Problem 67 - College Algebra

Question

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{array}{c} x^{2}+y^{2}=16 \\ x-y=4 \end{array} $$

Step-by-Step Solution

Answer

From this process, we get two intersection points which satisfy both equations. The points can be verified by substituing into both the original equations.

Step 1: Graph the Equations

First, graph both equations on the same rectangular coordinate system. The equation \(x^{2}+y^{2}=16\) represents a circle with center at (0,0) and radius 4. The equation \(x-y=4\) is a line with slope 1 and y-intercept -4.

Step 2: Find Intersection Points

Observe your graph and find the points of intersection of the circle and the line. These are the points that satisfy both equations. This can be solved algebraically by substituting the second equation into the first. Solving \(x^{2}+y^{2}=16\) using \(x = y + 4\) (from rearranging \(x-y=4\) ), will give 2 possible solutions.

Step 3: Show that the ordered pairs satisfy both equations

Substitute the obtained ordered pairs into both original equations to verify they satisfy both. If the left side of the equation equals the right side after substituting, then the point satisfies the equation.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane formed by two perpendicular lines called axes. These axes are labeled as the x-axis (horizontal) and y-axis (vertical). The point where these axes intersect is called the origin, with coordinates (0, 0). Each point in this plane can be identified by an ordered pair (x, y), which tells you how far along the x-axis and how far up or down the y-axis the point is from the origin.
When graphing equations, we plot points that satisfy these equations on the coordinate plane. This allows us to visually see how different equations relate to one another. In this exercise, both the circle and the line are graphed in the same coordinate system, making it easier to see their intersection points.

Circle Equation

A circle's equation in a rectangular coordinate system is given by the formula \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle and the center is at the origin (0, 0). This standard form makes it easy to graph a circle because all you need is the radius to draw a perfect circle around the origin.
In this exercise, the circle is defined by the equation \(x^2 + y^2 = 16\), meaning it has a radius of 4, since \(r^2 = 16\). We choose points that satisfy this equation, plot them in the coordinate system, and then draw a smooth curve to form the circle. This visual representation helps us identify intersection points with other figures, like lines.

Line Equation

The equation of a line in its simplest form is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope \(m\) describes the steepness of the line, while the intercept \(b\) is where the line crosses the y-axis.
For this problem, the line equation is given as \(x - y = 4\). Rearranging it to the form \(y = x - 4\), we see that the slope \(m\) is 1, and the y-intercept \(b\) is -4. This indicates that the line rises one unit for every unit it moves right. Graphing this line on the coordinate system allows us to see how it interacts with the circle.

Points of Intersection

Finding the points of intersection involves determining where two graphs intersect, or share common points. For our exercise, these are points where the circle defined by \(x^2 + y^2 = 16\) and the line \(x - y = 4\) cross each other.
Substitute \(y = x - 4\) (from the rearranged line equation) into the circle equation to find the intersection algebraically. The substitution simplifies the circle equation to \(x^2 + (x-4)^2 = 16\). Solve this quadratic equation to find possible values for \(x\). These \(x\) values will help us determine corresponding \(y\) values from the line equation, giving us intersection points.

• The points of intersection are the solutions that satisfy both equations simultaneously.

These points can then be verified by substitution back into the original equations, confirming they satisfy both and thus are true points of intersection.