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Section 1

The Line

Determine the equation of the line whose slope and $y$ -intercept are given.$$2,(0,4).$$

Determine the equation of the line whose slope and $y$ -intercept are given.$$-3,(0,0).$$

Determine the equation of the line whose slope and $y$ -intercept are given.$$0,(0,-2).$$

Determine the equation of the line whose slope and $y$ -intercept are given.$$1 / 2,(0,1 / 4).$$

Determine the equation of the line whose slope and $y$ -intercept are given.$$1 / 4,(0,3).$$

Find the slope, $x$ -intercept and $y$ -intercept of the given line.$$3 x-4 y=12.$$

Find the slope, $x$ -intercept and $y$ -intercept of the given line.$$2 x+4 y-9=0.$$

Find the slope, $x$ -intercept and $y$ -intercept of the given line.$$4 y=5 x+12.$$

Find the slope, $x$ -intercept and $y$ -intercept of the given line.$$1.3 x+4.7 y+11.2=0.$$

Find the slope, $x$ -intercept and $y$ -intercept of the given line.$$16-4 y=35.$$

Give an equation of the line with the given slope and passing through the given point.$$-3,(2,5)$$

Give an equation of the line with the given slope and passing through the given point.$$1 / 2,(-2,0)$$

Give an equation of the line with the given slope and passing through the given point.$$1 / 6,(0,0)$$

Give an equation of the line with the given slope and passing through the given point.$$14,(-1,0.3)$$

Give an equation of the line with the given slope and passing through the given point.$$4.1, (3, 0)$$

Find equation for the given line.A vertical line passing through $(-1,8).$

Find equation for the given line.A horizontal line passing through $(2,-6).$

Find equation for the given line.A line with intercepts $(0,4)$ and$ ( -2,0 ).$

Find equation for the given line.A vertical line passing through 4$ (12,2).$

Find equation for the given line.A horizontal line passing through $(-3,8).$

Plot the lines found in Exercise $1$.

Plot the lines found in Exercise $2$.

Plot the lines found in Exercise $3$.

Plot the lines found in Exercise $4$.

Plot the lines found in Exercise $5$.

Plot the lines found in Exercise $6$.

Plot the lines found in Exercise $7$.

Plot the lines found in Exercise $8$.

Plot the lines found in Exercise $9$.

Plot the lines found in Exercise $10$.

Plot the lines found in Exercise $11$.

Plot the lines found in Exercise $12$.

Plot the lines found in Exercise $13$.

Plot the lines found in Exercise $14$.

Plot the lines found in Exercise $15$.

Plot the lines found in Exercise $16$.

Plot the lines found in Exercise $17$.

Plot the lines found in Exercise $18$.

Plot the lines found in Exercise $19$.

Plot the lines found in Exercise $20$.

(a) Find the slope of the line whose equation is $2 x-5 y=6 .$ Find the $x$ and $y$ intercepts of the line. Plot the line. (b) Find the equation of the line parallel to the line given in (a) and passing through $(-1,7) .$ Plot the line on the same set of axes.

(a) Find the slope of the line whose equation is $3 x+7 y+42=0$. Find the $x$ and $y$ -intercepts of the line. Plot the line. (b) Find the equation of the line parallel to the line given in (a) and four units above it. Plot the line on the same set of axes.

(a) Find an equation for the horizontal line passing through the $y$ -intercept of the line in $41(\text { a). }(b)$ Find an equation for the vertical line passing through the $x$ -intercept of the line in $41(b).$

(a) Find an equation for the horizontal line passing through the $y$ -intercept of the line in $42(a) .$ (b) Find an equation for the vertical line passing through the $x$ -intercept of the line in $42(b).$

Find the equations of two lines parallel to the line $y=-3,$ and 4 units from it.

Find the equations of two lines parallel to $x=2$, and 6 units from it.

Find the slope of the line passing through each pair of points.$$(1,-2) \text { and }(1,-1.4).$$

Find the slope of the line passing through each pair of points.$$(2,-9) \text { and }(12,5).$$

Find the slope of the line passing through each pair of points.$$(-1 / 4,2 / 5) \text { and }(0,0).$$

Find the slope of the line passing through each pair of points.$$(1,4) \text { and }(2,4).$$

Find the slope of the line passing through each pair of points.$$(12,16) \text { and }(12,-73).$$

Find the slope of the line passing through each pair of points.$$(0,3) \text { and }(-6,0).$$

Find the slope of the line passing through each pair of points.$$(1 / 2,-2) \text { and }(1 / 4,-1 / 4).$$

Find the slope of the line passing through each pair of points.$$(0,-9) \text { and }(1 / 2,3).$$

Find the slope of the line passing through each pair of points.$$(-1 / 3,2 / 3) \text { and }(0,0).$$

Find the slope of the line passing through each pair of points.$$(1,-5) \text { and }(2,-5).$$

Find the slope of the line passing through each pair of points.$$(1 / 2,16) \text { and }(1 / 2,-73).$$

Find the slope of the line passing through each pair of points.$$(0,4) \text { and }(-7,0).$$

Find the equations of the lines and plot the lines from Exercise $47$.

Find the equations of the lines and plot the lines from Exercise $48$.

Find the equations of the lines and plot the lines from Exercise $49$.

Find the equations of the lines and plot the lines from Exercise $50$.

Find the equations of the lines and plot the lines from Exercise $51$.

Find the equations of the lines and plot the lines from Exercise $52$.

Find the equations of the lines and plot the lines from Exercise $53$.

Find the equations of the lines and plot the lines from Exercise $54$.

Find the equations of the lines and plot the lines from Exercise $55$.

Find the equations of the lines and plot the lines from Exercise $56$.

Find the equations of the lines and plot the lines from Exercise $57$.

Find the equations of the lines and plot the lines from Exercise $58$.

Determine an equation for the line parallel to $y=3 x-7$ and passing through the point $(1,-5)$.

Determine an equation for the line (a) parallel (b) perpendicular to $2 x-5 y=9$ and passing through the point $(-2,-4)$.

Determine an equation for the line (a) parallel (b) perpendicular to $3 x+7 y=11$ and passing through the point $(1,-3)$.

(a) Plot the line $4 x+6 y+12=0$. Find the area of the triangle formed by the line, the $x$ -axis and the $y$ -axis. (b) Repeat for $A x+B y+C=0$ for $A$$B, C$ positive.

Find an equation of a line whose $y$ -intercept is 4 and such that the area of the triangle formed by the line and the two axes is 20 square units.( Two possible answers.)

The area of a triangle formed by a line and the two axes is 40 and the slope of the line is $-5 .$ Find an equation for the line. (Two possible answers.)

(a) Find the length of the portion of the line $5 x+12 y=84$ that is cut off by the two axes. (b) Repeat for $A x+B y+C=0$ for $A, B, C$ positive.

(a) Plot the points (-1,-7),(4,2) and $(8,4) .$ (b) Do they lie on the same line? (c) How can you tell without plotting?

In 1990 the Massachusetts Non-Resident State Income Tax calls for a tax of $5 \%$ on earned income and $10 \%$ on unearned income. Suppose a person has total income of $\$ 40,000 dollars of which amount $x$ is earned. Find her $\operatorname{tax}, t,$ as a function of $x$.

When the price for a color television is $\$ 240,$ the average monthly sales for this item at a department store is $450 .$ For each $\$ 10$ increase in price, the average monthly sales fall by 20 units. What is the average monthly sales if the price is $\$ 400$ per color television?

When the price is $\$ 50$ per radio, a producer will supply 100 radios each month for sale. For each $\$ 2$ increase in price the producer will supply an additional 6 radios. How many radios are supplied if their per unit price is $\$ 72 ?$

Plot each of the following lines on the same set of axes. (a) $y=2 x$(b) $y=2(x-3)$ (c) $y=2(x+3)$ (d) How are these lines related?

Plot each of the following lines on the same set of axes. (a) $y=2 x$(b) $y-4=2 x(\text { c) } y+4=2 x$ (d) How are these lines related?

Plot each of the following lines on the same set of axes. (a) $y=2 x$(b) $y-4=2(x-3)$ (c) $y+4=2(x-3)$ (d) $y-4=2(x+3)$(e) $y+4=2(x+3)$ (f) How are these lines related?

In general, how are the lines $y=m x+b$ and $y-k=m(x-h)+b \mathrm{re}-$ lated? ( $m, b, h,$ and $k$ are constants.)

Show if $a b \neq 0,$ then the line with intercepts $(a, 0)$ and $(0, b)$ has the equa$\operatorname{tin} \frac{x}{a}+\frac{y}{b}=1$.

Using the previous exercise, determine the equation of the line with intercepts (a) (3,0),(0,6) (b) (2,0),(0,-4) (c) $(1 / 2,0),(0,2 / 3)$

Find the point on the line $y=2 x+3$ that is equidistant from the points $(-5,6)$ and $(0,0)$.

Given the two parallel lines $y=m x+b$ and $y=m x+B,$ determine the perpendicular distance between these two lines.

Let $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ be in any two points in the plane. (a) Plot these points, (b) Obtain the right triangle formed by drawing a horizontal line from $A$ and a vertical line through $B$. What are the coordinates of the point at which these two lines intersect? (c) Using the theorem of Pythagoras, derive the distance formula.