Chapter 5

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{r}2 x^{2}+y>4 \\ x<0 \\ y<2\end{array}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}1.5 x+2 y=3.75 \\ 7.5 x+10 y=18.75\end{array}\right.$$

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x+y=4 \\ x^{2}-y=2\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}2 x+y>2 \\ 6 x+3 y<2\end{array}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}0.05 x-0.03 y=0.21 \\ 0.07 x+0.02 y=0.16\end{array}\right.$$

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}3 x-7 y+6=0 \\ x^{2}-y^{2}=4\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}5 x-3 y>-6 \\ 5 x-3 y<-9\end{array}\right.$$

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}0.02 x-0.05 y=-0.19 \\ 0.03 x+0.04 y=0.52\end{array}\right.$$

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x^{2}+y^{2}=25 \\ 2 x+y=10\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \geq-3 \\ y \leq 1-x^{2}\end{array}\right.$$

Solve the system of equations. $$\left\\{\begin{aligned} 3 x-3 y+6 z &=7 \\\\-x+y-2 z &=3 \\ 2 x+3 y-4 z &=8 \end{aligned}\right.$$

The graphs of the two equations appear to be parallel. Are they? Justify your answer by using elimination to solve the system. $$\left\\{\begin{array}{l}200 y-x=200 \\ 199 y-x=-198\end{array}\right.$$

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x-2 y=4 \\ x^{2}-y=0\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x-y^{2}>0 \\ y>(x-3)^{2}-4\end{array}\right.$$

Solve the system of equations. $$\left\\{\begin{aligned} 4 x+3 y &=7 \\ x-2 y+z &=0 \\\\-2 x+4 y-2 z &=13 \end{aligned}\right.$$

The graphs of the two equations appear to be parallel. Are they? Justify your answer by using elimination to solve the system. $$\left\\{\begin{array}{l}25 x-24 y=0 \\ 13 x-12 y=120\end{array}\right.$$

Solve the system by the method of substitution. $$\left\\{\begin{array}{r}x^{2}+y^{2}=9 \\ x-y=-5\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 16 \\\ x^{2}+y^{2}<1\end{array}\right.$$

Solve the system of equations. $$\left\\{\begin{aligned} x &+3 w=4 \\ 2 y-z-w &=0 \\ 3 y-2 w &=1 \\ 2 x-y+4 z &=5 \end{aligned}\right.$$

Use the given statements to write a system of equations. Solve the system by elimination. The sum of a number \(x\) and a number \(y\) is \(13 .\) The difference of \(x\) and \(y\) is 3 .

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}y=x^{4}-2 x^{2}+1 \\ y=1-x^{2}\end{array}\right.$$

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 25 \\ x^{2}+y^{2} \geq 9\end{array}\right.$$

Solve the system of equations. $$\left\\{\begin{array}{r}x+y+z+w=6 \\ 2 x+3 y-w=0 \\ -3 x+4 y+z+2 w=4 \\\ x+2 y-z+w=0\end{array}\right.$$

Use the given statements to write a system of equations. Solve the system by elimination. The sum of a number \(a\) and a number \(b\) is 43 . The difference of \(a\) and \(b\) is \(-27 .\)

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}y=x^{3}-2 x^{2}+x-1 \\ y=-x^{2}+3 x-1\end{array}\right.$$

Optimal Profit A fruit grower raises crops \(\mathrm{A}\) and \(\mathrm{B}\). The profit is $$\$ 185$$ per acre for crop \(\mathrm{A}\) and $$\$ 245$$ per acre for crop \(\mathrm{B}\). Research and available resources indicate the following constraints. \- The fruit grower has 150 acres of land for raising the crops. \(-\) It takes 1 day to trim an acre of crop \(A\) and 2 days to trim an acre of crop \(\mathrm{B}\), and there are 240 days per year available for trimming. \- It takes \(0.3\) day to pick an acre of crop \(\mathrm{A}\) and \(0.1\) day to pick an acre of crop \(\mathrm{B}\), and there are 30 days per year available for picking. What is the optimal acreage for each fruit? What is the optimal profit?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x>y^{2} \\ x

Find two systems of equations that have the ordered triple as a solution. (There are many correct answers.) $$(3,-1,2)$$

Use the given statements to write a system of equations. Solve the system by elimination. The sum of twice a number \(r\) and a number \(s\) is 8 . The difference of \(r\) and \(s\) is 7 .

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x y-2=0 \\ y=\sqrt{x-1}\end{array}\right.$$

Optimal Profit The costs to a store for two models of Global Positioning System (GPS) receivers are $$\$ 80$$ and $$\$ 100$$. The $$\$ 80$$ model yields a profit of $$\$ 25$$ and the $$\$ 100$$ model yields a profit of $$\$ 30 .$$ Market tests and available resources indicate the following constraints. \- The merchant estimates that the total monthly demand will not exceed 200 units. \- The merchant does not want to invest more than $$\$ 18,000$$ in GPS receiver inventory. What is the optimal inventory level for each model? What is the optimal profit?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}x<2 y-y^{2} \\ 0

Find two systems of equations that have the ordered triple as a solution. (There are many correct answers.) $$\left(-\frac{1}{2},-2,4\right)$$

Use the given statements to write a system of equations. Solve the system by elimination. The difference of a number \(m\) and twice a number \(n\) is 1 . The sum of two times \(m\) and \(n\) is 22 .

Solve the system by the method of substitution. $$\left\\{\begin{array}{l}x y=3 \\ y=\sqrt{x-2}\end{array}\right.$$

Optimal Cost A farming cooperative mixes two brands of cattle feed. Brand \(X\) costs $$\$ 30$$ per bag, and brand \(Y\) costs $$\$ 25$$ per bag. Research and available resources have indicated the following constraints. \- Brand \(\mathrm{X}\) contains two units of nutritional element \(\mathrm{A}\), two units of element \(\mathrm{B}\), and two units of element \(\mathrm{C}\). \- Brand Y contains one unit of nutritional element A, nine units of element \(\mathrm{B}\), and three units of element \(\mathrm{C}\). \- The minimum requirements for nutrients \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) are 12 units, 36 units, and 24 units, respectively. What is the optimal number of bags of each brand that should be mixed? What is the optimal cost?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \leq \sqrt{3 x}+1 \\ y \geq x+1\end{array}\right.$$

Find two systems of equations that have the ordered triple as a solution. (There are many correct answers.) $$(1,-5,-3)$$

Airplane Speed An airplane flying into a headwind travels the 1800 -mile flying distance between Los Angeles, California and South Bend, Indiana in 3 hours and 36 minutes. On the return flight, the distance is traveled in 3 hours. Find the air speed of the plane and the speed of the wind, assuming that both remain constant.

Solve the system graphically. $$\left\\{\begin{array}{r}-x+2 y=2 \\ 3 x+y=15\end{array}\right.$$

Optimal Cost A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs $$\$ 1000$$ and each model B vehicle costs $$\$ 1500$$. Mission strategies and objectives indicate the following constraints. \- A total of at least 20 vehicles must be used. 4\. A model A vehicle can hold 45 boxes of supplies. A model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. \- A model A vehicle can hold 20 refugees. A model \(\mathrm{B}\) vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{lr}y<\sqrt{2 x}+3 \\ y> & x+3\end{array}\right.$$

Find two systems of equations that have the ordered triple as a solution. (There are many correct answers.) $$\left(0,2, \frac{1}{2}\right)$$

Airplane Speed Two planes start from the same airport and fly in opposite directions. The second plane starts \(\frac{1}{2}\) hour after the first plane, but its speed is 50 miles per hour faster. Find the air speed of each plane if, 2 hours after the first plane departs, the planes are 2000 miles apart.

Solve the system graphically. $$\left\\{\begin{aligned} x+y &=0 \\ 3 x-2 y &=10 \end{aligned}\right.$$

Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2 & 2.5 \\ \hline \text { Painting } & 4 & 1 \\ \hline \text { Packaging } & 1 & 0.75 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500 hours, respectively. The profits per unit are \(\$ 50\) for model \(\mathrm{A}\) and \(\$ 75\) for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y-2 x \\ x \leq 1\end{array}\right.$$

Write three ordered triples of the given form. $$\left(a, a-5, \frac{2}{3} a+1\right)$$

Acid Mixture Ten gallons of a \(30 \%\) acid solution is obtained by mixing a \(20 \%\) solution with a \(50 \%\) solution. How much of each solution is required to obtain the specified concentration of the final mixture?

Solve the system graphically. $$\left\\{\begin{array}{r}x-3 y=-2 \\ 5 x+3 y=17\end{array}\right.$$