In Exercises 67–70, graph both equations in the same rectangular ...

Question

In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. x² + y² = 16, x-y = 4

Accepted Answer

Hello today we're going to be graphing the following equations and we're going to be finding the points of intersection. So what we are given is x squared plus y squared is equal to we have x minus Y is equal to five. Let's go ahead and start by graphing the first equation. So the first equation given to us is X squared plus y squared is equal to 25. This is the graph of a circle in standard form where the center of the circle is located at the origin and that's going to be of the form X squared plus Y squared is equal to R squared. So since we know that the center is going to be at the origin, we can immediately write down that the center of the circle is going to be at 00. And we need to go ahead and identify the radius of the circle. Well the radius is represented by R or r squared. And in the equation given to us, 25 takes up the spot of r squared. So we can say that for the radius R squared is equal to 25 but let's go ahead and sulfur are in order to sulfur are we're going to take the square root of both sides of this equation And we're going to get arm is equal to five. So this is the equation of a circle whose center is at the origin and has a radius of five. So if I draw access over here, The centre of the circle is going to be at 00. And if it has a radius of five then that means that they're going to be four points total, we're going to have a point up here at zero comma five, we're going to have a point down here at zero comma negative five. We're going to have a point to the right at five comma zero and we're going to have a point to the left at negative five comma zero. And if you connect the four dots together, you get a roughly drawn circle. So now let's go ahead and identify what the graph is for the second equation. So the second equation given to us was x minus y is equal to five. Now this is the form of the equation of a line but not in standard form. So what we need to do is go ahead and put this into the form, why is equal to m x plus B. And in order to start that, we're gonna first subtract X to both sides of the equation and that's going to give us negative y is equal to negative X plus five. And finally we want to get y by itself. So we're going to buy everything by negative one and this is going to give us y is equal to x minus five. So this is the equation of a line. One where five, negative five is the y intercept. So we have b equal to negative five and we have a slope of one. So if this line is starting at negative five on the y intercept, that means that it's going to be starting at the bottomless point of the graph of the circle. So we've already identified one of those intersections of both of the equations of the graphs. And if it has a slope of one, it's eventually going to make its way up to 5:00 for the equation of the line. And if we connect those dots together, this is going to be liquid a graph for the equation of the line. So we have two intersecting points for both of these equations, one at zero comma negative five and one F five comma zero. And the graph that accurately represents this is going to be the graph of B. So that means that B is going to be the answer to this problem. So, I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video

In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. x² + y² = 16, x-y = 4

Key Concepts

Graphing Equations

Points of Intersection

Verification of Solutions

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