 5 -9

x 1.5:

y 1.5 - 5 -3.5

Graphical Solution: Using R

While the algebraic solution gives us the points of intersection, a graphical approach can provide a clear visual understanding of where these intersections occur. We will use R software's ggplot2 and cowplot packages to plot the two functions:

Dashed line: x-5

Solid line: 3 - 2x^2

Here is how you can do it:

Install and load the necessary packages: Create the data frames for the two functions: Plot the functions using ggplot2 and cowplot to see the points of intersection:

Based on the plot, it is clear that the solution points are near x 0 and x 2. We can verify these solutions with the algebraic method:

Verification Using the Algebraic Solution

Let's verify the solutions by substituting x 0 and x 2 into the equations:

x 0:

y 0 - 5 -5

y 3 - 2(0)^2 3

x 2:

y 2 - 5 -3

y 3 - 2(2)^2 -5

This confirms that the points of intersection are indeed at x 0 and x 2.

Alternative Methods for Solving Equations

Another method to solve the equations is by breaking each equation into two parts based on the value of x:

For y x - 5: When x 5, y 0. When x > 5, y 5 - x. For y 3 - 2x^2: When x 1.5, y 5 - 2x. When x > 1.5, y 2x - 3.

By solving the appropriate sets of equations for the intervals, we can further confirm the points of intersection. For example, in the interval [1.5, 5], the equations intersect at x 1.5, and in the interval [5, ∞), they intersect at x 5. Plugging these values back into either equation, we confirm the solutions.

For verification, you can also use a graphing calculator to plot the original equations and observe where the lines intersect. Both methods will yield integer solutions.

Additional Notes

Consider the hints provided:

x - 5 0 when x 5 3 - 2x 0 when x -1.5

Think about x -2, x 0, and x 6. What is interesting about those three values? Using a number line, mark 5 and -1.5 clearly to understand the intervals better.

By combining algebraic and graphical methods, you can thoroughly solve and understand the intersection of equations, ensuring a comprehensive understanding of how these solutions are derived.