Chapter 3

Solve each system of inequalities by graphing. \(4 y-2 x>4\) \(3 x+y>3\)

FARMING For Exercises \(34-37\) , use the following information. Dean Stadler has 20 days in which to plant corn and soybeans. The corn can be planted at a rate of 250 acres per day and the soybeans at a rate of 200 acres per day. He has 4500 acres available for planting these two crops. How much of each should Mr. Stadler plant if the profit on corn is \(\$ 29\) per acre and the profit on soybeans is \(\$ 24\) per acre? What is the maximum profit?

Write a system of inequalities that has no solution.

Solve each system of equations by using either substitution or elimination. \(8=0.4 m+1.8 n\) \(1.2 m+3.4 n=16\)

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(3 y-x=-2\) \(y-\frac{1}{3} x=2\)

Solve each system of inequalities by graphing. \(3 x+y \geq 1\) \(2 y-x \leq-4\)

Solve each system of equations by using either substitution or elimination. \(s+3 t=27\) \(2 t=19-\frac{1}{2} s\)

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(2 y-4 x=3\) \(\frac{4}{3} x-y=-2\)

Create a system of inequalities that forms a bounded region.

CHALLENGE Find the area of the region defined by \(|x|+|y| \leq 5\) and \(|x|+|y| \geq 2 .\)

Solve each system of equations by using either substitution or elimination. \(2 f+2 g=18\) \(\frac{1}{6} f+\frac{1}{3} g=1\)

To use a TI-83/84 Plus to solve a system of equations, graph the equations. Then, select INTERSECT, which is option 5 under the CALC menu, to find the coordinates of the point of intersection to the nearest hundredth. \(y=0.125 x-3.005\) \(y=-2.58\)

Mr. Talbot is writing a science test. It will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each for a total of 100 points. He wants to have twice as many multiple-choice questions as true/false. Write a system of equations that represents the number of each type of question.

To use a TI-83/84 Plus to solve a system of equations, graph the equations. Then, select INTERSECT, which is option 5 under the CALC menu, to find the coordinates of the point of intersection to the nearest hundredth. \(3.6 x-2 y=4\) \(-2.7 x+y=3\)

Which One Doesn't Belong? Given the following system of inequalities, which ordered pair does not belong? Explain your reasoning. $$y \leq \frac{1}{2} x+5 \quad y<-3 x+7 \quad y \geq-\frac{1}{3} x-2$$

Mr. Talbot is writing a science test. It will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each for a total of 100 points. He wants to have twice as many multiple-choice questions as true/false. How many of each type of question will be on the test?

To use a TI-83/84 Plus to solve a system of equations, graph the equations. Then, select INTERSECT, which is option 5 under the CALC menu, to find the coordinates of the point of intersection to the nearest hundredth. \(y=0.18 x+2.7\) \(y=-0.42 x+5.1\)

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(f(-1)\)

REVIEW To be a member of the marching band, a student must have a GPA of at least 2.0 and must have attended at least five after-school practices. Choose the system of inequalities that best represents this situation. $$ \begin{array}{ll}{\mathbf{F} \quad x \geq 2} & {\mathbf{H} x<2} \\ {y \geq 5} & {y<5} \\ {\mathbf{G} x \leq 2} & {\mathbf{J} \quad x>2} \\ {y \leq 5} & {y>5}\end{array} $$

Mr. Talbot is writing a science test. It will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each for a total of 100 points. He wants to have twice as many multiple-choice questions as true/false. If most of his students can answer true/false questions within 1 minute and multiple-choice questions within 1\(\frac{1}{2}\) minutes, will they have enough time to finish the test in 45 minutes?

Give an example of a system of equations that is consistent and independent.

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(f\left(\frac{1}{2}\right)\)

Megan exercises every morning for 40 minutes. She does a combination of step aerobics, which burns about 11 Calories per minute, and stretching, which burns about 4 Calories per minute. Her goal is to burn 335 Calories during her routine. Write a system of equations that represents Megan’s morning workout.

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{4 x-y=-20} \\ {x+2 y=13}\end{array} $$

Explain why a system of linear equations cannot have exactly two solutions.

ACT/SAT For a game she's playing, Liz must draw a card from a deck of 26 cards, one with each letter of the alphabet on it, and roll a six-sided die. What is the probability that Liz will roll an odd number and draw a letter in her name? $$ \begin{array}{llll}{A} & {\frac{2}{3}} & {B} & {\frac{1}{13}} & {C} & {\frac{1}{26}} & {D} & {\frac{3}{52}}\end{array} $$

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(g(1)\)

Megan exercises every morning for 40 minutes. She does a combination of step aerobics, which burns about 11 Calories per minute, and stretching, which burns about 4 Calories per minute. Her goal is to burn 335 Calories during her routine. How long should she do each activity in order to burn 335 Calories?

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{3 x-4 y=-2} \\ {5 x+2 y=40}\end{array} $$

REVIEW Which of the following best describes the graphs of \(y=3 x-5\) and \(4 y=12 x+16 ?\) F The lines have the same \(y\) -intercept. G The lines have the same \(x\) -intercept. H The lines are perpendicular. J The lines are parallel.

Find each value if \(f(x)=6 x+2\) and \(g(x)=3 x^{2}-x\). \(g(-3)\)

Give a system of equations that is more easily solved by substitution and a system of equations that is more easily solved by elimination.

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{4 x+5 y=7} \\ {3 x-2 y=34}\end{array} $$

Solve each system of inequalities by graphing. $$ \begin{array}{l}{2 y+x \geq 4} \\ {y \geq x-4}\end{array} $$

Ocean tides are caused by gravitational forces exerted by the Moon. Tides are also influenced by the size, boundaries, and depths of ocean basins and inlets. The highest tides on Earth occur in the Bay of Fundy in Nova Scotia, Canada. During the middle of the tidal range, the ocean shore is 30 meters from a rock bluff. The tide causes the shoreline to advance 8 meters and retreat 8 meters throughout the day. Write and solve an equation describing the maximum and minimum distances from the rock bluff to the ocean during high and low tide.

Solve each system of equations by graphing. $$ \begin{array}{l}{y=2 x+1} \\ {y=-\frac{1}{2} x-4}\end{array} $$

Make a conjecture about the solution of a system of equations if the result of subtracting one equation from the other is \(0=0\) .

Which of the following best describes the graph of the equations? \(4 y=3 x+8\) \(-6 x=-8 y+24\) A. The lines are parallel. B. The lines have the same \(x\)-intercept. C. The lines are perpendicular. D. The lines have the same \(y\)-intercept.

Solve each system of inequalities by graphing. $$ \begin{array}{l}{3 x-2 y \leq-6} \\ {y \leq \frac{3}{2} x-1}\end{array} $$

Juanita and Jamal are solving the system \(2 x-y=6\) and \(2 x+y=10 .\) Who is correct? Explain your reasoning.

Solve each system of equations by graphing. $$ \begin{array}{l}{2 x+y=-3} \\ {6 x+3 y=-9}\end{array} $$

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{4 x+5 y=20} \\ {5 x+4 y=7}\end{array} $$

Solve the system of equations. \(\frac{1}{x}+\frac{3}{y}=\frac{3}{4} \quad\) (Hint: Let \(m=\frac{1}{x}\) and \(n=\frac{1}{y})\) . \(\frac{3}{x}-\frac{2}{y}=\frac{5}{12}\)

Solve each system of equations by graphing. $$ \begin{array}{l}{2 x-y=6} \\ {-x+8 y=12}\end{array} $$

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{6 x+y=15} \\ {x-4 y=-10}\end{array} $$

To rent an inflatable trampoline for parties, it costs \(\$ 75\) an hour plus a set-up/tear-down fee of \(\$ 200 .\) Write an equation that represents this situation in slope-intercept form.

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{3 x+8 y=23} \\ {x-y=4}\end{array} $$

In order to practice at home, Tadeo purchased a basketball and a volleyball that cost a total of \(\$ 67,\) not including tax. If the price of the basketball \(b\) is \(\$ 4\) more than twice the cost of the volleyball \(v\) which system of linear equations could be used to determine the price of each ball? A. \(b+v=67\) \(b=2 v-4\) B. \(b+v=67\) \(b=2 v+4\) C. \(b+v=4\) \(b=2 v-67\) D. \(b+v=4\) \(b=2 v+67\)

Find each value if \(f(x)=4 x+3\) and \(g(x)=5 x-7\). $$ f(-2) $$

Nathan has 50 baseball cards in his collection from the \(1950^{\prime} \mathrm{s}\) and \(1960^{\prime} \mathrm{s}\) . His goal is to buy 2 more cards each month. Write an equation that represents how many cards Nathan will have in his collection in \(x\) months if he meets his goal.