Solve each nonlinear system of equations for real solutions....

Key Concepts

Elimination Method

The elimination method is used to reduce the number of variables in a system of equations by subtracting or adding equations to cancel out common terms. This approach is particularly useful in systems where symmetry or similar algebraic structures exist, allowing one to derive relationships between the remaining variables and simplify the problem.

Geometric Interpretation of Equations

Understanding the geometric meaning of equations aids in visualizing the problem. In many nonlinear systems, equations represent geometric shapes such as circles or other conics, and solving the system involves finding the intersection points of these shapes. This perspective can provide insights into the number and nature of the solutions.

Nonlinear Systems of Equations

A nonlinear system involves equations in which the variables appear in non-linear combinations, such as powers or products. These systems can have multiple, unique, or even no real solutions, making the process of solving them involve a range of algebraic or numerical techniques tailored to handle nonlinearities.

Transcript

00:01 In this problem, we're being asked to solve the given nonlinear system of equations.

00:05 Well, to do that, i'm going to use the substitution method.

00:09 So i'm going to look at this first equation, x squared plus y squared is equal to 1.

00:14 And i'm just going to get x squared by itself.

00:16 To do that, i'm just going to subtract y squared from both sides of the equation.

00:22 So now we're going to have x squared is equal to 1 minus y squared.

00:27 So here's what i'm going to do.

00:28 I'm going to take 1 minus y squared and plug it.

00:31 It into x squared in that second equation because then i'll only have an equation with y's.

00:37 So let's see what we're have.

00:38 Well, we're going to have 1 minus y squared plus, well, i have that quantity of y plus 3 squared.

00:44 So i'm going to rewrite that as y plus 3 times y plus 3, and that's equal to 4.

00:50 So now what we have to do is we need to foil our binomials out.

00:55 So we're going to have 1 minus y squared plus, well, y times y is positive y squared, y times 3 is positive 3y, 3x, 3x, 3x, and 3 times 3 is positive 3y, and 3 times 3 is positive 9, and it's all equal to 4.

01:10 So now let's combine our like terms.

01:13 Well, i have negative y squared plus y squared, that's 0, they cancel.

01:18 I have 3y plus 3y, which is 6y, and then i have 1 plus 9, which is positive 10, and it's equal to 4...

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