The solid form of nitrogen is a useful substance. It is used as a refrigerant in medical research, and it can also be used to remove pollutants and metal ions from waste water. Nitrogen freezes into solid form at a temperature of \(-210 \degree\)F.
Suppose your lab is kept at a temperature of \(65 \degree\)F, and your equipment can lower the temperature of nitrogen by \(5 \degree\)F per minute. How long will it take for nitrogen gas to solidify? The answer to this question is the solution of the equation
\begin{equation*}
65 - 5t = -210
\end{equation*}
In this section we’ll solve equations that involve negative numbers.
Recall that we solve an equation by undoing in reverse order the operations performed on the variable. To solve the equation
\begin{equation*}
2x+6 = 24
\end{equation*}
we first subtract 6 from both sides, then we divide both sides by 2.
\begin{align*}
2x + 6 \amp = 24 \amp \amp \blert{\text{Subtract 6 from both sides.}}\\
\underline{ ~~~~~~\blert{-6}} \amp = \underline{\blert{-6}}\\
2x \amp = 18 \amp \amp \blert{\text{Divide both sides by 2.}}\\
\\
\dfrac{2x}{\blert{2}} \amp = \dfrac{18}{\blert{2}}\\
x \amp = 9
\end{align*}
The solution is \(x = 9\text{.}\) If you like, you can review solving equations in Section 2.5 and Section 5.4.
A Quick Refresher.
Consider the expression
\begin{equation*}
8 - 2x
\end{equation*}
We might naturally read this expression as "8 subtract \(2x\text{,}\)" but we can also thnk of it as the sum of 8 and \(-2x\text{.}\)
In fact, we can think of any string of terms as a sum, where the \(~+~\) and \(~-~\) symbols tell us the sign of the term that follows. For example,
\begin{align*}
-5 - 4 ~~~~~~ \amp \text{is the sum of} ~~ -5 ~~ \text{and} ~~ -4\\
3a - 9 ~~~~~~ \amp \text{is the sum of} ~~~~~ 3a ~~ \text{and} ~~ -9\\
-7t + 6 ~~~~~~ \amp\text{is the sum of} ~~ -7t ~~ \text{and} ~~ +6
\end{align*}
We’ll use this idea to solve equations involving negative numbers.
Example7.4.1.
Solve the equation \(~8-2x = -6\)
Solution.
We start by asking ourselves what operations have been performed on \(x\text{.}\)
In Example 7.4.1, be sure to divide both sides by \(-2\text{,}\) not by 2. To reverse the operation "multiply by \(-2\text{,}\)" we divide by the same number, \(-2\text{.}\)
Checkpoint7.4.3.
Solve the equation \(~10 - 4x = 2\)
Operations performed on \(x\)
Steps for Solution
1.
1.
2.
2.
Step 1:
Step 2:
Check:
Answer.
2
Activity7.4.1.Solving Equations.
Use a calculator as needed to solve the equations.
Solve \(~-4.4 = -6.8 - 0.2m\)
Operations performed on \(x\)
Steps for Solution
1.
1.
2.
2.
Step 1:
Step 2:
Check:
Solve \(~-3.6 - 2.8t = -45.6\)
Operations performed on \(x\)
Steps for Solution
1.
1.
2.
2.
Step 1:
Step 2:
Check:
In Example 7.4.4, we use three steps to solve the equation.
Example7.4.4.
Solve \(~15 = \dfrac{-3x}{4} - 9\)
Solution.
We analyze the equation and formulate a plan for its solution.
Operations performed on \(x\)
Steps for Solution
1. Multiplied by \(-3\)
1. Add 9
2. Divided by 4
2. Multiply by 4
3. Subtracted 9
3. Divide by \(-3\)
Now we carry out the plan.
\begin{align*}
15 \amp = \dfrac{-3x}{4}- 9 ~~~~~~~~ \blert{\text{Add 9 to both sides}}\\
\underline{\hphantom{00}\blert{+9}} \amp = \underline{\blert{+9}}\\
24 \amp = \dfrac{-3x}{4}~~~~~~~~~~~ \blert{\text{Multiply both both sides by 4.}}\\
\blert{4}(24) \amp = \left(\dfrac{-3x}{4}\right)\blert{4}\\
96 \amp = -3x ~~~~~~~~~~~~ \blert{\text{Divide both sides by}~ -3.}\\
\dfrac{96}{\blert{-3}} \amp = \dfrac{-3x}{\blert{-3}}\\
-32 \amp = x
\end{align*}
The solution is \(~-32\text{.}\) (You should check the solution.)
Checkpoint7.4.5.
Solve \(~\dfrac{5-2x}{6} = -3\)
Operations performed on \(x\)
Steps for Solution
1.
1.
2.
2.
3.
3.
Step 1:
Step 2:
Check:
Answer.
\(11.5\)
Subsection7.4.2The Cartesian Coordinate System
Now that you can work with signed numbers, you can study graphs that use both positive and negative numbers. Recall that a graph shows the relationship between the values of two variables.
We used a number line to keep track of operations on signed numbers. To display the values of two variables we need two number lines. We draw a second number line perpendicular to the first one, as shown below. The values on this second number line increase from bottom to top, and the two lines intersect at the zero points of each.
The horizontal number line is called the \(x\)-axis, and the vertical number line is the \(y\)-axis. The point where the two axes intersect is called the origin. The two axes divide the plane into four regions called quadrants, which are numbered 1 through 4 counter-clockwise around the origin.
This system for locating points in the plane was inspired by the French philosopher and mathematician Rene Descartes, who lived in the seventeenth century. It is called a Cartesian coordinate system in his honor.
Subsection7.4.3Plotting Points
Now we can use both positive and negative values when we graph an equation. We’ll begin by plotting points on a Cartesian coordinate system. In Section 5.5 we plotted points from a table of values for two variables, and the same method works here.
How to Plot Points.
The first variable (the input variable) tells us the location of the point in the horizontal direction.
The second variable (the output variable) tells us its location in the vertical direction.
We’ll plot each pair of values in the table below as a point on a graph.
\(~x~\)
\(2\)
\(-3\)
\(-5\)
\(4\)
\(0\)
\(-3\)
\(~y~\)
\(6\)
\(4\)
\(-1\)
\(-3\)
\(-2\)
\(0\)
The first pair of values is \(x=2\) and \(y=6\text{,}\) which we write as \((2, 6)\text{.}\) We call the numbers 2 and 6 the coordinates of the point; 2 is the \(x\)-coordinate and 6 is the \(y\)-coordinate. The coordinates of a point describe its location in the plane.
To plot the point \((2, 6)\) we first find 2 on the \(x\)-axis (to the right of the origin), and from there move 6 units directly up, as shown at right.
The expression (2, 6) is called an ordered pair, because the order of the numbers is important. The first number in the pair is always the value of the input variable, and the second number is the value of the output variable.
Coordinates.
The location of a point in the plane is given by an ordered pair \((a, b)\text{,}\) where \(a\) is the \(x\)-coordinate of the point, and \(b\) is the \(y\)-coordinate.
The signs of the coordinates tell us which direction to move when plotting the point.
Starting at the origin, we move along the \(x\)-axis to the right if the \(x\)-coordinate is positive, and to the left if the \(x\)-coordinate is negative.
From this location on the \(x\)-axis, we move up if the \(y\)-coordinate is positive, and down if the \(y\)-coordinate is negative.
Example7.4.6.
Plot the points \((-3,4)\) and \((-5,-1)\text{.}\)
Solution.
To plot \((-3, 4)\text{,}\) we move 3 units to the left of the origin, then 4 units up. This point lies in the second quadrant. To plot \((-5, -1)\text{,}\) we move 5 units to the left, then 1 unit down into the third quadrant. Both points are shown below.
Checkpoint7.4.7.
Plot the point \((4,-3).\)
Answer.
Note7.4.8.
The signs of the coordinates determine in which of the four quadrants the point will appear. If one (or both) of its coordinates is zero, the point lies on one of the two axes.
Example7.4.9.
Plot the points \((0,-2)\) and \((-3,0)\text{.}\)
Solution.
If the \(x\)-coordinate is 0, the point lies on the \(y\)-axis, like the point \((0,-2)\) shown at right. If the \(y\)-coordinate is 0, the point lies on the \(x\)-axis, like the point \((-3,0)\text{.}\)
Checkpoint7.4.10.
Plot the following points on the grid.
\(\displaystyle (-6,2)\)
\(\displaystyle (-5,-7)\)
\(\displaystyle (8,-4)\)
\(\displaystyle (1,7)\)
\(\displaystyle (0,-9)\)
\(\displaystyle (-2,0)\)
Answer.
Subsection7.4.4Graphing Lines
At the start of this section we looked at an equation for freezing nitrogen. If \(T\) represents the temperature of the nitrogen after it has been cooling for \(t\) minutes, we have the equation
\begin{equation*}
T = 65 - 5t
\end{equation*}
The graph of this equation looks like this. (You can review graphing equations in Section 5.5.)
Remember that each point on the graph tells us something about the variables involved. For instance, the point \((0,65)\) tells us that the temperature of the nitrogen started at \(65\degree\)F.
Example7.4.11.
What does the point \((10, 15)\) tell us?
Solution.
At the point \((10, 15)\) we have \(t=10\) and \(T=15\text{.}\) After 10 minutes the temperature of the nitrogen is \(15\degree\)F.
Checkpoint7.4.12.
When does the nitrogen reach a temperature of \(0\degree\)F?
Answer.
At \(t=13\text{,}\) or at 13 minutes.
The simplest and most useful type of equation relating two variables has a very simple and useful graph. These equations are called linear equations because their graphs are straight lines.
where \(a\) and \(b\) are constants, is a linear equation. Its graph is a straight line.
In Section 5.5 we used three steps to graph an equation.
Steps for Graphing an Equation.
Make a table of values. Choose values for the input variable and use the equation to find the values of the output variable.
Choose appropriate scales and label the axes.
Plot the points from the table, and connect them with a smooth curve.
Example7.4.13.
Graph the equation \(~y = -2x + 6\)
Solution.
\(\blert{\text{Step 1} ~~}\) We begin by making a table of values. We should choose both positive and negative values for the input variable, \(x\text{.}\) Then we calculate the output values by substituting the \(x\)-values into the equation.
\(x\)
Calculation
\(y\)
Ordered Pair
\(-3\)
\(y = -2(\alert{-3}) + 6 = 12\)
\(12\)
\((-3,12)\)
\(-2\)
\(y = -2(\alert{-2}) + 6 = 10\)
\(10\)
\((-2,10)\)
\(-1\)
\(y = -2(\alert{-1}) + 6 = 8\)
\(8\)
\((-1,8)\)
\(0\)
\(y = -2(\alert{0}) + 6 = 6\)
\(6\)
\((0,6)\)
\(1\)
\(y = -2(\alert{1}) + 6 = 4\)
\(4\)
\((1,4)\)
\(2\)
\(y = -2(\alert{2}) + 6 = 2\)
\(12\)
\((2,2)\)
\(3\)
\(y = -2(\alert{3}) + 6 = 0\)
\(0\)
\((3,0)\)
\(4\)
\(y = -2(\alert{4}) + 6 = -2\)
\(-2\)
\((4,-2)\)
\(\uparrow\)
\(\hphantom{0000}\)
\(\uparrow\)
\(\hphantom{0000}\)
we choose these values
\(\hphantom{0000}\)
we calculate these values
\(\hphantom{0000}\)
\(\blert{\text{Step 2} ~~}\)Next, we sketch a Cartesian coordinate system and scale the axes. For this graph, we choose scales from 5 to 5 on the \(x\)-axis, and from 5 to 15 on the \(x\)-axis.
\(\blert{\text{Step 3} ~~}\)Then we plot each of the points in the table and connect them with a smooth curve.
You should find that all of the points lie on a straight line, as shown in the figure.
Note7.4.14.
If we choose more \(x\)-values and plot more points, we will find that the line extends forever in either direction. To indicate this we draw an arrow head at each end of the line, as shown in the figure.
Checkpoint7.4.15.
Graph the equation \(~y = 3x - 9\)
\(\blert{\text{Step 1} ~~}\) Complete the table.
\(x\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(3\)
\(4\)
\(y\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\blert{\text{Step 2} ~~}\) Label the scales on the axes.
\(\blert{\text{Step 3} ~~}\) Plot the points from the table and connect them with a straight line.
Answer.
Activity7.4.2.Graphing Lines.
How many points do we need to plot in order to draw an accurate graph? Perhaps you can see that to graph a line we really only need two points (and a ruler or straight-edge). It is a good idea to plot three points, though, as a check.
Graph the equation \(~y = 2x+4\)
\(\blert{\text{Step 1} ~~}\) Complete the table.
\(x\)
\(-3\)
\(0\)
\(1\)
\(y\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\blert{\text{Step 2} ~~}\) Label the scales on the axes.
\(\blert{\text{Step 3} ~~}\) Plot the points from the table and connect them with a straight line. Extend your line to the edges of the grid.
If the coefficient of \(x\) is a fraction, we can simplify the calculations by choosing \(x\)-values with the fraction in mind.
Graph the equation \(~y = \dfrac{3}{4}x - 2\)
\(\blert{\text{Step 1} ~~}\) Complete the table.
\(x\)
\(-4\)
\(0\)
\(4\)
\(y\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\hphantom{00}\)
\(\blert{\text{Step 2} ~~}\) Label the scales on the axes.
\(\blert{\text{Step 3} ~~}\) Plot the points from the table and connect them with a straight line. Extend your line to the edges of the grid.
Subsection7.4.5Applications of Linear Equations
Linear equations describe many common situations.
Example7.4.16.
Byron borrowed $6000 from his uncle to help pay for his college education. Now that he has graduated and has a job, he is paying back the loan at $100 per month. (His uncle is not charging Byron any interest on the loan.)
Write an equation showing the amount of money, \(y\text{,}\) that Byron still owes his uncle after \(x\) months.
Graph your equation.
Use your graph to find out how long will it take Byron to pay off his debt.
Solution.
Byron subtracts $100 from his debt each month, so
\begin{equation*}
y = 6000 - 100x
\end{equation*}
We choose values for \(x\) and make a table showing several points on the graph. Then we plot the points and connect them with a straight line.
\(~~x~~\)
\(y\)
\(0\)
\(6000\)
\(10\)
\(5000\)
\(20\)
\(4000\)
Byron has paid off his debt when the amount he still owes his uncle is $0, or when \(y=0\text{.}\) On the graph, the point with \(y\)-coordinate 0 has \(x\)-coordinate 60. Thus, it will take Byron 60 months, or 5 years, to repay his uncle.
Note7.4.17.
Note You can check your answer to part (c) of the previous Example by solving the equation
\begin{equation*}
6000 - 100x = 0
\end{equation*}
Checkpoint7.4.18.
Chloe’s All-Night Diner buys coffee in 50-pound containers. They use 2 pounds of coffee per day.
Write an equation for the number of pounds of coffee, \(C\text{,}\) left \(d\) days after they open a new container.
Graph your equation on the grid.
\(~~d~~\)
\(C\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
Chloe’s orders more coffee when they have 8 pounds left in the container. Use your graph to find out how long that will take.
Write and solve an equation to verify your answer to part (c).
Answer.
\(\displaystyle C = 50 - 2d\)
21 days
\(\displaystyle 8 = 50 - 2d\)
Subsection7.4.6Vocabulary
Cartesian coordinate system
\(x\)-axis
\(y\)-axis
origin
quadrants
input variable
output variable
\(x\)-coordinate
\(y\)-coordinate
ordered pair
linear equation
Exercises7.4.7Practice 7-4
Exercise Group.
Complete the table for the expressions in Problems 1–4.
1.
\(2-5x\)
Operations performed on \(x\)
Steps to isolate \(x~~~~\)
1.
1.
2.
2.
2.
\(-1-6y\)
Operations performed on \(y\)
Steps to isolate \(y~~~~\)
1.
1.
2.
2.
3.
\(\dfrac{z}{3} + 4\)
Operations performed on \(z\)
Steps to isolate \(z~~~~\)
1.
1.
2.
2.
4.
\(\dfrac{w-7}{2}\)
Operations performed on \(w\)
Steps to isolate \(w~~~~\)
1.
1.
2.
2.
Exercise Group.
For Problems 5-18, solve the equation. Show your work.
5.
\(3-9x = -15\)
6.
\(-6x-5 = 19\)
7.
\(-3x+7 = -26\)
8.
\(-2-8x = 38\)
9.
\(\dfrac{-7x}{2}+14 = 18\)
10.
\(\dfrac{5x}{3}-9 = -11\)
11.
\(7+\dfrac{x}{-2} = -2\)
12.
\(8+\dfrac{x}{-9} = -1\)
13.
\(-17 = \dfrac{-6x}{5}-5\)
14.
\(15 = -3+\dfrac{2x}{-3}\)
15.
\(\dfrac{x+8}{-4} = -3\)
16.
\(\dfrac{x-5}{8} = -7\)
17.
\(\dfrac{6-5x}{8} = -3\)
18.
\(\dfrac{-3-4x}{2} = -1\)
Exercise Group.
For Problems 19-24,
Identify the unknown quantity and choose a variable to represent it.
Find something in the problem that can be described in two ways and write an equation.
Solve the equation and answer the question in the problem.
19.
Starting at a depth of \(-45\) feet, a diver begins descending at a rate of \(-15\) feet per minute. How long will it take him to reach a depth of \(-159\) feet?
20.
At noon the temperature was 16°, and it has been growing colder at a rate of 3° per hour. How long will it be before the temperature reaches \(-26°\text{?}\)
21.
Alida has to let her chocolate pie cool to room temperature before cutting it. The pie comes out of the oven at 350°, and room temperature is 75°. If the pie cools on average 11° per minute, how long must Alida wait?
22.
Dean is cycling down Mt. Whitney, and descends at a rate of 80 feet per minute. If he started at an elevation of 9600 feet, how long will it be before he reaches an elevation of 1200 feet?
23.
Lois spent Saturday afternoon playing hand after hand of Fizbin with her sister. At 3 o’clock Lois had a score of 60 points, but by 5 o’clock her score was \(-96\text{.}\) If Lois lost 12 hands after 3 o’clock and didn’t win any, how much is each hand worth?
24.
Jean-Paul started a tab at The Common Grounds coffee house by paying the owner $50. One month later, Jean-Paul owed the owner $72.40. If a cup of coffee costs $1.70, how many cups did Jean-Paul drink?
Exercise Group.
For Problems 25-28, plot the points.
25.
\((2,3),~(2,-3),~(-2,3),~(-2,-3)\)
26.
\((0,4),~(4,0),~(-4,0),~(0,-4)\)
27.
\((1,5),~(5,1),~(-5,1),~(-1,5)\)
28.
\((2,2),~(6,6),~(-2,-2),~(-6,-6)\)
Exercise Group.
Answer the questions in Problems 29-30.
29.
If a point lies in the third quadrant, what do you know about its coordinates?
If a point lies in the second quadrant, what do you know about its coordinates?
30.
If a point lies on the \(x\)-axis, what do you know about its coordinates?
If a point lies on the \(y\)-axis, what do you know about its coordinates?
Exercise Group.
For Problems 31-42,
Make a table showing three solutions to the equation.
Graph the equation.
31.
\(y=x\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
32.
\(y=2x\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
33.
\(y=x - 1\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
34.
\(y=x + 2\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
35.
\(y=-2x\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
36.
\(y=-3x\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
37.
\(y=-x - 5\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
38.
\(y=-x + 4\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
39.
\(y=2x+3\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
40.
\(y=-3x-1\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
41.
\(y=\dfrac{1}{2}x - 4\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
42.
\(y=\dfrac{2}{3}x + 2\)
\(~x~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(~y~\)
\(\hphantom{000}\)
\(\hphantom{000}\)
\(\hphantom{000}\)
43.
Eloise makes $20 an hour as a computer programmer. Let \(x\) represent the number of hours Eloise works in a week, and let \(y\) represent her earnings.
Write an equation expressing \(y\) in terms of \(x\text{.}\)
Complete the table and graph your equation.
\(~x~\)
\(~y~\)
0
\(\hphantom{000}\)
10
\(\hphantom{000}\)
20
\(\hphantom{000}\)
40
\(\hphantom{000}\)
How many hours must Eloise work in order to make $500? Write and solve an equation, then locate the corresponding point on your graph.
44.
Anatole sells hand-tooled leather belts for $15 each. Let \(x\) represent the number of belts hesells in a week, and let \(y\) represent his revenue.
Write an equation expressing \(y\) in terms of \(x\text{.}\)
Complete the table and graph your equation.
\(~x~\)
\(~y~\)
0
\(\hphantom{000}\)
5
\(\hphantom{000}\)
10
\(\hphantom{000}\)
20
\(\hphantom{000}\)
How many belts must Anatole sell in order to make $600? Write and solve an equation, then locate the corresponding point on your graph.
45.
The equation that relates the temperature in degrees Fahrenheit to the temperature in degrees Celsius is
\begin{equation*}
F=1.8C+32
\end{equation*}
Graph the equation.
\(~C~\)
\(~F~\)
\(-20\)
\(\hphantom{000}\)
\(-10\)
\(\hphantom{000}\)
\(0\)
\(\hphantom{000}\)
\(10\)
\(\hphantom{000}\)
\(20\)
\(\hphantom{000}\)
What Celsius temperature is equal to 86° F? Write and solve an equation, then locate the corresponding point on your graph.
46.
On the planet Kaldor in the Gamma Quadrant, there are no seasons. The temperature at noon on any day is given by the formula
\begin{equation*}
T = 50 + 0.2h
\end{equation*}
where \(h\) is your elevation in Kaldorian feet, and \(T\) is the temperature in Kaldorian degrees.
Graph the equation.
\(~h~\)
\(~T~\)
\(-1000\)
\(\hphantom{000}\)
\(-500\)
\(\hphantom{000}\)
\(0\)
\(\hphantom{000}\)
\(1000\)
\(\hphantom{000}\)
\(1500\)
\(\hphantom{000}\)
At what elevation is the temperature on Kaldor 150 degrees? Write and solve an equation, then locate the corresponding point on your graph.
47.
Stuart invested $800 in a computer and word processor and now makes $5 a page typing research papers. Let \(x\) represent the number of pages Stuart has typed, and let \(y\) represent his profit from his business venture.
Write an equation expressing \(y\) in terms of \(x\text{.}\)
Complete the table and graph your equation.
\(~x~\)
\(~y~\)
0
\(\hphantom{000}\)
50
\(\hphantom{000}\)
100
\(\hphantom{000}\)
200
\(\hphantom{000}\)
How many pages must Stuart type in order to break even? Write and solve an equation to answer this question, then verify the answer on your graph.
48.
The Haley Light Opera ended last season $1200 in debt, but they are selling tickets to their summer series at $20 each. Let \(x\) represent the number of tickets they sell, and let \(y\) represent their balance.
Write an equation expressing \(y\) in terms of \(x\text{.}\)
Complete the table and graph your equation.
\(~x~\)
\(~y~\)
0
\(\hphantom{000}\)
20
\(\hphantom{000}\)
40
\(\hphantom{000}\)
100
\(\hphantom{000}\)
How many tickets must they sell in order to erase their debt? Write and solve an equation to answer this question, then verify the answer on your graph.
49.
An archaeological expedition gathers at a remote village at an altitude of 750 feet. They descend on foot into the valley below, at approximately 120 feet in elevation per hour. Let \(x\) represent the number of hours they have been hiking, and let \(y\) represent their elevation.
Write an equation expressing \(y\) in terms of \(x\text{.}\)
Complete the table and graph your equation.
\(~x~\)
\(~y~\)
0
\(\hphantom{000}\)
2
\(\hphantom{000}\)
5
\(\hphantom{000}\)
7
\(\hphantom{000}\)
How long will it take them to reach their site, which lies 210 feet below sea level on the valley floor? Write and solve an equation to answer this question, then verify the answer on your graph.
50.
A group of adventurers would like to set a new elevation record for a hot-air balloon flight. The temperature on the ground is 24°C, and the temperature decreases by about 8°C per kilometer above the earth’s surface. Let \(x\) represent the balloon’s altitude in kilometers, and let \(y\) represent the temperature.
Write an equation expressing \(y\) in terms of \(x\text{.}\)
Complete the table and graph your equation.
\(~x~\)
\(~y~\)
0
\(\hphantom{000}\)
2
\(\hphantom{000}\)
5
\(\hphantom{000}\)
10
\(\hphantom{000}\)
What is the altitude at the top of the troposphere, where the temperature is 120C? Write and solve an equation to answer this question, then verify the answer on your graph.
51.
Plot all the points whose \(y\)-coordinate is \(3\text{.}\) What do you get?
52.
Plot all the points whose \(x\)-coordinate is \(-3\text{.}\) What do you get?
Exercise Group.
For Problems 53-62, evaluate the expression for the given values of the variables.
