Section 1
Lines in the Plane
Match each equation with its form.(a) $A x+B y+C=0$(b) $x=a$(c) $y=b$(d) $y=m x+b$(e) $y-y_1=m\left(x-x_1\right)$(i) vertical line(ii) slope-intercept form(iii) general form(iv) point-slope form(v) horizontal line
In Exercises 2 and 3, fill in the blank.For a line, the ratio of the change in $y$ to the change in $x$ is called the ________ of the line.
In Exercises 2 and 3, fill in the blank.Two lines are ________ if and only if their slopes are equal.
What is the relationship between two lines whose slopes are -3 and $\frac{1}{3}$ ?
What is the slope of a line that is perpendicular to the line represented by $x=3$ ?
Give the coordinates of a point on the line whose equation in point-slope form is $y-(-2)=\frac{1}{2}(x-5)$.
In Exercises 7 and 8, identify the line that has the indicated slope.(a) $m=\frac{2}{3}$(b) $m$ is undefined.(c) $m=-2$(GRAPH CAN'T COPY)
In Exercises 7 and 8, identify the line that has the indicated slope.(a) $m=0$(b) $m=-\frac{3}{4}$(c) $m=1$(GRAPH CAN'T COPY)
In Exercises 9 and 10, estimate the slope of the line.(GRAPH CAN'T COPY)
In Exercises 11 and 12, sketch the lines through the point with the indicated slopes on the same set of coordinate axes.Point$(2,3)$Slopes(a) 0(b) 1(c) 2(d) -3
In Exercises 11 and 12, sketch the lines through the point with the indicated slopes on the same set of coordinate axes.Point$(-4,1)$Slopes(a) 3(b) -3(c) $\frac{1}{2}$(d) Undefined
In Exercises 13-16, find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.)$(0,-10),(-4,0)$
In Exercises 13-16, find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.)$(2,4),(4,-4)$
In Exercises 13-16, find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.)$(-6,-1),(-6,4)$
In Exercises 13-16, find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.)$(-3,-2),(1,6)$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(2,1) & m=0\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(3,-2) & m=0\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(1,5) & m \text { is undefined. }\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(-4,1) & m \text { is undefined. }\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(0,-9) & m=-2\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(-5,4) & m=2\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(7,-2) & m=\frac{1}{2}\end{array}$$
In Exercises 17-24, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.)$$\begin{array}{ll}\text { Point } & \text { Slope } \\(-1,-6) & m=\frac{1}{2}\end{array}$$
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$(0,-2), \quad m=3$
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$(-3,6), \quad m=-2$
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible. $(2,-3), \quad m=-\frac{1}{2}$
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$(-2,-5), \quad m=\frac{3}{4}$
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$(6,-1), \quad m$ is undefined
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$(-10,4), \quad m$ is undefined
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$\left(-\frac{1}{2}, \frac{3}{2}\right), \quad m=0$
The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible.$(2.3,-8.5), \quad m=0$
The median player salary for the New York Yankees was $$\$1.6$$ million in 2001 and $$\$5.2$$ million in 2009. Write a linear equation giving the median salary $y$ in terms of the year $x$. Then use the equation to predict the median salary in 2017.
The median player salary for the Dallas Cowboys was $$\$ 441,300$$ in 2000 and $$\$ 1,326,720$$ in 2008. Write a linear equation giving the median salary $y$ in terms of the year $x$. Then use the equation to predict the median salary in 2016 .
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph.$x-2 y=4$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph. $3 x+4 y=1$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph.$2 x-5 y+10=0$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph.$4 x-3 y-9=0$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph. $x=-6$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph.$y=12$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph.$3 y+2=0$
In Exercises 35-42, determine the slope and $y$-intercept (if possible) of the linear equation. Then describe its graph.$2 x-5=0$
In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).$5 x-y+3=0$
In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).$2 x+3 y-9=0$
In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).$5 x-2=0$
In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).$3 x+7=0$
In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).$3 y+5=0$
In Exercises 43-48, (a) find the slope and $y$-intercept (if possible) of the equation of the line algebraically, and (b) sketch the line by hand. Use a graphing utility to verify your answers to parts (a) and (b).$-11-8 y=0$
In Exercises 49 and 50, find the slope-intercept form of the equation of the line shown.(GRAPH CAN'T COPY)
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $(5,-1),(-5,5)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$(4,3),(-4,-4)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible). $(-8,1),(-8,7)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$(-1,4),(6,4)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$\left(2, \frac{1}{2}\right),\left(\frac{1}{2}, \frac{5}{4}\right)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$(1,1),\left(6,-\frac{2}{3}\right)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$\left(-\frac{1}{10},-\frac{3}{5}\right),\left(\frac{9}{10},-\frac{9}{5}\right)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$\left(\frac{3}{4}, \frac{3}{2}\right),\left(-\frac{4}{3}, \frac{7}{4}\right)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$(1,0.6),(-2,-0.6)$
Finding the Slope-Intercept Form In Exercises 51-60, write an equation of the line that passes through the points. Use the slope-intercept form (if possible). If not possible, explain why and use the general form. Use a graphing utility to graph the line (if possible).$(-8,0.6),(2,-2.4)$
In Exercises 61 and 62, use a graphing utility to graph the equation using each viewing window. Describe the differences in the graphs.$y=0.5 x-3$$$\begin{aligned}&\begin{array}{|l|}\hline \mathrm{Xmin}=-5 \\\mathrm{Xmax}=10 \\\mathrm{Xscl}=1 \\\mathrm{Ymin}=-1 \\\mathrm{Ymax}=10 \\\mathrm{Yscl}=1 \\\hline\end{array}\\&\begin{array}{|l|}\hline \mathrm{Xmin}=-2 \\\mathrm{Xmax}=10 \\\mathrm{Xscl}=1 \\\mathrm{Ymin}=-4 \\\mathrm{Ymax}=1 \\\mathrm{Yscl}=1 \\\hline\end{array}\\&\begin{array}{|l|}\hline \mathrm{Xmin}=-5 \\\mathrm{Xmax}=10 \\\mathrm{Xscl}=1 \\\mathrm{Ymin}=-7 \\\mathrm{Ymax}=3 \\\mathrm{Yscl}=1 \\\hline\end{array}\end{aligned}$$
In Exercises 61 and 62, use a graphing utility to graph the equation using each viewing window. Describe the differences in the graphs.$y=-8 x+5$$$\begin{aligned}&\begin{array}{|l|}\hline \mathrm{Xmin}=-5 \\\mathrm{Xmax}=5 \\\mathrm{Xscl}=1 \\\mathrm{Ymin}=-10 \\\mathrm{Ymax}=10 \\\mathrm{Yscl}=1 \\\hline\end{array}\\&\begin{array}{|l|}\hline \mathrm{Xmin}=-5 \\\mathrm{Xmax}=10 \\\mathrm{Xscl}=1 \\\mathrm{Ymin}=-80 \\\mathrm{Ymax}=80 \\\mathrm{Yscl}=20 \\\hline\end{array}\\&\begin{array}{|l|}\hline \mathrm{X} \min =-5 \\\mathrm{X} \max =13 \\\mathrm{Xscl}=1 \\\mathrm{Y} \min =-2 \\\mathrm{Ymax}=10 \\\mathrm{Yscl}=1 \\\hline\end{array}\end{aligned}$$
In Exercises 63-66, determine whether the lines $L_1$ and $L_2$ passing through the pairs of points are parallel, perpendicular, or neither.$\begin{aligned} & L_1:(0,-1),(5,9) \\ & L_2:(0,3),(4,1)\end{aligned}$
In Exercises 63-66, determine whether the lines $L_1$ and $L_2$ passing through the pairs of points are parallel, perpendicular, or neither.$\begin{aligned} & L_1:(-2,-1),(1,5) \\ & L_2:(1,3),(5,-5)\end{aligned}$
In Exercises 63-66, determine whether the lines $L_1$ and $L_2$ passing through the pairs of points are parallel, perpendicular, or neither.$\begin{aligned} & L_1:(3,6),(-6,0) \\ & L_2:(0,-1),\left(5, \frac{7}{3}\right)\end{aligned}$
In Exercises 63-66, determine whether the lines $L_1$ and $L_2$ passing through the pairs of points are parallel, perpendicular, or neither.$\begin{aligned} & L_1:(4,8),(-4,2) \\ & L_2:(3,-5),\left(-1, \frac{1}{3}\right)\end{aligned}$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(2,1), 4 x-2 y=3$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(-3,2), x+y=7$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$\left(-\frac{2}{3}, \frac{7}{8}\right), \quad 3 x+4 y=7$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$\left(\frac{2}{5},-1\right), \quad 3 x-2 y=6$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(-3.9,-1.4), 6 x+2 y=9$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(-1.2,2.4), 5 x+4 y=1$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(3,-2), x-4=0$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. $(3,-1), y-2=0$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(-4,1), y+2=0$
In Exercises 67-76, write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.$(-2,4), x+5=0$
In Exercises 77 and 78, the lines are parallel. Find the slope-intercept form of the equation of line $y_2$.(GRAPH CAN'T COPY)
In Exercises 79 and 80, the lines are perpendicular. Find the slope-intercept form of the equation of line $y_2$.(GRAPH CAN'T COPY)
In Exercises 81-84, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results.(a) $y=2 x$(b) $y=-2 x$(c) $y=\frac{1}{2} x$
In Exercises 81-84, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results.(a) $y=\frac{2}{3} x$(b) $y=-\frac{3}{2} x$(c) $y=\frac{2}{3} x+2$
In Exercises 81-84, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results.(a) $y=-\frac{1}{2} x$(b) $y=-\frac{1}{2} x+3$(c) $y=2 x-4$
In Exercises 81-84, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that each slope appears visually correct. Use the slopes of the lines to verify your results.(a) $y=x-8$(b) $y=x+1$(c) $y=-x+3$
The "rise to run" ratio of the roof of a house determines the steepness of the roof. The rise to run ratio of the roof in the figure is 3 to 4 . Determine the maximum height in the attic of the house if the house is 32 feet wide.(FIGURE CAN'T COPY)
When driving down a mountain road, you notice warning signs indicating that it is a " $12 \%$ grade." This means that the slope of the road is $-\frac{12}{100}$. Approximate the amount of horizontal change in your position if you note from elevation markers that you have descended 2000 feet vertically.(FIGURE CAN'T COPY)
The graph shows the sales $y$ (in millions of dollars) of the Coca-Cola Bottling Company each year $x$ from 2000 through 2008, where $x=0$ represents 2000. (Source: Coca-Cola Bottling Company)(FIGURE CAN'T COPY)(a) Use the slopes to determine the years in which the sales showed the greatest increase and greatest decrease.(b) Find the equation of the line between the years 2000 and 2008.(c) Interpret the meaning of the slope of the line from part (b) in the context of the problem.(d) Use the equation from part (b) to estimate the sales of the Coca-Cola Bottling Company in 2010. Do you think this is an accurate estimate? Explain.
The table shows the profits $y$ (in millions of dollars) for Buffalo Wild Wings for each year $x$ from 2002 through 2008 , where $x=2$ represents 2002 . (Source: Buffalo Wild Wings Inc.)$$\begin{array}{|c|c|}\hline \text { Year, } \boldsymbol{x} & \text { Profits, } \boldsymbol{y} \\\hline 2 & 3.1 \\3 & 3.9 \\4 & 7.2 \\5 & 8.9 \\6 & 16.3 \\7 & 19.7 \\8 & 24.4 \\\hline\end{array}$$(a) Sketch a graph of the data.(b) Use the slopes to determine the years in which the profits showed the greatest and least increases.(c) Find the equation of the line between the years 2002 and 2008.(d) Interpret the meaning of the slope of the line from part (c) in the context of the problem.(e) Use the equation from part (c) to estimate the profit for Buffalo Wild Wings in 2010. Do you think this is an accurate estimate? Explain.
In Exercises 89-92, you are given the dollar value of a product in 2009 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value $V$ of the product in terms of the year $t$. (Let $t=9$ represent 2009.)$$\begin{array}{lc}2009 \text { Value } & \text { Rate } \\\$ 2540 & \$ 125 \text { increase per year }\end{array}$$
In Exercises 89-92, you are given the dollar value of a product in 2009 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value $V$ of the product in terms of the year $t$. (Let $t=9$ represent 2009.)$$\begin{array}{lc}2009 \text { Value } & \text { Rate } \\\$ 156 & \$ 4.50 \text { increase per year }\end{array}$$
In Exercises 89-92, you are given the dollar value of a product in 2009 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value $V$ of the product in terms of the year $t$. (Let $t=9$ represent 2009.)$$\begin{array}{lc}2009 \text { Value } & \text { Rate } \\\$ 20,400 & \$ 2000 \text { increase per year }\end{array}$$
In Exercises 89-92, you are given the dollar value of a product in 2009 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value $V$ of the product in terms of the year $t$. (Let $t=9$ represent 2009.)$$\begin{array}{lc}2009 \text { Value } & \text { Rate } \\\$ 245,000 & \$ 5600 \text { decrease per year }\end{array}$$
A school district purchases a highvolume printer, copier, and scanner for $$\$ 25,000$$. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $$\$ 2000$$.(a) Write a linear equation giving the value $V$ of the equipment for each year $t$ during its 10 years of use.(b) Use a graphing utility to graph the linear equation representing the depreciation of the equipment, and use the value or trace feature to complete the table. Verify your answers algebraically by using the equation you found in part (a).$$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline V & & & & & & & & & & & \\\hline\end{array}$$
Recall that water freezes at $0^{\circ} \mathrm{C}\left(32^{\circ} \mathrm{F}\right)$ and boils at $100^{\circ} \mathrm{C}\left(212^{\circ} \mathrm{F}\right)$.(a) Find an equation of the line that shows the relationship between the temperature in degrees Celsius $C$ and degrees Fahrenheit $F$.(b) Use the result of part (a) to complete the table.$$\begin{array}{|l|l|l|l|l|l|l|}\hline C & & -10^{\circ} & 10^{\circ} & & & 177^{\circ} \\\hline F & 0^{\circ} & & & 68^{\circ} & 90^{\circ} & \\\hline\end{array}$$
A contractor purchases a bulldozer for $$\$ 36,500$$. The bulldozer requires an average expenditure of $$\$ 9.25$$ per hour for fuel and maintenance, and the operator is paid $$\$ 18.50$$ per hour.(a) Write a linear equation giving the total $\operatorname{cost} C$ of operating the bulldozer for $t$ hours. (Include the purchase cost of the bulldozer.)(b) Assuming that customers are charged $$$ 65$$ per hour of bulldozer use, write an equation for the revenue $R$ derived from $t$ hours of use.(c) Use the profit formula $(P=R-C)$ to write an equation for the profit gained from $t$ hours of use.(d) Use the result of part (c) to find the break-even point (the number of hours the bulldozer must be used to gain a profit of 0 dollars).
A real estate office handles an apartment complex with 50 units. When the rent per unit is $$\$ 580$$ per month, all 50 units are occupied. However, when the rent is $$\$ 625$$ per month, the average number of occupied units drops to 47 . Assume that the relationship between the monthly rent $p$ and the demand $x$ is linear.(a) Write the equation of the line giving the demand $x$ in terms of the rent $p$.(b) Use a graphing utility to graph the demand equation and use the trace feature to estimate the number of units occupied when the rent is $$\$ 655$$. Verify your answer algebraically.(c) Use the demand equation to predict the number of units occupied when the rent is lowered to $$\$ 595$$. Verify your answer graphically.
In 1990, Penn State University had an enrollment of 75,365 students. By 2009, the enrollment had increased to 87,163. (Source: Penn State Fact Book)(a) What was the average annual change in enrollment from 1990 to 2009 ?(b) Use the average annual change in enrollment to estimate the enrollments in 1995, 2000, and 2005.(c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the context of the problem.
Using the results of Exercise 97, write a short paragraph discussing the concepts of slope and average rate of change.
In Exercises 99 and 100, determine whether the statement is true or false. Justify your answer.The line through $(-8,2)$ and $(-1,4)$ and the line through $(0,-4)$ and $(-7,7)$ are parallel.
In Exercises 99 and 100, determine whether the statement is true or false. Justify your answer.If the points $(10,-3)$ and $(2,-9)$ lie on the same line, then the point $\left(-12,-\frac{37}{2}\right)$ also lies on that line.
In Exercises 101-104, use a graphing utility to graph the equation of the line in the form$$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0 \text {. }$$Use the graphs to make a conjecture about what $a$ and $b$ represent. Verify your conjecture. $\frac{x}{5}+\frac{y}{-3}=1$
In Exercises 101-104, use a graphing utility to graph the equation of the line in the form$$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0 \text {. }$$Use the graphs to make a conjecture about what $a$ and $b$ represent. Verify your conjecture.$\frac{x}{-6}+\frac{y}{2}=1$
In Exercises 101-104, use a graphing utility to graph the equation of the line in the form$$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0 \text {. }$$Use the graphs to make a conjecture about what $a$ and $b$ represent. Verify your conjecture.$\frac{x}{4}+\frac{y}{-\frac{2}{3}}=1$
In Exercises 101-104, use a graphing utility to graph the equation of the line in the form$$\frac{x}{a}+\frac{y}{b}=1, \quad a \neq 0, b \neq 0 \text {. }$$Use the graphs to make a conjecture about what $a$ and $b$ represent. Verify your conjecture.$\frac{x}{\frac{1}{2}}+\frac{y}{5}=1$
In Exercises 105-108, use the results of Exercises 101-104 to write an equation of the line that passes through the points.$x$-intercept: $(2,0)$$y$-intercept: $(0,3)$
In Exercises 105-108, use the results of Exercises 101-104 to write an equation of the line that passes through the points.$x$-intercept: $(-5,0)$$y$-intercept: $(0,-4)$
In Exercises 105-108, use the results of Exercises 101-104 to write an equation of the line that passes through the points.$\begin{aligned} & x \text {-intercept: }\left(-\frac{1}{6}, 0\right) \\ & y \text {-intercept: }\left(0,-\frac{2}{3}\right)\end{aligned}$
In Exercises 105-108, use the results of Exercises 101-104 to write an equation of the line that passes through the points.$x$-intercept: $\left(\frac{3}{4}, 0\right)$ $y$-intercept: $\left(0, \frac{4}{5}\right)$
In Exercises 109 and 110, determine which equation(s) may be represented by the graphs shown. (There may be more than one correct answer.)(GRAPH CAN'T COPY)(a) $2 x-y=-10$(b) $2 x+y=10$(c) $x-2 y=10$(d) $x+2 y=10$
In Exercises 109 and 110, determine which equation(s) may be represented by the graphs shown. (There may be more than one correct answer.)(GRAPH CAN'T COPY)(a) $2 x+y=5$(b) $2 x+y=-5$(c) $x-2 y=5$(d) $x-2 y=-5$
In Exercises 111 and 112, determine which pair of equations may be represented by the graphs shown.(GRAPH CAN'T COPY)(a)$$\begin{aligned}& 2 x-y=5 \\& 2 x-y=1\end{aligned}$$(b)$$\begin{aligned}& 2 x+y=-5 \\& 2 x+y=1\end{aligned}$$(c)$$\begin{aligned}& 2 x-y=-5 \\& 2 x-y=1\end{aligned}$$(d)$$\begin{aligned}& x-2 y=-5 \\& x-2 y=-1\end{aligned}$$
In Exercises 111 and 112, determine which pair of equations may be represented by the graphs shown.(GRAPH CAN'T COPY)(a)$$\begin{aligned}& 2 x-y=2 \\& x+2 y=12\end{aligned}$$(b)$$\begin{aligned}& x-y=1 \\& x+y=6\end{aligned}$$(c)$$\begin{aligned}& 2 x+y=2 \\& x-2 y=12\end{aligned}$$(d)$$\begin{aligned}& x-2 y=2 \\& x+2 y=12\end{aligned}$$
Does every line have both an $x$-intercept and a $y$-intercept? Explain.
Can every line be written in slope-intercept form? Explain.
Does every line have an infinite number of lines that are parallel to it? Explain.
Match the description with its graph. Determine the slope of each graph and how it is interpreted in the given context. [The graphs are labeled (i), (ii), (iii), and (iv).](i) 40 (GRAPH CAN'T COPY)(ii) 125 (GRAPH CAN'T COPY)(iii) 25 (GRAPH CAN'T COPY)(iv) 600 (GRAPH CAN'T COPY)(a) You are paying $$\$ 10$$ per week to repay a $$\$ 100$$ loan.(b) An employee is paid $$\$ 12.50$$ per hour plus $$\$ 1.50$$ for each unit produced per hour.(c) A sales representative receives $$\$ 30$$ per day for food plus $$\$ 0.35$$ for each mile traveled.(d) A computer that was purchased for $$\$ 600$$ depreciates $$\$ 100$$ per year.
In Exercises 117-122, determine whether the expression is a polynomial. If it is, write the polynomial in standard form.$x+20$
In Exercises 117-122, determine whether the expression is a polynomial. If it is, write the polynomial in standard form.$3 x-10 x^2+1$
In Exercises 117-122, determine whether the expression is a polynomial. If it is, write the polynomial in standard form.$4 x^2+x^{-1}-3$
In Exercises 117-122, determine whether the expression is a polynomial. If it is, write the polynomial in standard form.$2 x^2-2 x^4-x^3+2$
In Exercises 117-122, determine whether the expression is a polynomial. If it is, write the polynomial in standard form.$\frac{x^2+3 x+4}{x^2-9}$
In Exercises 117-122, determine whether the expression is a polynomial. If it is, write the polynomial in standard form.$\sqrt{x^2+7 x+6}$
In Exercises 123-126, factor the trinomial.$x^2-6 x-27$
In Exercises 123-126, factor the trinomial.$x^2-11 x+28$
In Exercises 123-126, factor the trinomial.$2 x^2+11 x-40$
In Exercises 123-126, factor the trinomial.$3 x^2-16 x+5$
To work an extended application analyzing the numbers of bachelor's degrees earned by women in the United States from 1985 through 2007, visit this textbook's Companion Website. (Source: National Center for Education Statistics)