Problem 38 - 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. $$ \begin{array}{l} y=4 x+3 \\ x^{2}+y^{2}=81 \end{array} $$

Step-by-Step Solution

Answer

The graphs intersect at approximately \((1.7647, 10.06)\) and \((-5.647, -19.588)\).

Step 1: Understand the Equations

The first equation is a linear equation, \( y = 4x + 3 \), which represents a straight line. The second equation, \( x^2 + y^2 = 81 \), is the equation of a circle with a center at the origin \((0,0)\) and a radius of 9.

Step 2: Rearrange the Linear Equation

For graphing purposes, both equations are already solved for \(y\). The linear equation is \( y = 4x + 3 \). We will graph it directly without any further rearrangement.

Step 3: Graph the Circle

To graph the circle represented by \( x^2 + y^2 = 81 \), plot points that satisfy this equation. This circle has a center at \((0,0)\) and a radius of 9. Draw the circle on the coordinate plane.

Step 4: Graph the Line

Take the linear equation \( y = 4x + 3 \) and graph it on the same coordinate plane. The y-intercept is 3, and the slope is 4, meaning the line rises 4 units for every 1 unit it moves to the right.

Step 5: Find the Points of Intersection

Both equations are graphed on the coordinate plane. To find the intersection points:1. Substitute \( y = 4x + 3 \) into \( x^2 + y^2 = 81 \) to obtain: \[ x^2 + (4x + 3)^2 = 81 \]2. Expand and simplify the equation, then solve for \(x\): \[ x^2 + (16x^2 + 24x + 9) = 81 \] \[ 17x^2 + 24x + 9 = 81 \] \[ 17x^2 + 24x - 72 = 0 \]3. Use the quadratic formula to find \(x\): \[ x = \frac{-24 \pm \sqrt{24^2 - 4(17)(-72)}}{2 \times 17} \] \[ x = \frac{-24 \pm \sqrt{2304 + 4896}}{34} \] \[ x = \frac{-24 \pm 84}{34} \] \[ x = 1.7647 \, ext{and} \, x = -5.647 \].

Step 6: Find Corresponding y-values

For \( x = 1.7647\):\( y = 4(1.7647) + 3 = 10.06\)For \( x = -5.647\):\( y = 4(-5.647) + 3 = -19.588\).

Step 7: Identify and Label the Intersection Points

The points of intersection are approximately \((1.7647, 10.06)\) and \((-5.647, -19.588)\). These points should be labeled on the graph as the points where the circle and line intersect.

Key Concepts

Linear Equation

A linear equation is one of the simplest yet most fundamental concepts in algebra. It is an equation that forms a straight line when graphed on a coordinate plane. A typical linear equation in two variables, such as the one given, is of the form \( y = mx + b \). Here, \( m \) is the slope of the line, indicating its steepness, and \( b \) represents the y-intercept, where the line crosses the y-axis.

This equation, \( y = 4x + 3 \), means that for every step you move to the right along the x-axis, the y-value increases by 4 units. The line crosses the y-axis at 3, which is the value of \( b \). By understanding these components, you can easily graph the line by identifying its steepness and starting point.

Quadratic Formula

The quadratic formula is a powerful tool used to solve equations of the form \( ax^2 + bx + c = 0 \). It is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

When we come across quadratic equations like \( 17x^2 + 24x - 72 = 0 \), plugging them into the quadratic formula allows us to find the x-values that satisfy the equation.

In our exercise, this formula helped us find the x-values where the line and circle intersect. Understanding the discriminant, \( b^2 - 4ac \), is crucial as it indicates the nature of the roots — real and distinct, real and repeated, or complex. Here, a positive discriminant signaled two distinct real roots, leading us to the intersection points on the graph.

Coordinate Geometry

Coordinate geometry, also known as Cartesian geometry, provides a connection between algebra and geometry through graphs. It involves plotting points, lines, and curves on a coordinate plane defined by an x-axis and y-axis.

In our case, coordinate geometry allows us to visualize the equations \( y = 4x + 3 \) and \( x^2 + y^2 = 81 \) via graphs. The meeting points of these graphs are the solutions to the system of equations. These coordinates represent the solution set where each value satisfies both the linear and the circular equation. This graphical representation provides a clear and visual understanding, making it easier to find and label intersection points.

Graphing Techniques

Graphing is a valuable skill in solving and understanding mathematical equations. To effectively graph a linear equation like \( y = 4x + 3 \), one should start by marking the y-intercept on the graph. Then, use the slope, \( m = 4 \), to rise 4 units for every 1 unit moved right, plotting points along this path to draw the line.

For graphing a circle, such as \( x^2 + y^2 = 81 \), identify its center at \((0,0)\) and plot a circle with a radius of 9. Understanding these techniques ensures the accurate construction of graphs, allowing us to clearly identify the intersection points by seeing where the line crosses the circle. Practice in these graphing techniques enhances one's ability to solve problems involving multiple equations visually and accurately.