Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs. $$ \begin{array}{l} y=x \\ x^{2}+y^{2}=4 \end{array} $$

Step-by-Step Solution

Answer

The points of intersection are \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Step 1: Analyze the Equations

We have two equations: one is a linear equation \( y = x \), and the other is a circle equation \( x^2 + y^2 = 4 \). The linear equation represents a line through the origin with a slope of 1, and the circle equation represents a circle centered at the origin with a radius of 2.

Step 2: Solve Equations Simultaneously

To find the intersection points, substitute \( y = x \) into the circle equation \( x^2 + y^2 = 4 \). This gives \( x^2 + x^2 = 4 \). Simplifying, we get \( 2x^2 = 4 \), so \( x^2 = 2 \). Solving for \( x \), we get \( x = \pm\sqrt{2} \).

Step 3: Determine Corresponding y-values

Since \( y = x \), the \( y \) values corresponding to \( x = \sqrt{2} \) and \( x = -\sqrt{2} \) are \( y = \sqrt{2} \) and \( y = -\sqrt{2} \), respectively. Thus, the points of intersection are \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Step 4: Plot the Equations

Plot the line defined by the equation \( y = x \). Next, plot the circle defined by \( x^2 + y^2 = 4 \). The line will intersect the circle and pass through two points on the circle.

Step 5: Label the Points of Intersection

Mark and label the points \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\) on the plot as points of intersection of the line and the circle.

Key Concepts

Linear Equation

A linear equation is one of the simplest forms of mathematical expressions. It consists of only one or two variables without any exponent. The general form of a linear equation in two dimensions is usually written as \( y = mx + c \). In this equation, \( m \) represents the slope of the line, and \( c \) is the y-intercept, where the line crosses the y-axis. In our exercise, the linear equation is \( y = x \).

• The slope \( m = 1 \) indicates that for every unit increase in \( x \), \( y \) increases by the same amount.
• The y-intercept \( c = 0 \) tells us that the line passes through the origin.

This straightforward relationship means that the line moves diagonally through the origin at a 45-degree angle relative to both the x and y axes, creating a perfect diagonal line.

Circle Equation

The equation of a circle in a two-dimensional plane is \( x^2 + y^2 = r^2 \). This equation consists of squared terms that define how far the points on the circle are from the center. In our exercise, the circle equation is \( x^2 + y^2 = 4 \).

• Here, the center of the circle is found at the origin \((0, 0)\).
• The radius \( r \) of the circle is \( \sqrt{4} = 2 \), which determines how wide or extended the circle is.

Understanding the parts of this equation helps to identify the limits within which the circle will be drawn on a graph, encapsulating all points that are exactly two units away from the center, in every direction.

Solving Simultaneous Equations

Solving simultaneous equations involves finding a set of values for variables that satisfy all given equations at once. In this problem, we have a linear equation and a circle equation:- \( y = x \) (linear equation)- \( x^2 + y^2 = 4 \) (circle equation)To find intersection points, we substitute \( y = x \) into the circle equation:\[x^2 + (x)^2 = 4\]which simplifies to\[2x^2 = 4\]Leading to\[x^2 = 2\]Solving this, we find \( x = \pm \sqrt{2} \). Since the line equation also says \( y = x \), then \( y \) is also \( \sqrt{2} \) and \( -\sqrt{2} \). These solutions provide the points of intersection, \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Graph Plotting

Graph plotting is a visual method to represent algebraic equations. For effective plotting, each variable's relationship with the other(s) is depicted on a coordinate plane.To plot a graph:1. Identify the equations you need to represent. In this exercise, it’s the line \( y = x \) and the circle \( x^2 + y^2 = 4 \).2. For the line \( y=x \), draw a straight line through the origin, making sure it follows a slope of \( 1 \).3. For the circle \( x^2 + y^2 = 4 \), use the center \( (0,0) \) and radius \( 2 \) to draw a perfectly round shape enclosing all points \( 2 \) units away from the center.By following these steps, anyone can recreate the same visual representation of the relationship between the equations.

Intersection Points

Intersection points are where the graphs of given equations meet. These points satisfy all equations simultaneously and can be critical in understanding the relationship between different equations.In this exercise:

• The intersection points are determined by solving simultaneously the equations \( y = x \) and \( x^2 + y^2 = 4 \).
• The solutions \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\) are labeled on the graph as the exact spots where both the line and the circle intersect.

Labeling these intersection points on the graph allows a visual understanding of where the two different shapes physically meet on a coordinate plane. Such clear identification helps in visualizing solutions and understanding their practical implications more deeply.