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Two cheeseburgers and one small order of French fries from a fast-food restaurant contain a total of \( 830 \) calories. Three cheeseburgers and two small orders of French fries contain a total of \( 1360 \) calories.Find the caloric content of each item.

In Exercises 41-54, sketch the graph and label the vertices of the solution set of the system of inequalities. \( \left\\{\begin{array}{l} x^2 + y^2 \le 36\\\ x^2 + y^2 \ge 9\end{array}\right. \)

In Super Bowl I, on January 15, 1967, the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to 10. The total points scored came from 13 different scoring plays, which were a combination of touchdowns, extra- point kicks, and field goals, worth 6, 1, and 3 points, respectively. The same number of touchdowns and extra-point kicks were scored. There were six times as many touchdowns as field goals. How many touchdowns, extra-point kicks, and field goals were scored during the game?

In the 2008 Women's NCAA Final Four Championship game, the University of Tennessee Lady Volunteers defeated the University of Stanford Cardinal by a score of 64 to 48. The Lady Volunteers won by scoring a combination of two- point baskets, three-point baskets, and one-point free throws. The number of two-points baskets was two more than the number of free throws. The number of free throws was two more than five times the number of three-point baskets. What combination of scoring accounted for the Lady Volunteers' 64 points?

A small corporation borrowed \( \$800,000 \) to expand its line of toys. Some of the money was borrowed at \( 8\% \), some at \( 9\% \), and some at \( 10\% \). How much was borrowed at each rate if the annual interest owed was \( \$67,000 \) and the amount borrowed at \( 8\% \) was five times the amount borrowed at \( 10\% \)?

A small fast-food restaurant invests \( \$10,000 \) to produce a new food item that will sell for \( \$3.99 \). Each item can be produced for \( \$1.90 \). (a) How many items must be sold to break even? (b) How many items must be sold to make a profit of \( \$12,000 \)?

You are offered two jobs selling dental supplies. One company offers a straight commission of \( 6\% \) of sales. The other company offers a salary of \( \$500 \) per week plus \( 3\% \) of sales. How much would you have to sell in a week in order to make the straight commission offer better?

A chemist needs 10 liters of a \( 25\% \) acid solution. The solution is to be mixed from three solutions whose concentrations are \( 10\% \), \( 20\% \), and \( 50\% \). How many liters of each solution will satisfy each condition? (a) Use 2 liters of the \( 50\% \) solution. (b) Use as little as possible of the \( 50\% \) solution. (c) Use as much as possible of the \( 50\% \) solution.

Applying Kirchhoff's Laws to the electrical network in the figure, the currents \( I_1, I_2 \), and \( I_3 \) are the solution of the system. \( \left\\{\begin{array}{l} I_1 - I_2 + I_3 = 0\\\ 3 I_1 + 2 I_2 \hspace{1cm} = 7\\\ \hspace{1cm} 2 I_2 + 4 I_3 = 8\end{array}\right. \) find the currents.

A system of pulleys is loaded with 128 -pound and 32 -pound weights (see figure). The tensions \(t_{1}\) and \(t_{2}\) in the ropes and the acceleration \(a\) of the 32 -pound weight are found by solving the system of equations \(\left\\{\begin{aligned} t_{1}-2 t_{2} \qquad &=0 \\ t_{1} \qquad -2 a&=128 \\\ t_{2}+a &=32 \end{aligned}\right.\) where \(t_{1}\) and \(t_{2}\) are measured in pounds and \(a\) is measured in feet per second squared. (Figure Cant Copy) (a) Solve this system. (b) The 32-pound weight in the pulley system is replaced by a 64-pound weight. The new pulley system will be modeled by the following system of equations. \(\left\\{\begin{aligned} t_{1}-2 t_{2} \qquad &=0 \\ t_{1} \qquad -2 a&=128 \\\ t_{2}+a &=64 \end{aligned}\right.\) Solve this system and use your answer for the acceleration to describe what (if anything) is happening in the pulley system.