Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 31-38, plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{aligned} &y=x \\ &x^{2}+y^{2}=4 \end{aligned} $$

Step-by-Step Solution

Answer

The graphs intersect at \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Step 1: Identify the type of equations

The first equation, \(y = x\), represents a straight line. The second equation, \(x^2 + y^2 = 4\), represents a circle with a radius of 2, centered at the origin.

Step 2: Find points of intersection analytically

To find the points of intersection, substitute \(y = x\) from the first equation into the second equation: \(x^2 + x^2 = 4\), which simplifies to \(2x^2 = 4\). Solve for \(x\): \(x^2 = 2\) leading to \(x = \pm \sqrt{2}\).

Step 3: Solve for corresponding y-values

Use \(y = x\) to find the corresponding \(y\) values for the \(x\) values calculated. When \(x = \sqrt{2}\), \(y = \sqrt{2}\). When \(x = -\sqrt{2}\), \(y = -\sqrt{2}\).

Step 4: Graph the equations on the coordinate plane

Plot the straight line \(y = x\) and the circle \(x^2 + y^2 = 4\) on the same coordinate plane. The line will cross through the origin at a 45-degree angle and the circle will have a radius of 2, centered at (0, 0).

Step 5: Identify points of intersection

By analyzing the graph, confirm that the graphs intersect at the points \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

Key Concepts

Equation Graphing

Equation graphing involves plotting equations on a coordinate plane. It is like bringing equations to life as visual representations, which can be studied to gain insights into their behavior. When graphed:

• A line such as \( y = x \) appears as a diagonal line crossing through the origin, creating a 45-degree angle with the axes.
• A circle such as \( x^2 + y^2 = 4 \) forms a perfect round shape. Its center is at the origin (0, 0) and it spreads outward equally since all its points are equally distant from the center. The equation \( x^2 + y^2 = 4 \) represents a circle with a radius of 2.

Graphing both equations on the same plane helps to visualize their intersection points, which tells us where the equations' solutions overlap.

Points of Intersection

Points of intersection are locations where two or more graphs meet on a coordinate plane. They are significant because they represent solutions that satisfy multiple equations simultaneously.
To find these points:

• Substitute one equation into another. For example, substituting \( y = x \) into \( x^2 + y^2 = 4 \) turns the problem into a single-variable one: \( 2x^2 = 4 \).
• Solve for the variable, which might involve algebraic techniques such as factoring or using the quadratic formula. Here, solving \( x^2 = 2 \) gives two possible solutions: \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
• Use the initial equation to find corresponding \( y \)-values, giving points \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2}, -\sqrt{2})\).

These points give an understanding of where and how the equations interact.

Mathematical Analysis

Mathematical analysis allows us to dissect complex problems into smaller, manageable parts. This technique facilitates a deeper understanding of each component and how it contributes to the whole.
In solving systems of equations:

• We begin by identifying the type and nature of each equation, such as a line or a circle in this case.
• The process involves substitution to reduce complexity, which transforms two equations into one simplified form.
• Solving the simplified equation often requires using square roots, as in finding \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
• Reversing the substitution helps to find corresponding solutions for other variables, crucial in identifying intersection points.

By systematically following these steps, not only do we find solutions, but we also reveal the interactions and relationships between multiple equations.