Solutions to Problem Set 6

Math 101-06

9-15-2017

[Lines]

1. Find the slope of the line passing through the points $(3, -5)$ and $(7, 9)$ .

$$\dfrac{-5 - 9}{3 - 7} = \dfrac{-14}{-4} = \dfrac{7}{2}.\quad\halmos$$


2. Find the slope of the line passing through the points $(6, 7)$ and $(6, -4)$ .

$$\dfrac{7 - (-4)}{6 - 6} \quad\hbox{is undefined}.\quad\halmos$$


3. Find the slope and the y-intercept of the line

$$y = 8 x - 5.$$

The slope is 8 and the y-intercept is -5.


4. Find an equation for the line with slope 10 and y-intercept -4.

$$y = 10 x - 4.\quad\halmos$$


5. Find the slope and the y-intercept of the line

$$3 x + 11 y = 9.$$

Solve for y:

$$\eqalign{ 3 x + 11 y & = 9 \cr 11 y & = -3 x + 9 \cr \noalign{\vskip2pt} y & = -\dfrac{3}{11} x + \dfrac{9}{11} \cr}$$

The slope is $-\dfrac{3}{11}$ and the y-intercept is $\dfrac{9}{11}$ .


6. Find the slope and the y-intercept of the line

$$7 x - 5 y - 10 = 0.$$

Solve for y:

$$\eqalign{ 7 x - 5 y - 10 & = 0 \cr 7 x - 10 & = 5 y \cr \noalign{\vskip2pt} \dfrac{7}{5} x - 2 & = y \cr}$$

The slope is $\dfrac{7}{5}$ and the y-intercept is -2.


7. Find the slope and the y-intercept of the line

$$x = 8.$$

The line is vertical, so its slope is undefined and it has no y-intercept.


8. For the following line, find the x-intercept and the y-intercept, and use them to sketch the graph of the line.

$$4 x - 3 y = -12.$$

Setting $y = 0$ gives $4 x = -12$ , so $x = -3$ is the x-intercept.

Setting $x = 0$ gives $-3 y = -12$ , so $y = 4$ is the y-intercept.

$$\hbox{\epsfysize=1.5in \epsffile{lines1.eps}}\quad\halmos$$


9. For the following line, find the x-intercept and the y-intercept, and use them to sketch the graph of the line.

$$5 x + 2 y = 10.$$

Setting $y = 0$ gives $5 x = 10$ , so $x = 2$ is the x-intercept.

Setting $x = 0$ gives $2 y = 10$ , so $y = 5$ is the y-intercept.

$$\hbox{\epsfysize=1.5in \epsffile{lines2.eps}}\quad\halmos$$


10. For the following line, find the x-intercept and the y-intercept, and use them to sketch the graph of the line.

$$y = -4.$$

The line is horizontal, so there is no x-intercept, and $y = -4$ is the y-intercept.

$$\hbox{\epsfysize=1.5in \epsffile{lines3.eps}}\quad\halmos$$


11. Find the equation of the line which passes through the points $(3, 5)$ and $(7, 14)$ .

The slope is

$$\dfrac{5 - 14}{3 - 7} = \dfrac{-9}{-4} = \dfrac{9}{4}.$$

The line is

$$y - 5 = \dfrac{9}{4} (x - 3), \quad\hbox{or}\quad y - 14 = \dfrac{9}{4} (x - 7), \quad\hbox{or}\quad y = \dfrac{9}{4} x - \dfrac{7}{4}.\quad\halmos$$


12. Find the equation of the line which passes through the points $(-7, 2)$ and $(3, 15)$ .

The slope is

$$\dfrac{2 - 15}{-7 - 3} = \dfrac{13}{10}.$$

The line is

$$y - 2 = \dfrac{13}{10} (x + 7), \quad\hbox{or}\quad y - 15 = \dfrac{13}{10} (x - 3), \quad\hbox{or}\quad y = \dfrac{13}{10} x + \dfrac{111}{10}.\quad\halmos$$


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