Elementary Algebra

1.

1. Tuition at Woodrow University is $400 plus $30 per unit. Write an equation for tuition, \(W\text{,}\) in terms of the number of units, \(u\text{,}\) and fill in the table below.

   \begin{equation*} W=\hphantom{0000} \end{equation*}

2. At Xavier College, the tuition, \(X\text{,}\) is $200 plus $30 per unit. Write an equation \(X\) and fill in the table.

   \begin{equation*} X=\hphantom{0000} \end{equation*}

3. At the Yardley Institute, the tuition, \(Y\text{,}\) is $30 per unit. Write an equation \(Y\) and fill in the table.

   \begin{equation*} Y=\hphantom{0000} \end{equation*}

   \(u\)	\(W\)	\(X\)	\(Y\)
   \(3\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(5\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(8\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(10\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(12\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)

4. Graph all three equations on the grid.
   

5. Find the slope and the \(y\)-intercept for each equation.

   \begin{align*} \text{W: slope}~=~\hphantom{0000}~\amp y\text{-intercept}~=~\\ \text{X: slope}~=~\hphantom{0000}~\amp y\text{-intercept}~=~\\ \text{Y: slope}~=~\hphantom{0000}~\amp y\text{-intercept}~=~ \end{align*}

6. How are your results from part (e) reflected in the graphs of the equations?

2.

1. Anouk is traveling by train across Alaska at 60 miles per hour. Write an equation for the distance, \(A\text{,}\) Anouk has traveled in terms of hours, \(h\text{,}\) and fill in the table below.

   \begin{equation*} A=\hphantom{0000} \end{equation*}

2. Boris is traveling by snowmobile at 30 miles per hour. Write an equation for Boris' distance, \(B\text{,}\) and fill in the table.

   \begin{equation*} B=\hphantom{0000} \end{equation*}

3. Chaka is traveling in a small plane at 100 miles per hour. Write an equation for Chaka's distance, \(C\text{,}\) and fill in the table.

   \begin{equation*} C=\hphantom{0000} \end{equation*}

   \(h\)	\(A\)	\(B\)	\(C\)
   \(3\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(5\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(8\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(10\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
   \(16\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)

4. Graph all three equations on the grid.
   

5. Find the slope and the \(y\)-intercept for each equation.

   \begin{align*} \text{A: slope}~=~\hphantom{0000}~\amp y\text{-intercept}~=~\\ \text{B: slope}~=~\hphantom{0000}~\amp y\text{-intercept}~=~\\ \text{C: slope}~=~\hphantom{0000}~\amp y\text{-intercept}~=~ \end{align*}

6. How are your results from part (e) reflected in the graphs of the equations?

3.

Let's summarize the results of the Activity. In part 1 we graphed three equations:

All of these equations have the same , namely 30, but different \(y\)-intercepts. In each case the \(y\)-intercept is the same as the term in the equation. (This makes sense, because we find the\(y\)intercept by setting \(u=0\text{.}\))

In part 2 we graphed

All of these equations pass through the origin, so their \(y\)-intercepts are all , but each has a different . In each case, the slope is the same as the coefficient of \(h\) in the equation. This also makes sense if you think about it: if we increase \(h\) by one hour, then \(A\) increases by 60 miles, \(B\) increases by 30 miles, and \(c\) increases by 100 miles.

1.

1. Write an equation for the line whose \(y\)-intercept is \((0,-3)\) and whose slope is \(\dfrac{-2}{3}\text{.}\)

2. Use the slope-intercept method to graph the equation.

   \(\blert{\text{Plot the}~y\text{-intercept.}}\)

   \(\begin{array}{l} \blert{\text{Use the slope}~m =\dfrac{\Delta y}{\Delta x}=\dfrac{-2}{3}}\\ \blert{\text{to plot another point.}} \end{array}\)

   \(\begin{array}{l} \blert{\text{Use the slope}~m =\dfrac{\Delta y}{\Delta x}=\dfrac{2}{-3}}\\ \blert{\text{to plot another point.}} \end{array}\)

2.

1. State the slope and \(y\)-intercept of the equation \(y=-3x+4\)

2. Graph the equation by the slope-intercept method.

   Hint: Write the slope as a fraction,

   \begin{equation*} m= \dfrac{\Delta y}{\Delta x}=\dfrac{-3}{1} =\dfrac{3}{-1} \end{equation*}

3.

On Memorial Day weekend, Arturo drives from his home to a cabin on Diamond Lake. His distance in miles from Diamond Lake after \(x\) hours of driving is given by the equation \(y=450-50x\text{.}\)

1. What are the slope and the \(y\)-intercept of the graph of this equation?

2. Graph the equation.

3. What does the \(y\)-intercept tell you about the problem?

   What does the slope tell you about the problem?