# Two variables in one equation....

• greg_rack
In summary: I have just come up with a solution... but maybe a bit unorthodox.That third grade at the denominator confused me, I know the process of rationalizing but not if the denominator is elevated to some exponent.By thinking on it, I managed to reach the solution, but I'm not satisfied with the process.In summary, the student tried to solve an equation but got stuck on the third grade equation. After thinking on it, he was able to solve it by using rationalizing the denominator. He is not satisfied with the process however, and wants to post the whole solution.
greg_rack
Gold Member
Homework Statement
When simplified, 1/(1-√2)^3 is written in the form a+b√2 where a and b are integers;

what is the value of b?
Relevant Equations
No equations needed
Now, I obviously started by equalling the two expressions, but ending up with a 2-variables equation which of course can't be solved...
I cannot really understand from where could I take the second equation involving a and b to be able to put the two into a system of equations.

greg_rack said:
Homework Statement:: When simplified, 1/(1-√2)^3 is written in the form a+b√2 where a and b are integers;

what is the value of b?
Relevant Equations:: No equations needed

Now, I obviously started by equalling the two expressions, but ending up with a 2-variables equation which of course can't be solved...
I cannot really understand from where could I take the second equation involving a and b to be able to put the two into a system of equations.
The idea is not to set the original expression equal to ##a + b\sqrt 2##. It's to work with the given expression until it has the form that is requested. What you need to do is to get rid of the ##1 -\sqrt 2## factors in the denominator. You can do this by what is called rationalizing the denominator, by multiplying by the conjugate of the denominator.
For example, ##\frac 2{a + \sqrt b} = \frac{2(a - \sqrt b)}{(a + \sqrt b)(a - \sqrt b)} = \frac{2(a - \sqrt b)}{a^2 -b}##. In the last expression, the denominator no longer has any radicals in it. In the 2nd expression, the original fraction was multiplied by 1 in the form of the conjugate of ##a + \sqrt b## over itself. It's always legitimate to multiply an expression by 1.

Lnewqban
Mark44 said:
The idea is not to set the original expression equal to ##a + b\sqrt 2##. It's to work with the given expression until it has the form that is requested. What you need to do is to get rid of the ##1 -\sqrt 2## factors in the denominator. You can do this by what is called rationalizing the denominator, by multiplying by the conjugate of the denominator.
For example, ##\frac 2{a + \sqrt b} = \frac{2(a - \sqrt b)}{(a + \sqrt b)(a - \sqrt b)} = \frac{2(a - \sqrt b)}{a^2 -b}##. In the last expression, the denominator no longer has any radicals in it. In the 2nd expression, the original fraction was multiplied by 1 in the form of the conjugate of ##a + \sqrt b## over itself. It's always legitimate to multiply an expression by 1.
Got it, but still have a doubt...how do I treat the cubic power at the denominator?

greg_rack said:
Got it, but still have a doubt...how do I treat the cubic power at the denominator?
What have you done so far?

PeroK said:
What have you done so far?
I have just come up with a solution... but maybe a bit unorthodox.
That third grade at the denominator confused me, I know the process of rationalizing but not if the denominator is elevated to some exponent.
By thinking on it, I managed to reach the solution, but I'm not satisfied with the process.

Why don't you just post the whole solution you constructed, and we can tell you if there's anything wrong with it.

archaic and Delta2

## What is the definition of "two variables in one equation"?

Two variables in one equation refers to a mathematical equation that contains two unknown quantities or variables. These variables are typically represented by letters and the equation shows the relationship between them.

## Why is it important to have two variables in one equation?

Having two variables in one equation allows us to solve for one variable in terms of the other, which can help us better understand the relationship between the two variables. It also allows us to solve for both variables simultaneously, which can be useful in real-world applications.

## What are some common examples of equations with two variables?

Some common examples include linear equations, quadratic equations, and systems of equations. For example, y = mx + b is a linear equation with two variables (x and y), while x^2 + y^2 = r^2 is a quadratic equation with two variables (x and y).

## How do you solve an equation with two variables?

To solve an equation with two variables, you typically need to have two equations with the same two variables. You can then use algebraic methods such as substitution or elimination to solve for the values of the variables. Graphing the equations can also help in finding the solution.

## What are some real-world applications of equations with two variables?

Equations with two variables are commonly used in fields such as physics, engineering, economics, and statistics. They can be used to model and analyze various real-world situations, such as motion, population growth, and supply and demand. They are also used in data analysis to determine the relationship between two variables.

• Precalculus Mathematics Homework Help
Replies
3
Views
3K
• Precalculus Mathematics Homework Help
Replies
5
Views
912
• Precalculus Mathematics Homework Help
Replies
2
Views
1K
• Precalculus Mathematics Homework Help
Replies
10
Views
1K
• Precalculus Mathematics Homework Help
Replies
14
Views
2K
• Introductory Physics Homework Help
Replies
20
Views
1K
• Precalculus Mathematics Homework Help
Replies
14
Views
3K
• Aerospace Engineering
Replies
0
Views
545
• Thermodynamics
Replies
8
Views
968
• Precalculus Mathematics Homework Help
Replies
5
Views
2K