Solve each system. Use any method you wish. { 5 x y+13 y^2+36=0 x y+7 y^2=6 .

Name: Problem 38, Chapter 12: Algebra and Trigonometry
Uploaded: 2021-07-28T18:24:33-08:00
Duration: 4 min 12 s
Channel: Ziya Ogron
Description: Solve each system. Use any method you wish. { 5 x y+13 y^2+36=0 x y+7 y^2=6 .

Solve each system. Use any method you wish. { 5 x y+13...

Key Concepts

Factoring

Factoring is the process of rewriting an algebraic expression as a product of simpler expressions. In the context of solving equations, factoring can reveal the roots or solutions when the equation is set to zero, especially when dealing with quadratic or higher degree terms.

Systems of Equations

Systems of equations involve finding values for two or more variables that satisfy all given equations simultaneously. They can be linear or nonlinear, and solving them often requires methods like substitution or elimination to reduce the number of variables in the equations.

Nonlinear Equations

Nonlinear equations include terms that are not simply first degree in the variables, such as product terms or squared terms. They often lead to curves or more complex graphs, and solving a system of nonlinear equations can require techniques that differ from those used for linear systems.

Substitution Method

The substitution method is a strategy where one of the equations in the system is solved for one variable in terms of the other variables, and that expression is substituted into the remaining equations. This reduces the system to fewer equations with fewer variables, making it more manageable to solve.

Elimination Method

The elimination method involves adding or subtracting equations from each other to cancel out one of the variables. This method is useful when the equations are structured such that one variable can be easily eliminated, simplifying the process of finding the solution.

Quadratic Equations

Quadratic equations are polynomial equations of degree two and typically appear when reducing systems or dealing with squared terms. Common methods for solving quadratic equations include factoring, completing the square, and using the quadratic formula, each of which can be instrumental in finding the solutions to parts of a nonlinear system.

Transcript

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

00:06 So because our x, y, and y squares line up, i'm going to solve this system of equation by elimination so that i can eliminate xy to x, y, to help me find what y is.

00:19 So therefore, our first equation, i'm going to move the 36 to the other side.

00:23 So we have 5xy plus 13 y squared equals to negative 3 .3.

00:31 And then the second equation to eliminate the y, the xy, i need to have a negative 5 in front so that 5 plus negative 5 is going to become 0.

00:41 So therefore our second equation here, i'm actually going to multiply by negative 5.

00:46 So therefore we have negative 5 xy minus 35 y squared equals to negative 30.

00:55 So now i'm going to add these two equations together.

00:57 So we know that the 5xy and negative 5xy becomes 0.

01:05 And 13 minus 35, that's going to give you negative 22 y squared.

01:14 And negative 36 minus 30 is going to give you negative 66.

01:18 And when you divide both sides by negative 22, y square is going to equal to 3.

01:28 So therefore, now i'm going to solve for y.

01:33 So when you take the square root both sides, y is going to be equal to either positive square of three or negative square of three.

01:41 So to find the x value, you're going to plug into one of these equations.

01:46 So i'm going to plug into the second equation here.

01:49 So we have x times y.

01:51 Y is going to be the square root three plus seven times y again is going to be the square root three, squared equals to six.

02:05 So i'm going to solve this equation.

02:07 So this is going to give us the screw 3x plus the square is 3...

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