Steve bought a Blu-Ray player for $269 and a number of discs at $14 each. Write an expression for Steve’s total bill, \(B\) (before tax), in terms of the number of discs he bought, \(d\text{.}\)
\begin{equation*}
B = \text{starting value} + \text{rate} \times d
\end{equation*}
where Steve’s bill started with the Blu-Ray player or $269, and then increased by a number of discs at a rate of $14 each. Substituting those values, we have
At 6 am the temperature was 50\(\degree\text{,}\) and it has been falling by 4\(\degree\) every hour. Write an equation for the temperature, \(T\text{,}\) after \(h\) hours.
\begin{equation*}
T = \text{starting value} + \text{rate} \times h
\end{equation*}
The temperature started at 50\(\degree\text{,}\) and then decreased each hour at the rate of 4\(\degree\) per hour, so we subtract \(4h\) from 50 to get
Kyli’s electricity company charges her $6 per month plus $0.10 per kilowatt hour (kWh) of energy she uses. Write an equation for Kyli’s electric bill, \(E\text{,}\) if she uses \(w\) kWh of electricity.
Salewa saved $5000 to go to school full time. She spends $200 per week on living expenses. Write an equation for Salewa’s savings, \(S\text{,}\) after \(w\) weeks.
As a student at City College, Delbert pays a $50 registration fee plus $15 for each unit he takes. Write an equation that gives Delbert’s tuition, \(T\text{,}\) if he takes \(u\) units.
Greta’s math notebook has 100 pages, and she uses on average 6 pages per day for notes and homework. How many pages, \(P\text{,}\) will she have left after \(d\) days?
Asa has typed 220 words of his term paper, and is still typing at a rate of 20 words per minute. How many words, \(W\text{,}\) will Asa have typed after \(m\) more minutes?
The temperature in Nome was \(-12 \degree\) F at noon. It has been rising at a rate of \(2 \degree\) F per hour all day. Write an equation for the temperature, \(T\text{,}\) after \(h\) hours.
Francine borrowed money from her mother, and she owes her $750 right now. She has been paying off the debt at a rate of $50 per month. Write an equation for Francine’s financial status, \(F\text{,}\) in terms of \(m\text{,}\) the number of months from now.
Choose both positive and negative values for \(x\text{.}\) Calculate the \(y\)-value for each \(x\)-value by substituting the \(x\)-value into the equation.
Next, sketch a Cartesian coordinate system with appropriate scales on the \(x\)- and \(y\)-axes. Plot each of the points in the table of values and connect them with a straight line. The completed graph is shown at right.
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.
Write an equation showing the amount of money, \(y\text{,}\) that Byron still owes his uncle after \(x\) months.
Stuart invested $800 in a computer 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.
Write an equation for \(y\) in terms of \(x\text{.}\)
Ludmilla earns a commission of 5% of her real estate sales. Let \(x\) represent her sales in thousands of dollars, and let \(y\) represent the commission she earns from her sales, in thousands of dollars.
Write an equation for \(y\) in terms of \(x\text{.}\)
Recall that to solve an equation we want to "isolate" the variable on one side of the equals sign. We "undo" each operation performed on the variable by performing the opposite operation on both sides of the equation.
\begin{align*}
-3x+1 \amp \gt 7 \amp \amp ~\blert{\text{ Subtract 1 from both sides.}}\\
-3x \amp \gt 6 \amp \amp\
\begin{array}{l}
\blert{\text{Divide both sides by }{-3}\text{, and reverse }}\\
\blert{\text{the direction of the inequality.}}
\end{array}\\
x \amp \lt -2
\end{align*}
The solutions are all the numbers less than \(-2\text{.}\) The graph of the solutions is shown below.
\begin{align*}
-3 \amp \lt 2x-5 \le 6 \amp \amp \blert{\text{Add 5 on all three sides of the inequality.}}\\
2 \amp \lt 2x \le 11 \amp \amp \blert{\text{Divide each side by 2.}}\\
1 \amp \lt x \le \dfrac{11}{2} \amp \amp \blert{\text{Notice that we did not reverse the inequality.}}
\end{align*}
The solutions are all the numbers greater than 1 but less than 5.5. The graph of the solutions is shown below.
Recall that a solid dot on a number line indicates that the number is part of the solution; an open dot means that the number is not part of the solution.
We show that substituting \(-5\) for \(x\) makes the equation true. When we substitute a negative number for a variable, we should enclose the number in parentheses.
SubsubsectionA.1.2.3Solutions of an equation in two variables
A solution of an equation in two variables \(x\) and \(y\) is written as an ordered pair, \((x,y)\text{.}\) For example, the solution \((-2,5)\) means that \(x=-2\) and \(y=5\text{.}\)
The points \((-3,-2)\) and \((-1,-6)\) lie on the graph, so they represent solutions of the equation. The points \((-5,0)\) and \((1,-4)\) do not lie on the graph, so they are not solutions of the equation.
First, we find the \(x\)- and \(y\)-intercepts of the graph. To find the \(y\)-intercept, we substitute \(0\) for \(x\) and solve for \(y\text{:}\)
\begin{align*}
3(\alert{0})+2y \amp =7 \amp \amp \blert{\text{Simpify the left side.}}\\
2y \amp =7 \amp \amp \blert{\text{Divide both sides by 2.}}\\
y \amp =\dfrac{7}{2}=3\dfrac{1}{2}
\end{align*}
The \(y\)-intercept is the point \(\left(0, 3\dfrac{1}{2}\right)\text{.}\) To find the \(x\)-intercept, we substitute \(0\) for \(y\) and solve for \(x\text{:}\)
\begin{align*}
3x+2(\alert{0}) \amp =7 \amp \amp \blert{\text{Simpify the left side.}}\\
3x \amp =7 \amp \amp \blert{\text{Divide both sides by 3.}}\\
x \amp =\dfrac{7}{3}=2\dfrac{1}{3}
\end{align*}
The \(x\)-intercept is the point \(\left(2\dfrac{1}{3}, 0\right)\text{.}\)
The \(T\)-intercept is \((0,-12)\text{.}\) This point tells us that when \(h=0, T=-12\text{,}\) or the temperature at noon was \(-12 \degree\text{.}\) To find the \(h\)-intercept, we set \(T=0\) and solve for \(h\text{.}\)
\begin{align*}
\alert{0} \amp = -12+2h \amp \amp \blert{\text{Add 12 to both sides.}}\\
12 \amp = 2h \amp \amp \blert{\text{Divide both sides by 2.}}\\
6 \amp =h
\end{align*}
The \(h\)-intercept is the point \((6,0)\text{.}\) This point tells us that when \(h=6, T=0\text{,}\) or the temperature will reach zero degrees at six hours after noon, or 6 pm.
\begin{align*}
2x-3y \amp = 8 \amp \amp \blert{\text{Subtract }2x \text{ from both sides.}} \\
-3y \amp = 8-2x \amp \amp \blert{\text{Divide both sides by} -3.}\\
y \amp =\dfrac{8-2x}{-3} \amp \amp \blert{\text{Divide each term of the numerator by} -3.}\\
y \amp =\dfrac{8}{3}-\dfrac{2}{3}x
\end{align*}
Slope is a type of ratio that compares vertical distance per unit of horizontal distance. We use ratios for comparison in other situations, for example, when shopping we might compute price per unit.
You are choosing between two brands of iced tea. Which is a better bargain: a 28-ounce bottle of Teatime for $1.82, or a 36-ounce bottle of Leafdream for $2.25?
The trail to Lookout Point gains 780 feet in elevation over a distance of 1.3 miles. The trail to Knife Edge gains 950 feet in elevation over a distance of 1.6 miles. Which trail is steeper?
Choose two points on the line, and calculate the ratio of vertical change to horizontal change. Use the grid lines on the graph, but don’t forget to note the scales on the axes.
The slope is the ratio \(\dfrac{\Delta h}{\Delta t}\text{.}\) The variable on the horizontal axis increases by 4 units, from 2 to 6, so \(\Delta t=4\text{.}\) The variable on the vertical axis increases by 8 grid lines, but each grid line represents 2 units, so \(\Delta h=16\text{.}\) Thus, the slope is \(\dfrac{\Delta h}{\Delta t} = \dfrac{16}{4}=4\text{.}\)
Choose two points on the line, and calculate the ratio of vertical change to horizontal change. Use the grid lines on the graph, but don’t forget to note the scales on the axes.
The slope is the ratio \(\dfrac{\Delta V}{\Delta t}\text{.}\) The horizontal variable, \(t\text{,}\) increases by 6 grid lines, but each grid line represents 2 units, so \(\Delta t=12\text{.}\) The vertical variable, \(V\text{,}\) decreases by 3 grid lines, or 6 units, so \(\Delta V=-6\text{.}\) Thus, \(\dfrac{\Delta V}{\Delta t} = \dfrac{-6}{12}=\dfrac{-1}{2}\text{.}\)
Hint: Find two points that lie on the intersubsection of grid lines, so that it’s easy to read their coordinates. For example, you could use \((2, 300)\) and \((8, 600)\text{.}\)
Lynette is saving money for the down payment on a new car. The figure below shows the amount \(A\) she has saved, in dollars, \(w\) weeks after the first of the year.
Because the \(y\)-intercept \((0,b)\) is the "starting value" of a linear model, and its rate of change is measured by its slope,\(m\text{,}\) the equation for a linear model
\begin{equation*}
y = \text{starting value} + \text{rate} \times x
\end{equation*}
The temperature inside a pottery drying oven starts at 70 degrees and is rising at a rate of 0.5 degrees per minute. Write a function for the temperature, \(H\text{,}\) inside the oven after \(t\) minutes.
A perfect score on a driving test is 120 points, and you lose 4 points for each wrong answer. Write a function for your score, \(S\text{,}\) if you give \(n\) wrong answers.
Monica has saved $7800 to live on while she attends college. She spends $600 a month. Write a function for the amount, \(S\text{,}\) in Monica’s savings account after \(t\) months.
Jesse opened a new doughnut shop in an old store-front. He invested $2400 in remodeling and set-up, and he makes about $400 per week from the business. Write a function giving the shop’s financial standing, \(F\text{,}\) after \(w\) weeks.
It doesn’t matter which point is \(P_1\) and which is \(P_2\text{,}\) so we choose \(P_1\) to be \((-6,2)\text{.}\) Then \((x_1,y_1)=(-6,2)\) and \((x_2,y_2)=(3,-1)\text{.}\) Thus,
Step 2: We use the slope, \(\dfrac{\Delta y}{\Delta x} = \dfrac{3}{4}\text{,}\) to find another point on the line, as follows. Start at the point \((0,-2)\) and move 3 units up and 4 units to the right. Plot a second point here, at \((4,1)\text{.}\)
Step 3: Find a third point by writing the slope as \(\dfrac{\Delta y}{\Delta x} = \dfrac{-3}{-4}\text{:}\) from \((0,-2)\text{,}\) move down 3 units and 4 units to the left. Plot a third point here, at \((-4,-5)\text{.}\)
Step 2: Use the slope, \(\dfrac{\Delta y}{\Delta x} = \dfrac{-1}{2}\text{,}\) to find another point on the line, as follows. Start at the point \((-3,-2)\) and move 1 unit down and 2 units to the right. Plot a second point here, at \((-1,-3)\text{.}\)
Step 3: Find a third point by writing the slope as \(\dfrac{\Delta y}{\Delta x} = \dfrac{1}{-2}\text{:}\) from \((-3,-2)\text{,}\) move 1 unit up and 2 units to the left. Plot a third point here, at \((-5,-1)\text{.}\)
