Quiz 4 (Solution)

Date: MATH 1090-4 - Fall 2003

1. Graph the set determined by $3x+2y=6$. Make sure that your axes have labels and unit marks!

   We choose two points on the line, plot the points on the plane and then draw the line through them. For instance, we take $(2,0)$ and $(0,3)$.

   Figure 1: Graph of the line for exercise 1

   

2. For each of the following lines, find the slope and intercepts (both $x$ and $y$-intercepts). If any of these elements does not exist, state it clearly.
   1. $x=-3$.

      This is a vertical line so the slope is undefined. Being a vertical line, the $x$-intercept is the point $(-3,0)$ and there is no $y$-intercept.

   2. $2x-3y=1$.

      We start by rewriting the equation in slope-intercept form.

      $$\begin{split}2x-3y&=1\\ 2x-1&=3y\\ \frac{2}{3} x -\frac{1}{3} &=y \end{split}$$

      
      
      Then, the slope is $\frac{2}{3}$ and the $y$-intercept is $(0,-\frac{1}{3})$. To find the $x$-intercept we solve $2x-1=0$, so that $x=\frac{1}{2}$ and the $x$-intercept is $(\frac{1}{2},0)$.

   3. $y=-3$.

      Since the formula is already in slope-intercept form, we see that the slope is 0 (the line is horizontal) and the $y$-intercept is $(0,-3)$. Since the line is horizontal, there is no $x$-intercept.

3. Decide if the lines $y=3x-2$ and $x+3y=1$ are parallel, perpendicular, or neither.

   Since the first line is already given in slope-intercept form we see that its slope is $3$. Next, we rewrite the second line in slope-intercept form:

   $$\begin{split}x+3y&=1\\ 3y&=-x+1\\ y&=-\frac{1}{3} x + 1 \end{split}$$

   
   
   Thus we see that the slope of the second line is $-\frac{1}{3}$.

   Since the slopes are different we conclude that the lines are not parallel. But, since the product of the slopes is $3\cdot (-\frac{1}{3}) = -1$ we see that the lines are perpendicular.

4. The water company charges its customers, monthly, according to the following scheme: a base cost of $7 and an additional $ 0.01 for each gallon of water used.
   1. Write an equation for the monthly charge, $C$, as a function of the amount of water used, $W$.

      $C(W) = 7 + 0.01 \cdot W$.

   2. If, in addition, there is a 7% tax on water bills, what is the new equation for the charge as a function of the amount of water used?

      The cost is the same as before, plus a 7%, that is

      $$CT(W) = C(T) + 0.07 \cdot C(T) = (1+ 0.07)\cdot C(T) = 1.07 \cdot ( 7 + 0.01 W) = 7.49 + 0.0107 W$$