Solve each system. Use any method you wish. { x+y+1=0 x^2+y^2+6 y-x=-5 .

Name: Problem 29, Chapter 12: Algebra and Trigonometry
Uploaded: 2021-07-28T16:45:57-08:00
Duration: 5 min 8 s
Channel: Ziya Ogron
Description: Solve each system. Use any method you wish. { x+y+1=0 x^2+y^2+6 y-x=-5 .

Solve each system. Use any method you wish. { x+y+1=0 ...

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial plus a constant. This technique is useful for rewriting conic sections or solving quadratic equations by revealing their vertex form, which can simplify the process of finding the variable’s value.

Quadratic Equations

Quadratic equations, which are polynomial equations of degree two, often arise when solving nonlinear systems or after substituting one variable in terms of another. The solutions can be found using factoring, the quadratic formula, or completing the square.

Systems of Equations

Systems of equations involve finding common solutions to multiple equations simultaneously. They can include linear equations, nonlinear equations, or a mix of both, and solving them requires methods that reduce complexity such as elimination or substitution.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation(s). This technique simplifies the system by reducing the number of variables, making it easier to solve.

Transcript

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

00:05 So i'm going to use the substitution method because we can either isolate the x or the y from the first equation.

00:13 So therefore, the first equation is the same thing as x equals to subtract y and subtract 1.

00:24 So that's negative y minus 1.

00:26 And now i'm going to substitute that x equals into our second equation, wherever the x is.

00:32 So right here and right there.

00:34 So therefore, our second equation is going to become negative y minus one squared plus y squared plus 6y minus our x here is minus negative y minus equals to negative five.

01:05 So now let's simplify this.

01:08 So we know that negative y minus 1 means you're multiplying negative y times negative y minus 1.

01:18 So you have to foil this.

01:20 So that's going to give you y squared plus y plus another y plus one square, which is 1.

01:28 So therefore, negative y minus 1 squared, same thing as y squared plus 2y plus 1.

01:36 And then plus y squared plus 6y.

01:40 And now i'm going to distribute the negative.

01:42 So this will become positive y plus 1 equals to negative 5.

01:49 So now we're going to combine like terms.

01:51 So we have y square here and y square here.

01:55 So that's going to give us 2 y squared.

01:58 And then we have 2y plus 6y, which is 8y plus another y.