Problem 40 - Precalculus: Mathematical for Calculus

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. $$\left\\{\begin{array}{l}y=x^{2}-4 x \\\2 x-y=2\end{array}\right.$$

Step-by-Step Solution

Answer

Solutions occur at \((1.17, 0.34)\) and \((3.83, 5.66)\).

Step 1: Solve the first equation for y

The first equation of the system is given by \( y = x^2 - 4x \). This equation is already solved for \( y \). This equation represents a parabola.

Step 2: Solve the second equation for y

The second equation is \( 2x - y = 2 \). Rearrange this to solve for \( y \):\[ y = 2x - 2 \] This equation represents a straight line.

Step 3: Graph the first equation

Plot the graph of the parabola \( y = x^2 - 4x \). You can find the vertex at the point \( x = -\frac{b}{2a} = \frac{4}{2} = 2 \), by vertex formula. Substitute \( x = 2 \) into the equation to find \( y = 2^2 - 4 \times 2 = -4 \). Thus, the vertex of the parabola is at \( (2, -4) \) and the parabola opens upwards.

Step 4: Graph the second equation

Plot the graph of the line \( y = 2x - 2 \). To do this, you can identify two points that satisfy the equation. For example, if \( x = 0 \), then \( y = -2 \) giving the point \( (0, -2) \). If \( x = 1 \), then \( y = 0 \) giving the point \( (1, 0) \). Draw the line through these points.

Step 5: Identify the points of intersection

On the graph, observe the points where the line \( y = 2x - 2 \) intersects the parabola \( y = x^2 - 4x \). These intersections are the solutions to the system of equations. Use a graphing calculator or plot with exact scale to find exact intersection points.

Step 6: Find the intersection points and check precision

The points of intersection can be found graphically or with the aid of a graphing calculator. They occur at approximately \( x = 1.17, y = 0.34 \) and \( x = 3.83, y = 5.66 \). Verify these values by plugging into both equations to ensure correct precision to two decimal places.

Key Concepts

Systems of Equations

A system of equations refers to a collection of two or more equations with the same set of unknowns. The primary goal is to find the values of these unknowns that make all the equations true at the same time. In this exercise, the system consists of a parabola and a linear equation:

• The first equation is non-linear: \( y = x^2 - 4x \), a quadratic equation.
• The second equation is linear: \( 2x - y = 2 \).

To solve this system graphically, both equations are plotted on the same set of axes, and solutions are found at the points where they intersect. These points represent the values of \( x \) and \( y \) that simultaneously satisfy both equations. These solutions provide us with the intersection points in this coordinate system.

Parabola

In mathematics, a parabola is a U-shaped curve described by a quadratic equation. The specific equation \( y = x^2 - 4x \) gives a parabola that opens upwards.

• This equation can be rewritten as \( y = (x - 2)^2 - 4 \), revealing a vertex at \( (2, -4) \).
• The vertex formula, \( x = -\frac{b}{2a} \), helps us locate this vertex along the x-axis.

The graph of a parabola provides crucial insights into its shape and opening direction. In this system, its points of intersection with a linear graph outline where it meets the other curve.

Linear Equation

A linear equation is a simple equation that forms a straight line when graphed. The linear equation from our system, \( 2x - y = 2 \), can be rewritten in slope-intercept form as \( y = 2x - 2 \).

• This form shows the slope of the line is 2, indicating a steep incline.
• The y-intercept is -2, meaning the line crosses the y-axis at (0, -2).

Linear equations are straightforward to graph. In our case, plotting two points and drawing a straight line through them will accurately represent this equation on a graph. The solutions to the system appear where this line intersects the parabola.

Graphing Calculator

A graphing calculator is a valuable tool for students dealing with systems of equations, especially when accuracy is important. It aids by providing a visual depiction and allows for easy adjustment of scales to find precise solutions. In this specific task, a graphing calculator can:

• Graph both the parabola and linear equation on the same axes.
• Identify points of intersection by calculating where the graphs overlap.
• Offer more precision to confirm intersection points such as \( x = 1.17, y = 0.34 \) and \( x = 3.83, y = 5.66 \).

This tool significantly simplifies the graphical method, ensuring accurate solutions checked to the necessary decimal places.