Formula for multiple integral equations

Question

I need a formula to determine the value of $X$ & $Y$ in the below examples:

\begin{cases} 26.95 - 26.95X - Y = 26.22 \\ 80.82 - 80.82X - Y = 79.23 \\ 53.87 - 53.87X - Y = 52.71 \end{cases}

Formula should always take either 2 or 3 known equations as above and be able to determine the value of both $X$ and $Y$.

Answer 1

Generally, you would need to rewrite all equations in the form $aX + bY = c$ and apply Gaussian Elimination.

In the specific example you posted, however, you can do the following manipulations without matrices easily:

1. Solve for $Y$ in terms of $X$ from the first equation.
2. Plug this into the second equation and solve it for $X$.
3. Use this solution for $X$ to substitute into the formula from (1) to get $Y$.
4. Plug both into the 3rd equation if it exists.

Let's take your example. From the first equation, $$ Y = 26.95 - 26.22 - 26.95 X = 0.73-26.95X \tag{1} $$ which you can plug into the second equation: $$ \begin{split} 79.23 &= 80.82 - 80.82X - Y \\ &= 80.82 - 80.82X - (0.73-26.95X) \\ &= (80.82 - 0.73) - (80.82-26.95)X \\ X &= \frac{79.23 - 80.82 + 0.73}{- (80.82-26.95)} = \frac{-0.86}{-53.87} \end{split} $$ You can now use the value of $X$ and plug it into equation (1), getting $Y$ and plugging the result into the last equation. Can you finish it?