solving $c=2a+5b$,$d=3a+8b$ with substitution for $a,b$

Question

I want to solve \begin{cases}c=2a+5b\\d=3a+8b\end{cases} with substitution for $a,b$ but can't seem to express $a$ and $b$ with $c$ and $d$ only. Any hint welcome thank you

Answer 1

Let us try to find an expression for $a$ by itself in terms of $c,d$. In doing so, it should become clear the technique I used so you can do it again as well has how to continue to find similar for $b$.

We want to take some sort of multiples of each of the two lines so that when I add or subtract them, the $b$'s entirely disappear. To do this, let us multiply the first equation by $8$ and the second equation by $5$ (picked because these were the coefficients of $b$)

In doing so, we have now the equivalent system:

$$\begin{cases} 8(c) = 8(2a + 5b)\\ 5(d) = 5(3a + 8b)\end{cases}$$

and simplified

$$\begin{cases} 8c = 16a + 40b\\ 5d = 15a + 40b\end{cases}$$

Now, subtracting one from the other, we have an expression for $a$

$$(8c - 5d) = (16a + 40b) - (15 a + 40b)$$

$$8c - 5d = a$$

If we needed to, we could then divide by the coefficient of $a$ in this result to further organize/simplify things but in this case the coefficient happened to be $1$ so we don't need to do much more to clean it up.

Now, you could plug this expression in for $a$ in one of the original equations to find a similar expression for $b$ written only in $c$'s and $d$'s after some organizing, or you could perform steps if you so prefer to find the expression for $b$ in this way.

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It is worth letting you know that the technique I employed here is known by the name "The Elimination Method" which as the name suggests is performed by "eliminating" one or more of the unnecessary variables at a time, possibly requiring multiple steps.

As the problems get more complicated with more equations and more variables, this method does still work to some extent, but it may become less clear which steps should be used at what points to continue. There is a fully generalized approach which builds on the same idea that you can just follow the same pattern for any problem that requires little thought that you can read about on the wikipedia page for Gaussian Elimination. As you continue your studies, you will come across many scenarios where Gaussian Elimination is going to be very a very powerful tool that can be used as a major step in finding a solution to the problem, especially when studying Linear Algebra.