Use a graphing calculator to solve each system of equations. { x^2+y^2=68 y^2-3 x^2=4

Use a graphing calculator to solve each system of equations....

Key Concepts

Systems of Equations

This concept involves finding values for variables that satisfy multiple equations simultaneously. In the given problem, it means determining the intersection points of the curves represented by the equations.

Non-linear Equations

These are equations where the variables are raised to a power different from one, such as squared terms. Since both equations in the problem are quadratic, solving them involves methods specific to non-linear algebraic systems.

Graphical Analysis

Utilizing graphical representations, often via a graphing calculator, allows one to visually determine the points where the graphs of the equations intersect, which correspond to the solutions of the system.

Conic Sections

The equations represent specific types of conic sections. The equation x² + y² = 68 defines a circle, while the modified quadratic form in y² - 3x² = 4 can be interpreted as relating to another conic section. Understanding their geometric properties helps in analyzing the system.

Transcript

00:01 And this problem, we're being asked to solve a system of equations using our graphing calculator.

00:08 So the first thing we have to do is get our equation, x squared plus y squared equals 68, and solve for y.

00:16 So the top side of our circle, which is the positive and the bottom, which is the negative.

00:23 And we have the square root of 68 minus x squared.

00:29 Now getting our second equation, we've got y squared minus 3x squared equals 4.

00:38 And that tells us we're going to have a vertical hyperbola.

00:43 And so we're going to have the top side of the hyperbola, which is the plus, the bottom side of the hyperbola, which is the minus, and the square root of 4, 4 plus 3x squared.

01:02 Now we're ready to get our graphing calculator.

01:05 So we'll go get our graphing calculator here.

01:10 And we're going to plug in our equations.

01:12 So we're going to go to y equals and helps you out.

01:15 This will show you the buttons i'm pressing down here at the bottom.

01:22 So doing our first one, we're going to need a square root.

01:25 So there's our square root of 68 minus x squared.

01:37 And then we're going to go down to the next one.

01:38 We'll do the minus or the negative square root.

01:42 So we want to use this negative sign down here at the bottom.

01:44 We do not want to use the minus sign.

01:48 And we have the square root, again, of 68 minus x squared.