Problem 67 - Precalculus Essentials

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. $$\begin{aligned}x^{2}+y^{2} &=16 \\\x-y &=4\end{aligned}$$

Step-by-Step Solution

Answer

The points of intersections are the ordered pairs that solves both equations simultaneously.

Step 1: Graph the Equations

Start by graphing the two equations in the same coordinate system. The first equation \(x^{2}+y^{2} = 16\) represents a circle with radius of 4 units centered at the origin. The second equation \(x-y=4\) represents a straight line with a slope of 1 and a y-intercept of -4. Graph these using any graphing tool.

Step 2: Find Points of Intersection

The points where these graphs intersect are the solutions to this problem. Visually, you can see they intersect at two points, but to find these exactly, set the equations equal to each other and solve for \(x\) and \(y\).

Step 3: Validate the Ordered Pairs

The points of intersection found will be ordered pairs. Validate these pairs by substituting them into both equations and confirming they hold true in both instances.

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share the same variables. To solve a system means to find the values of the variables that satisfy all the equations simultaneously. In our exercise, we have two equations: the circle equation \(x^{2}+y^{2} = 16\) and the line equation \(x-y=4\). Our goal is to find the point or points where these two graphs intersect, which will be the solution to this system.

When dealing with systems of equations, you can represent the solutions graphically (as we do in this exercise) or solve them algebraically. Graphically, the solution is where the graphs of the equations intersect on a coordinate plane. Algebraically, you often substitute or eliminate variables to solve the equations. Both methods should yield the same set of solutions if done correctly, confirming their accuracy.

Circle Equation

A circle equation in a Cartesian coordinate system is typically written as \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \((h, k)\) is the center of the circle, and \(r\) is the radius. In our specific example, \(x^{2}+y^{2} = 16\), the center is at the origin \((0, 0)\) and the radius is 4 units.

The circle equation describes all the points that are equidistant from the center, forming a perfect round shape on the graph. To graph this circle, locate the center point, and use a compass or another tool to draw a circle with radius 4. This will show us all possible solutions, or points, that lie on the circle.

Line Equation

A line equation in the slope-intercept form is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this task, the equation \(x-y=4\) can be rearranged to the form \(y = x - 4\). Here, the slope \(m\) is 1, indicating a 45-degree angle rise over run, and the y-intercept is -4, where the line crosses the y-axis.

Lines on a graph create a straight path and can be extended infinitely in both directions. To graph the line \(x-y=4\), you can use two points. Start at the y-intercept point \((0, -4)\), and from there, use the slope to find another point. With these points, draw the line on the same graph as the circle to visually determine potential intersection points.

Intersection Points

Intersection points are where two or more graphs meet or cross one another on a coordinate plane. These points, if they exist, are the solutions to the system of equations. For our exercise, after graphing both the circle and the line, identify where they intersect to determine the ordered pairs.

To mathematically find these points, we set the equations equal to each other or substitute one equation into the other. Solving \(x - y = 4\) for \(y\) and substituting into \(x^{2}+y^{2}=16\), we facilitate finding the exact intersection points. Once found, verify them by substituting back into both original equations to ensure they satisfy both. If correct, these coordinates are indeed the solutions you're seeking.