# Direct solution for two unknowns in two equations

• MHB
• IanfromBristol
In summary, The conversation is about solving a physical problem using two equations with two unknowns. The coefficients can be calculated and the known values are boundary conditions. The speaker has tried substitution and elimination but it leads to a complicated quartic equation. They are looking for a method to solve it directly and avoid an iterative approach. Dan suggests using an approximation method instead of an exact solution. Ian agrees that the iterative method is easier to work out.
IanfromBristol
I am writing a computer program to solve a physical problem which at some part involves the following two equations and two unknowns;

(1) $C_1(x^4-y^4) + C_2x = Q_1$
(2) $C_1(y^4-x^4) + C_3y = Q_2$

C1, C2 & C3 are coefficients which can readily be calculated and do not rely on knowing the values of X or Y. Q1 and Q2 are known values (effectively boundary conditions). I've tried substitution and elimination but end up with a much more complicated expression which looks to be a quartic equation. I know I can solve this iteratively and it does converge quickly and is stable. However, is there any method that can solve it directly and thus avoid the iterative approach?

I see no way to avoid the quartic. There is a method to do this to get an exact answer (here) but I'd stick with approximation. It's much easier to work out.

-Dan

topsquark said:
I see no way to avoid the quartic. There is a method to do this to get an exact answer (here) but I'd stick with approximation. It's much easier to work out.

-Dan
Hi Dan, I took a look at that solution for the quartic and you're not wrong the iterative method (i.e. approximation) is far easier to work out!

Ian

## 1. What is a direct solution for two unknowns in two equations?

A direct solution for two unknowns in two equations refers to the process of finding the values of two variables that satisfy both equations simultaneously. This method is also known as solving a system of equations.

## 2. How do you solve a system of equations with two unknowns?

To solve a system of equations with two unknowns, you can use the substitution method or the elimination method. In the substitution method, you solve one equation for one variable and substitute it into the other equation. In the elimination method, you manipulate the equations to eliminate one variable and solve for the other.

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

Yes, a system of equations can have one solution, no solution, or infinitely many solutions. This depends on the relationship between the two equations. For example, if the two equations represent parallel lines, there will be no solution. If the two equations represent the same line, there will be infinitely many solutions.

## 4. What is the importance of solving a system of equations with two unknowns?

Solving a system of equations with two unknowns is important in many fields, including mathematics, science, engineering, and economics. It allows us to find the values of variables that satisfy multiple conditions, which is useful in real-world problem-solving and decision-making.

## 5. Can technology be used to solve a system of equations with two unknowns?

Yes, technology such as graphing calculators and computer software can be used to solve a system of equations with two unknowns. These tools use algorithms to find the solutions quickly and accurately, saving time and effort compared to solving by hand.

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