Chapter 8

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{r} -2 x+3 y=4 \\ 2 x+3 y=8 \end{array}\right. $$

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(8,7),(4,4) \\ &L_{2}:(2,1),(-1,5) \end{aligned} $$

Write a system of linear equations that is more efficiently solved by the method of elimination than by the method of substitution. (There are many correct answers.)

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{l} 2 x+5 y=15 \\ 2 x-5 y=5 \end{array}\right. $$

Two types of tickets are to be sold for a concert. One type costs \(\$ 15\) per ticket and the other type costs \(\$ 25\) per ticket. The promoter of the concert must sell at least 15,000 tickets, including at least 8000 of the \(\$ 15\) tickets and at least 4000 of the \(\$ 25\) tickets. Moreover, the gross receipts must total at least \(\$ 275,000\) in order for the concert to be held. Write a system of linear inequalities that describes the different numbers of tickets that can be sold.

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(-2,3),(-4,1) \\ &L_{2}:(-6,1),(-8,-1) \end{aligned} $$

Write a system of linear equations that is more efficiently solved by the method of substitution than by the method of elimination. (There are many correct answers.)

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{l} -x+4 y=7 \\ 3 x-12 y=-21 \end{array}\right. $$

A dietitian is asked to design a special dietary supplement using two different foods. Each ounce of food \(X\) contains 20 units of calcium, 15 units of iron, and 10 units of vitamin B. Each ounce of food Y contains 10 units of calcium, 10 units of iron, and 20 units of vitamin B. The minimum daily requirements in the diet are 280 units of calcium, 160 units of iron, and 180 units of vitamin B. Write a system of linear inequalities that describes the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that can be used in the diet.

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(12,0),(7,-2) \\ &L_{2}:(0,7),(-5,9) \end{aligned} $$

Consider the system of linear equations. $$ \left\\{\begin{array}{r} x+y=8 \\ 2 x+2 y=k \end{array}\right. $$ (a) Find the value(s) of \(k\) for which the system has an infinite number of solutions. (b) Find one value of \(k\) for which the system has no solution. (There are many correct answers.) (c) Can the system have a single solution for some value of \(k\) ? Why or why not?

In Exercises \(43-48\), write the equations of the lines in slope-intercept form. What can you conclude about the number of solutions of the system? $$ \left\\{\begin{array}{r} 7 x+6 y=-4 \\ 3.5 x+3 y=-2 \end{array}\right. $$

A furniture company can sell all the tables and chairs it produces. Each table requires 1 hour in the assembly center and \(1 \frac{1}{2}\) hours in the finishing center. Each chair requires 2 hours in the assembly center and \(\frac{3}{4}\) hour in the finishing center. The company's assembly center is available 12 hours per day, and its finishing center is available 18 hours per day. Write a system of linear inequalities that describes the different production levels. Graph the system.

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(-10,1),(-7,2) \\ &L_{2}:(5,-2),(6,-5) \end{aligned} $$

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ (-6,4),(-3,-4) $$

The sum of two numbers \(x\) and \(y\) is 40 and the difference of the two numbers is 10 . The system of equations that represents this situation is $$ \left\\{\begin{array}{l} x+y=40 \\ x-y=10 \end{array}\right. $$ Solve this system to find the two numbers.

The sum of two numbers \(x\) and \(y\) is 20 and the difference of the two numbers is 2 . The system of equations that represents this situation is $$ \left\\{\begin{array}{l} x+y=20 \\ x-y=2 \end{array}\right. $$ Solve the system graphically to find the two numbers.

An electronics company can sell all the HD TVs and DVD players it produces. Each HD TV requires 3 hours on the assembly line and \(1 \frac{1}{4}\) hours on the testing line. Each DVD player requires 2 hours on the assembly line and 1 hour on the testing line. The company's assembly line is available 20 hours per day, and its testing line is available 16 hours per day. Write a system of linear inequalities that describes the different production levels. Graph the system.

In Exercises 50-53, solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 3 x+3 y=7 \\ 3 x+5 y=3 \end{array}\right. $$

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ (4,6),(8,-2) $$

The sum of two numbers \(x\) and \(y\) is 50 and the difference of the two numbers is 20 . The system of equations that represents this situation is $$ \left\\{\begin{array}{l} x+y=50 \\ x-y=20 \end{array}\right. \text {. } $$ Solve this system to find the two numbers.

The sum of two numbers \(x\) and \(y\) is 35 and the difference of the two numbers is 11 . The system of equations that represents this situation is $$ \left\\{\begin{array}{l} x+y=35 \\ x-y=11 \end{array}\right. \text {. } $$ Solve the system graphically to find the two numbers.

Explain the meaning of the term half-plane. Give an example of an inequality whose graph is a half-plane.

In Exercises 50-53, solve the system by the method of elimination. $$ \left\\{\begin{array}{l} -4 x+3 y=18 \\ -6 x+y=-8 \end{array}\right. $$

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ \left(\frac{7}{2}, \frac{9}{2}\right),\left(\frac{4}{3},-3\right) $$

Find an equation of the line with slope \(m=2\) passing through the intersection of the lines \(x-2 y=3\) and \(3 x+y=16\).

In Exercises 51 and 52, the graphs of the two equations appear to be parallel. Determine whether the two lines are actually parallel. Does the system have a solution? If so, find the solution. $$ \left\\{\begin{array}{l} x-200 y=-200 \\ x-199 y=198 \end{array}\right. $$

Can the solution of a system of linear inequalities be a line? Explain.

In Exercises 50-53, solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 6 x+5 y=19 \\ 2 x+3 y=5 \end{array}\right. $$

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ \left(-\frac{3}{4},-\frac{7}{4}\right),\left(-1, \frac{5}{2}\right) $$

Find an equation of the line with slope \(m=-3\) passing through the intersection of the lines \(4 x+6 y=26\) and \(5 x-2 y=-15\).

In Exercises 51 and 52, the graphs of the two equations appear to be parallel. Determine whether the two lines are actually parallel. Does the system have a solution? If so, find the solution. $$ \left\\{\begin{array}{l} 25 x-24 y=0 \\ 13 x-12 y=24 \end{array}\right. $$

Can the solution of a system of linear inequalities be a single point? Explain.

In Exercises 50-53, solve the system by the method of elimination. $$ \left\\{\begin{array}{r} 4 x+5 y=35 \\ -3 x+2 y=-9 \end{array}\right. $$

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ (-3,6),(-3,2) $$

Describe any advantages of the method of substitution over the graphical method of solving a system of linear equations.

It is possible for a consistent system of linear equations to have exactly two solutions. Justify your answer.

Is it possible for a system of linear inequalities to have no solution? If so, write an example.

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ (6,2),(10,2) $$

Your instructor says, "An equation (not in standard form) such as \(2 x-3=5 x-9\) can be considered a system of equations." Create the system, and find the solution point. How many solution points does the "system" \(x^{2}-1=2 x-1\) have? Illustrate your results with a graph.

Write a system of linear equations with integer coefficients that has the unique solution \((3,-4)\). (There are many correct answers.)

In Exercises 55-58, rewrite the expression in exponential form. $$ 3 \cdot 3 \cdot 3 \cdot 3 $$

In Exercises 55-58, solve and graph the inequality. $$ x \leq 3 $$

Write a system of linear equations that has no solution. (There are many correct answers.)

In Exercises 55-58, rewrite the expression in exponential form. $$ (-6) \cdot(-6) \cdot(-6) $$

In Exercises 55-58, solve and graph the inequality. $$ x>-4 $$

Write a system of linear equations that has infinitely many solutions. (There are many correct answers.)

In Exercises 55-58, rewrite the expression in exponential form. $$ \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} $$

In Exercises 55-58, solve and graph the inequality. $$ x+5<6 $$

In Exercises 57-60, evaluate the expression. $$ \frac{2}{3}+\frac{1}{3} $$