A publishing company publishes a total of no more than $100$ books every year. At least $20$ of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

A farmer has three hundred acres of arable land on which she wants to plant cauliflower and cabbage. The farmer has $\$ 17,500$ available for planting and $\$ 12,000$ for fertilizer. Planting 1 acre of cauliflower costs $\$ 70$, and planting 1 acre of cabbage costs $\$ 35$. Fertilizer costs $\$ 25$ for 1 acre of cauliflower and $\$ 55$ for 1 acre of cabbage.

(a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region.

(b) Can the farmer plant one hundred fifty-five acres of cauliflower and one hundred fifteen acres of cabbage?

(c) Can the farmer plant one hundred fifteen acres of cauliflower and one hundred seventy-five acres of cabbage?

State whether the equation or system of equations is linear.

$$\left\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y+ \quad &=2 \\ y+2 z &=4 \end{aligned}\right.$$

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?

Use the substitution method to find all solutions of the system of equations.

$$\left\{\begin{array}{l}3 x+y=1 \\ 5 x+2 y=1\end{array}\right.$$

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place.

$$\left\{\begin{array}{l}y \geq x^3 \\ 2 x+y \geq 0 \\ y \leq 2 x+6\end{array}\right.$$

The sum of two numbers is twice their difference. The larger number is $6$ more than twice the smaller. Find the numbers.

A system of inequalities and several points are given. Determine which points are solutions of the system.

$$\left\{\begin{array}{l}3 x-2 y \leq 5 \\ 2 x+y \geq 3\end{array} ; \quad(0,0),(1,2),(1,1),(3,1)\right.$$

For what value of $k$ does the following system have infinitely many solutions?

$$\left\{\begin{aligned} k x+y+z &=0 \\ x+2 y+k z &=0 \\ -x \qquad+3 z &=0 \end{aligned}\right.$$

A system of linear equations is given. (a) Find the complete solution of the system, or show that there is no solution. (b) State whether the system is inconsistent, dependent, or neither.

$$\left\{\begin{array}{r}x+2 y+z=3 \\ x+3 y+2 z=3 \\ 2 x+3 y-z=8\end{array}\right.$$

A man invests $5000 in an account that pays 8.5% interest per year, compounded quarterly. (a) Find the amount after 3 years. (b) How long will it take for the investment to double?

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place.

$$\left\{\begin{aligned} x+y & \geq 12 \\ 2 x+y & \leq 24 \\ x-y & \geq-6 \end{aligned}\right.$$

Find two numbers whose sum is $34$ and whose difference is $10$ .

A farmer has 500 acres of arable land on which he wants to plant potatoes and corn. The farmer has $\$ 40,000$ available for planting and $\$ 30,000$ for fertilizer. Planting 1 acre of potatoes costs $\$ 90$, and planting 1 acre of corn costs $\$ 50$. Fertilizer costs $\$ 30$ for 1 acre of potatoes and $\$ 80$ for 1 acre of corn.

(a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region.

(b) Can the farmer plant $300$ acres of potatoes and $180$ acres of corn?

(c) Can the farmer plant $150$ acres of potatoes and $325$ acres of corn?