Solve each system. x^2+y^2 =9 x^2 =9-2 y |...

Name: Problem 23, Chapter 11: Precalculus: Graphs and Models
Uploaded: 2020-06-23T21:36:56-08:00
Duration: 3 min 45 s
Channel: Michael Jacobsen
Description: Solve each system. x^2+y^2 =9 x^2 =9-2 y

Key Concepts

Systems of Equations

A system of equations involves two or more equations with the same set of variables. The objective is to find the values that satisfy all the equations simultaneously. Systems can include linear or nonlinear equations and require strategic methods such as substitution or elimination for their resolution.

Substitution Method

This method involves solving one equation for one variable and then substituting that expression into the other equation(s). It is a powerful tool for reducing a system of equations to a single equation in one variable, simplifying the process of finding the solution.

Quadratic Equations

Quadratic equations are second-degree polynomial equations commonly expressed in the form ax² + bx + c = 0. They frequently appear when solving systems involving curves like circles and parabolas. Mastery of techniques such as factoring, completing the square, or using the quadratic formula is essential to solve these equations.

Conic Sections

Conic sections—including circles, parabolas, ellipses, and hyperbolas—are curves formed by the intersection of a plane with a cone. Understanding their geometric properties and equations is fundamental when analyzing systems where their intersections represent the solutions.

Transcript

00:01 In this video, we are going to solve the system of equations that x squared plus y squared equals 9 and x squared equals 9 minus 2y.

00:09 A great way to solve this system of equations is through substitution, since we know immediately that x squared that appears here and here can be swapped out with 9 minus 2y.

00:22 Let's begin immediately with that solution.

00:25 So we can write next in place of x squared 9 minus 2 .5.

00:30 2 .y plus y squared equals 9.

00:37 In solving this equation, let's first reorder the terms on the left -hand side in descending order on the degree so that we have y squared minus 2y plus 9 equals 9.

00:51 The next step we can take to solve this equation is to subtract 9 from both sides.

00:55 That'll obtain a 0 on one side so that we have y squared minus 2 times y equals 0.

01:02 This allows us to factor by factoring out the greatest common factor of y, leaving y minus 2 equal 0.

01:12 At this stage, we have two factors multiplying, resulting in the quantity of 0.

01:18 The only way that this could happen is if the first factor, y, is zero, or the second factor, y minus 2 is equal to 0.

01:29 In either case, we can go back to the first equation we have here and begin a substitution.

01:35 First let me write down that equation.

01:38 X squared is 9 minus 2 times y, and x squared equals 9 minus 2 times y.

01:45 So in the first case, we have immediately that y is equal to 0...

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