Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


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


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.

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer
Yeah I thought so. Since there were enough equations to solve for the cost of each item individually, there wasn't any real need to try and find a combination of them that would sum up to the final state. Thanks for the sanity check. – miles Oct 25 '12 at 10:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.