Use the elimination method to find all solutions of the system of equations. { 3 x^2+4 y=17

Use the elimination method to find all solutions of the...

Key Concepts

Nonlinear Systems

Nonlinear systems feature equations in which at least one of the terms is not linear, such as quadratic or higher degree terms. These systems require additional care when applying elimination or substitution techniques because the relationships between variables are more complex than in linear systems.

Quadratic Equation

A quadratic equation is an equation where the variable is raised to the second power, typically expressed in the form ax² + bx + c = 0. In the context of systems that include quadratic equations, solving these equations may involve applying methods like completing the square or the quadratic formula after reducing the system through elimination or substitution.

System of Equations

A system of equations consists of two or more equations that are solved simultaneously to find values for the variables that satisfy every equation. This concept is central to algebra and helps in modeling and solving problems where multiple conditions must be met at the same time.

Elimination Method

The elimination method is an algebraic technique used to solve systems of equations by adding or subtracting equations to cancel one of the variables. This process reduces the system to a simpler one, making it easier to isolate and solve for the remaining variables.

Transcript

00:01 So we're going to solve this system of two equations using the process of elimination.

00:06 So when we do elimination, we need to have one variable sort of matching up with the same variable and the other equation, so that if we were to add or subtract these two equations from each other, one variable would be eliminated in that process.

00:21 Now, we don't have that in the current state.

00:23 We do have x squared and y in both of them, but the coefficients aren't the same.

00:28 What we can do, though, is multiply, both equations by a constant in order to get the coefficients to equal the same thing.

00:36 There's a few ways to do this, but what will work really smoothly is to multiply this first equation by 2 and the second equation by negative 3.

00:45 So that will have 6x squared and negative 6x squared, and that will allow us to eliminate the x.

00:51 So let's do that.

00:52 We'll multiply this first equation by 2 and we have to multiply everything by 2.

00:57 So we get 6x squared plus 8 y equals 34.

01:04 And then we'll do this one by negative 3 to get negative 6 x squared minus 15 y equals negative 6.

01:17 And so then we're ready to add these two together so that we can eliminate.

01:22 So the x is get eliminated.

01:24 And then we have 8y minus 15 y.

01:27 So that's negative 7y.

01:30 And then we have 34 minus 6.

01:33 Which is 28...

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