Aragorn the Great has boasted to his hordes of followers that many a notorious villain has fallen to his awesome sword: His total of 560 victims consists of evil sorcerers, trolls, and orcs. These he has slain with a total of 620 mighty thrusts of his sword; evil sorcerers and trolls each requiring two thrusts (to the chest) and orcs each requiring one thrust (to the neck). When asked about the number of trolls he has slain, he replies, "I, the mighty Aragorn, despise trolls five times as much as I despise evil sorcerers. Accordingly, five times as many trolls as evil sorcerers have fallen to \(\mathrm{m} \mathrm{y}\) sword!" How many of each type of villain has he slain?

Suppose a system of equations has a unique solution. What must be true of the number of pivots in the reduced matrix of the system? Why?

Resource Allocation You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. How many quarts of each flavor should you make in order to use up all the eggs and cream? HINT [See Example 5.]

Use Gauss-Jordan row reduction to solve the given systems of equation. We suggest doing some by hand, and others using technology. HINT [See Examples 1-6.] $$ \begin{gathered} x+y+2 z=-1 \\ 2 x+2 y+2 z=2 \\ \frac{3}{5} x+\frac{3}{5} y+\frac{3}{5} z=\frac{2}{5} \end{gathered} $$

Things have not been going too well here at Accurate Accounting, Inc. since we hired Todd Smiley. He has a tendency to lose important documents, especially around April, when tax returns of our business clients are due. Today Smiley accidentally shredded Colossal Conglomerate Corp.'s investment records. We must therefore reconstruct them based on the information he can gather. Todd recalls that the company earned an \(\$ 8\) million return on investments totaling \(\$ 65\) million last year. After a few frantic telephone calls to sources in Colossal, he learned that Colossal had made investments in four companies last year: X, Y, Z, and W. (For reasons of confidentiality, we are withholding their names.) Investments in company X earned \(15 \%\) last year, investments in Y depreciated by \(20 \%\) last year, investments in \(Z\) neither appreciated nor depreciated last year,while investments in W earned \(20 \%\) last year. Smiley was also told that Colossal invested twice as much in company \(X\) as in company \(Z\), and three times as much in company \(\mathrm{W}\) as in company \(Z\). Does Smiley have sufficient information to piece together Colossal's investment portfolio before its tax return is due next week? If so, what does the investment portfolio look like?

Select one: Multiplying both sides of a linear equation by a nonzero constant results in a linear equation whose graph is (A) parallel to (B) the same as (C) not always parallel to (D) not the same as the graph of the original equation.