Solving x & y from 2 Equations in Terms of a & b

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In summary, Homework Equations involve solving algebra equations. The Attempt at a Solution found that adding the equations results in a + b. The correct answer is x = (3√3b + a) - √y.
  • #1
The legend
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Homework Statement



Find x and y in terms of a and b from the following equations

x * √x + y * √y = a
x * √y + y * √x = b

Homework Equations


the general algebra equations


The Attempt at a Solution



adding the equations

(x + y) * (√x + √y) = a + b
x + y =
a + b
_______
√x + √y​

x =
a + b + y√x + y√y)
______________________
√x + √y​

Now substituting in eqn 1

a + b + y√x + y√y
_________________ + y√y = a
√x + √y

a + b + y√x + y√y + y√x√y + y2 = a√x + a√y


... Now I don't know how to proceed..
 
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  • #2
To begin, add 3 times the second equation to the first one.

ehild
 
  • #3
I tried it out..

x * √x + y * √y = a
3*x * √y + 3*y * √x = b

adding and using (p + q)3 formula

(√x + √y)3 = 3b + a

therefore

√x = (3√3b + a) - √y
so
x√x = {3(√3b + a) - √y}3

Further solving

I get y =
-(3√3b + a)2 +- √(3√3b + a)4 + 4b * (3√3b + a)
_________________________________________________________________
2 * (3√3b + a)​


Is it right?(Because I haven't got its solutions.. and I have never got such complex answers!)
 
  • #4
The legend said:
I tried it out..

x * √x + y * √y = a
3*x * √y + 3*y * √x = b

adding and using (p + q)3 formula

(√x + √y)3 = 3b + a

Correct up to here, but all the others are wrong. What is x+y? The whole left side should be under the cubic root.

More: You can factor out √x √y from the second equation. As you know
(√x + √y), you get an equation for √x √y.

There will be a system of two equations, one for (√x + √y), the other for √x √y, which is easy to solve.

ehild
 
  • #5
Whew!
I tried it out your way and got the answer!(which I don't think I want to key down)

Though u say that the whole left side should be under cube root... actually it is(that is cube root of (√x + √y)3 so it will be just √x + √y)... so anything wrong? ( maybe I went wrong in the calculations)
 
  • #6
I mixed my hands... :) I meant the right side.

√x = (3√3b + a) - √y--this is wrong. You have the cubic root of 3 alone.

[tex]\sqrt{x}+\sqrt{y}=\sqrt[3]{3b+a}[/tex]
ehild
 
  • #7
I think that's what he implied by the use of brackets (even though it wasn't correctly expressed) :smile:
 
  • #8
Ok thanks ehild!
And also Mentallic got me right! I juggled up my use of brackets!
 

FAQ: Solving x & y from 2 Equations in Terms of a & b

What is the process for solving x and y from two equations in terms of a and b?

The process for solving x and y from two equations in terms of a and b is called simultaneous equations. This involves setting up two equations with the same variables (x and y) and solving for them at the same time.

What are the steps for solving simultaneous equations?

The steps for solving simultaneous equations are:
1. Rearrange the equations so that the variables (x and y) are on one side and the constants (a and b) are on the other side.
2. Choose a variable (x or y) to eliminate by multiplying one or both equations by a number that will make the coefficients of that variable the same.
3. Add or subtract the equations to eliminate the chosen variable.
4. Solve for the remaining variable.
5. Substitute the value found in step 4 into one of the original equations and solve for the other variable.
6. Check the solution by substituting the values of x and y into both equations to see if they are true.

What are the possible outcomes of solving simultaneous equations?

The possible outcomes of solving simultaneous equations are:
1. One unique solution where both x and y have a specific value.
2. No solution, which means the equations are parallel and will never intersect.
3. Infinite solutions, where the equations represent the same line and have an infinite number of intersection points.
4. Inconsistent solution, where the equations represent two parallel lines that do not intersect.

What are some tips for solving simultaneous equations?

Some tips for solving simultaneous equations are:
1. Choose the variable with the smallest coefficient to eliminate first.
2. If the coefficients of one variable are already equal, skip step 2 and proceed to step 3.
3. Use the addition method if the coefficients of the chosen variable are the same. Use the subtraction method if they are opposite.
4. Always check your solution by substituting the values into both equations to ensure they are true.
5. If the equations are not in standard form, use the distributive property to rearrange them.

How can simultaneous equations be used in real life?

Simultaneous equations are used in real life to solve problems involving multiple variables. They can be used in fields such as physics, engineering, and economics to analyze systems and make predictions. For example, simultaneous equations can be used to predict the cost of producing a certain number of products based on the cost of materials and labor. They can also be used to calculate the speed and distance of an object in motion.

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