System of Equations: Solving for (x,y)

In summary, the problem was to solve for (x,y) in a system of equations. The first equation involved simplifying with negative powers and rearranging to make y the subject. The second equation was solved by making y the subject. The final solution was (1.2, 2.4), which was verified by plugging in the values into the original equations.
  • #1
prophet05
12
0
[SOLVED] System of equations problem

I'm trying to brush up on my algebra and had difficulties with a problem.

Solve for (x,y): (x^-(2/3))(y^(2/3)) = (2x^(1/3))(y^-(1/3)) and 6 = x + 2y

I'm mainly having problems simplifying the first equation for y but got y=0. Any help?
 
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  • #2
[tex]x^{-a} = \frac{1}{x^a}[/tex]

[tex]\frac{x^a}{x^b} = x^{a-b}[/tex]

Start with that and let's see where it takes you.
 
  • #3
solving the second equation is easy enough - just make y the subject.

As for the first, I find it's easier to rewrite it without using negative powers.
[tex]x^{-2/3} = \frac{1}{x^{2/3}}[/tex]

From there, it's just a matter of rearranging to make y the subject again.
From there, you can find x using both equations.
 
  • #4
Perfect, I think it was the basic concepts that helped. Just to verify:

I was able to simplify the first equation to y = 2x. Using that...
y = 2(6-2y)
y = 12 - 4y
5y= 12
y=12/5

Plug into x = 6-2y...
x = 6 - 2(12/5)
x = 1.2

So, (1.2, 2.4)

That seems right. Thank you.
 
  • #5
Plug in those values back into the original question to see if the left hand side equals the right hand side to make sure you're right.
 
  • #6
(1.2,2.4)

[y=2x]
2.4 = 2 (1.2)
2.4 = 2.4 CHECK

[x=6-2y]
1.2 = 6 - 2(2.4)
1.2 = 1.2 CHECKEverything is verified. Thanks.
 

1. What is a system of equations problem?

A system of equations problem is a mathematical problem that involves finding the values of multiple variables that satisfy a set of equations. It is often used to model real-world situations and can have multiple solutions or no solutions at all.

2. How do I solve a system of equations problem?

There are several methods for solving a system of equations, including substitution, elimination, and graphing. The most common method is substitution, where you solve for one variable in one equation and substitute that value into the other equation to find the value of the other variable.

3. Can a system of equations have more than one solution?

Yes, a system of equations can have infinitely many solutions, or no solution at all. This depends on the equations and the number of variables involved. For example, two parallel lines will have no solution, while two overlapping lines will have infinitely many solutions.

4. How do I know if my solution to a system of equations problem is correct?

You can check your solution by substituting the values into each equation and seeing if they satisfy both equations. If the values make both equations true, then your solution is correct. You can also graph the equations to visually confirm the solution.

5. Are there real-world applications for solving systems of equations?

Yes, systems of equations are commonly used in fields such as economics, engineering, and physics to model real-world situations. For example, a company may use systems of equations to determine the optimal production levels for different products, taking into account production costs and demand.

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