In Problems 25-54, solve each system. Use any method you wish. { 2 x^2+y^2 =18 x y =4.

Name: Problem 25, Chapter 11: Precalculus: pearson new international edition
Uploaded: 2021-07-28T16:20:03-08:00
Duration: 5 min 35 s
Channel: Ziya Ogron
Description: In Problems 25-54, solve each system. Use any method you wish. { 2 x^2+y^2 =18 x y =4.

In Problems 25-54, solve each system. Use any method you...

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two and are typically solved by factoring, completing the square, or applying the quadratic formula. When quadratic forms appear in a system, they require careful handling to ensure all valid solutions are identified and extraneous solutions are eliminated.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others, and then substituting this expression into the other equation(s). This method is particularly useful in nonlinear systems where isolating a variable can simplify the process of finding solutions.

Nonlinear Equations

Nonlinear equations involve variables raised to a power other than one or include products of variables, making them more complex than linear equations. Solving systems that contain nonlinear equations often requires specific strategies, such as substitution to reduce the problem to a solvable form.

System of Equations

A system of equations is a set of two or more equations with the same unknowns. The goal is to find a common solution that satisfies all the equations simultaneously. Systems can be linear or nonlinear and may require different methods like substitution or elimination for their solution.

Transcript

00:01 For this problem, we're asked to solve the system of equation using any method.

00:06 So for this problem, i think it's easier if we solve it by using the substitution method where i can either isolate the x or the y looking at the second equation.

00:18 So with this second equation, i'm going to solve for y.

00:21 You can solve for x if you want to.

00:23 It doesn't really matter.

00:24 So i'm going to solve for y, so therefore this is the same thing as y equals to 4 over x.

00:29 And then i'm going to substitute 4 over x into the y to the first equation.

00:36 So therefore the first equation is going to become 2x squared plus 4 over x squared equals to 18.

00:48 So now we're going to simplify this.

00:52 So we have 2x squared plus and i'm going to square the numerator and the denominator.

00:57 So we have 16 over x squared equals to 18.

01:01 And this is a quadratic equation, so i'm going to set equal to 0.

01:06 So i'm going to subtract 2x squared on both sides.

01:10 So therefore we have 16 over x squared equals to negative 2x squared plus 18.

01:19 And then i'm going to get rid of the fraction by multiply both sides by x squared on both sides.

01:28 So therefore we have 16 equals to negative 2x to the 4.

01:34 Plus 18x squared.

01:37 And again, i can set that equal to zero again by subtracting 16.

01:42 So we have 0 equals to negative 2x to the fourth plus 18x squared minus 16.

01:53 And we can solve this equation in the form of a quadratic equation because we can rewrite this equation as, if we let another variable u equals to x squared.

02:09 So therefore, this equation is same thing as 0 equals to negative 2, u, which is x squared, plus another squared, that will give you x to the fourth, plus 18 x squared, x squared is u minus 16.

02:26 So as you can see, this is a quadratic equation now, so we can solve for you by factoring, or you can use the quadratic formula.

02:34 If you want to, but i think it's easier with factoring...

Please give Ace some feedback