Here is the example case. You might recognize this to be from Lewis Carroll's "A Tangled Tale". If you're already familiar, you can scroll down to the question.

## Example

Given that one glass of lemonade, 3 sandwiches, and 7 biscuits cost \$14 and that one glass of lemonade, 4 sandwiches, and 10 biscuits cost \$19, find the cost of the following:

(a) a glass of lemonade, a sandwich, and a biscuit
(b) 2 glasses of lemonade, 3 sandwiches, and 5 biscuits.

Lewis' approach is substitution, whereas I use an approach involving determinants.

I can rewrite the two cases as $$a+3b+7c=14$$ $$a+4b+10c=17$$ where I want to find $$a+b+c$$ and $$2a+3b+5c.$$

I'll demonstrate the second case.

Then I want to find $x,y$ such that $$x\langle1,3,7\rangle+y\langle1,4,10\rangle=\langle2,3,5\rangle$$ and this gives me three more equations $$x+y=2$$ $$3x+4y=3$$ $$7x+10y=5$$ However, we have three equations and two unknowns so using Cramer's rule with any two of the three equations will give the correct values for $x,y$. I'll use the first two equations. $$x=\frac{\begin{vmatrix}2&1\\3&4\end{vmatrix}}{\begin{vmatrix}1&1\\3&4\end{vmatrix}}=5$$ $$y=\frac{\begin{vmatrix}1&2\\3&3\end{vmatrix}}{\begin{vmatrix}1&1\\3&4\end{vmatrix}}=-3$$

So then $$2a+3b+5c=14(5)+17(-3)=19.$$

## Problem

Given $$a+2b+3c=4$$ $$5a+6b+7c=8$$ $$9a+c=2$$ $$-3a+14b+5c=6,$$ find $$7a+8b+9c.$$

However, I believe that has no solution.

-

Yes, your system of four equations is inconsistent (i.e. has no solution). The unique solution of the first three is $a=0,b=-1,c=2$, which violates the fourth.