91Ó°ÊÓ

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6. $$\left\\{\begin{aligned} 2 x-6 y &=10 \\ -3 x+9 y &=-15 \end{aligned}\right.$$

$$\left\\{\begin{array}{r} x>0 \\ y>0 \\ x+y<10 \\ x^{2}+y^{2}>9 \end{array}\right.$$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6. $$\left\\{\begin{aligned} u-30 v &=-5 \\ -3 u+80 v &=5 \end{aligned}\right.$$

Follow the hints and solve the systems. $$\left\\{\begin{array}{ll} x^{2}+x y=1 & \text { I Hint: Add the equations, and factor } \\ x y+y^{2}=3 & \text { the result.] } \end{array}\right.$$

Graph the solution set of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} x^{2}+y^{2} & \leq 8 \\ x & \geq 2 \\ y & \geq 0 \end{aligned}\right.$$

Graph the solution set of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{aligned} y & \geq x^{3} \\ y & \leq 2 x+4 \\ x+y & \geq 0 \end{aligned}\right.$$

Value of Coins A man has 14 coins in his pocket, all of which are dimes and quarters. If the total value of his change is \(\$ 2.75,\) how many dimes and how many quarters does he have?

Nutrition A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely 32 units of niacin and \(22,000\) units of retinol. She uses two types of commercial pellet foods. Food A contains 0.12 unit of niacin and 100 units of retinol per gram. Food B contains 0.20 unit of niacin and 50 units of retinol per gram. How many grams of each food does she feed this group of rats each day?

A chemist has three acid solutions at various concentrations. The first is \(10 \%\) acid, the second is \(20 \%,\) and the third is \(40 \% .\) How many milliliters of each should she use to make \(100 \mathrm{mL}\) of \(18 \%\) solution, if she has to use four times as much of the \(10 \%\) solution as the \(40 \%\) solution?

DISCUSS: Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$\left\\{\begin{array}{r} x+2 y>4 \\ -x+y<1 \\ x+3 y<9 \\ x<3 \end{array}\right.$$