Solve each system. Use any method you wish. { (1)/(x^4)-(1)/(y^4)=1 (1)/(x^4)+(1)/(y^4)=

step 1 Okay, here we have a nonlinear system of equations problem. Best thing to do was probably not ju

Solve each system. Use any method you wish. { ...

Key Concepts

Systems of Equations

This concept involves working with two or more equations simultaneously, where the solution must satisfy all equations. Techniques such as substitution, elimination, or transformation are used to find values for the variables that make each equation true.

Nonlinear Equations

Nonlinear equations are those that do not have variables raised only to the first power. They may involve exponents, reciprocals, or other functions of the variables. Solving such systems often requires careful algebraic manipulation to reduce them to a more manageable form.

Exponentiation and Reciprocals

Understanding how exponents and reciprocals interact is crucial. When variables are raised to a power and then inverted, as with 1/x^4, it is important to correctly manipulate these expressions and consider the implications of even powers, which can affect the domain of possible solutions.

Elimination Method

The elimination method is a technique used to remove one of the variables by adding or subtracting equations. This method simplifies the system to a single equation in one variable, making it easier to solve for one unknown before back-substituting to find the other.

Transcript

00:01 Okay, here we have a nonlinear system of equations problem.

00:05 Best thing to do was probably not just the substitution method because we have variables in the denominators.

00:10 It would just be too complicated to go through them and solve for each of them.

00:13 So we can just add these two together in a way, which would result in 1 over x to the fourth plus 1 of x to the fourth, which would just be 2 over x to the fourth.

00:28 The negative 1 over y to the 4th and positive 1 over y to the 4th and positive 1 over y to the 4th we'll just cancel out.

00:33 And then adding the 1 and 4 would result in 5.

00:36 Then we just solve for x, which would give us x.

00:40 So we just multiply both sides by x to the 4, which would get us 2 equals 5, x to the 4th.

00:49 Then we divide everything by 5.

00:53 Essentially we're dividing the left and the right by 5, which would result in x to the 4th equals 2 over 5.

01:02 Then we just take the fourth root of the x of the both sides again, which would give us x equals plus or minus fourth root of two over five.

01:15 Again, this is because x to the fourth, the four is an even exponent, so it would turn a positive or negative number into a positive number.

01:27 So we have to account for the positive square root of two over five and the negative score of or fourth root of two.

01:33 2 over 5 as well...

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