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

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two. They can be solved by methods such as factoring, completing the square, or using the quadratic formula. Recognizing and solving quadratic equations is essential in many contexts, including when they appear as the result of substituting one equation into another in a system.

Intersection of Curves

The intersection of curves refers to the set of points that satisfy multiple equations simultaneously, each representing a different curve. Understanding how curves like circles and parabolas intersect helps in visualizing and solving nonlinear systems of equations, as the real solutions correspond to the coordinates of the intersection points.

Nonlinear Systems of Equations

Nonlinear systems involve equations in which the variables are raised to powers other than one or appear in products or other non-linear functions. These systems often produce curves such as circles, parabolas, or hyperbolas when graphed, and solving them requires techniques that account for their non-linear nature.

Substitution Method

The substitution method is a technique for solving systems of equations in which one equation is solved for one variable and then substituted into the other equations. This method simplifies the system by reducing the number of variables in the equations, making it easier to isolate and solve for the remaining unknowns.

Transcript

00:01 So we're being asked to solve the given nonlinear system of equations.

00:05 Now, what i'm going to do is i'm going to use the substitution method to do this.

00:11 Now, i am going to use the second equation.

00:14 However, i'm not going to substitute that expression in for x because then i'm going to end up having the square rate and i'm going to get y to the fourth terms and that could get a little confusing.

00:22 So what i am going to do is i'm going to get y squared by itself by adding five to both sides of the equation.

00:29 So now what i have is that x plus 5 is equal to y squared.

00:36 So now what i can do is i can substitute x plus 5 in place of y squared in the first equation.

00:42 So what we're going to have is x squared plus x plus 5 is equal to 25.

00:49 Well, now we have a quadratic equation.

00:52 So i have to get it set equal to zero.

00:54 So to do that, i'm going to subtract 25 from both sides.

00:57 And now we're left with x squared plus x minus 20 equals zero so now i can solve this equation by factoring well what factors of negative 20 will add to positive 1 well that would be negative 5 and positive 4 so my two factors are x plus 5 and x minus 4 so now to solve i simply set both of the equations equal to 0 so we have x plus 5 equals 0 and x minus 4 equals 0 and now we solve well for the first equation, we must subtract 5 from both sides, so x could be negative 5, and from the second equation, we're going to add 4 to both sides, so x could equal to 4.

01:41 So now that we've found our values for x, we have to find their corresponding y values.

01:47 But remember, in that second equation, we already got y squared by itself.

01:51 So what i'm going to do is go to a different page, and remember, we had the equation x plus 5 is equal to y squared.

01:59 And looking back, remember, we just found we had for x values negative 5 and 4.

02:06 So first, i'm going to take negative 5 and substitute it in for x.

02:11 I'm also going to put over here, so i don't forget, our other solution is x is 4.

02:15 So when i substitute negative 5 in for x, we're going to have negative 5 plus 5 equals y squared.

02:23 Well, negative 5 plus 5 is 0...

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