2.1 The rectangular coordinate systems and graphs  (Page 6/21)

 Page 6 / 21

$4x-3=2y$

$3x+8y=9$

The x- intercept is $\text{\hspace{0.17em}}\left(3,0\right)\text{\hspace{0.17em}}$ and the y -intercept is $\text{\hspace{0.17em}}\left(0,\frac{9}{8}\right).$

$2x-\frac{2}{3}=\frac{3}{4}y+3$

For each of the following exercises, solve the equation for y in terms of x .

$4x+2y=8$

$y=4-2x$

$3x-2y=6$

$2x=5-3y$

$y=\frac{5-2x}{3}$

$x-2y=7$

$5y+4=10x$

$y=2x-\frac{4}{5}$

$5x+2y=0$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

$\left(-4,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-4\right)$

$d=\sqrt{74}$

$\left(2,-5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(7,4\right)$

$\left(5,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,6\right)$

$d=\sqrt{36}=6$

$\left(-4,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(10,3\right)$

Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.

$\left(19,12\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(41,71\right)$

$d\approx 62.97$

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

$\left(-5,-6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,2\right)$

$\left(-1,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(7,-4\right)$

$\left(3,\frac{-3}{2}\right)$

$\left(-5,-3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,-8\right)$

$\left(0,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,-9\right)$

$\left(2,-1\right)$

$\left(-43,17\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(23,-34\right)$

Graphical

For each of the following exercises, identify the information requested.

What are the coordinates of the origin?

$\left(0,0\right)$

If a point is located on the y -axis, what is the x -coordinate?

If a point is located on the x -axis, what is the y -coordinate?

$y=0$

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).

$\left(4,1\right)\left(-2,-3\right)\left(5,0\right)$

$\left(-1,2\right)\left(0,4\right)\left(2,1\right)$

not collinear

$\left(-3,0\right)\left(-3,4\right)\left(-3,-3\right)$

Name the coordinates of the points graphed.

$\left(-3,2\right),\left(1,3\right),\left(4,0\right)$

Name the quadrant in which the following points would be located. If the point is on an axis, name the axis.

$\begin{array}{l}a.\left(-3,-4\right)\\ b.\left(-5,0\right)\\ c.\left(1,-4\right)\\ d.\left(-2,7\right)\\ e.\left(0,-3\right)\end{array}$

For each of the following exercises, construct a table and graph the equation by plotting at least three points.

$y=\frac{1}{3}x+2$

 $x$ $y$ $-3$ 1 0 2 3 3 6 4

$y=-3x+1$

$2y=x+3$

 x y –3 0 0 1.5 3 3

Numeric

For each of the following exercises, find and plot the x- and y -intercepts, and graph the straight line based on those two points.

$4x-3y=12$

$x-2y=8$

$y-5=5x$

$3y=-2x+6$

$y=\frac{x-3}{2}$

For each of the following exercises, use the graph in the figure below.

Find the distance between the two endpoints using the distance formula. Round to three decimal places.

$d=8.246$

Find the coordinates of the midpoint of the line segment connecting the two points.

Find the distance that $\text{\hspace{0.17em}}\left(-3,4\right)\text{\hspace{0.17em}}$ is from the origin.

$d=5$

Find the distance that $\text{\hspace{0.17em}}\left(5,2\right)\text{\hspace{0.17em}}$ is from the origin. Round to three decimal places.

Which point is closer to the origin?

$\left(-3,4\right)$

Technology

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2 nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y -intercept, for each of the following graphs.

${\text{Y}}_{1}=-2x+5$

${\text{Y}}_{1}=\frac{3x-8}{4}$

${\text{Y}}_{1}=\frac{x+5}{2}$

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.

After graphing it, use the 2 nd CALC button and 2:zero button, hit enter. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x -intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x -intercept, hit enter. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x -intercept. Hit enter. At the bottom of your screen it will display the coordinates of the x- intercept or the “zero” to the y -value. Use this to find the x -intercept.

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x -intercept between your right and left boundaries. With other types of functions (more than one x -intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

${\text{Y}}_{1}=-8x+6$

${\text{Y}}_{1}=4x-7$

${\text{Y}}_{1}=\frac{3x+5}{4}\text{\hspace{0.17em}}$ Round your answer to the nearest thousandth.

Extensions

A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

If the road was made in the previous exercise, how much shorter would the man’s one-way trip be every day?

mi shorter

Given these four points: find the coordinates of the midpoint of line segments $\text{\hspace{0.17em}}\overline{\text{AB}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\overline{\text{CD}}.$

After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.

$\text{6}.0\text{42}$

Given the graph of the rectangle shown and the coordinates of its vertices, prove that the diagonals of the rectangle are of equal length.

In the previous exercise, find the coordinates of the midpoint for each diagonal.

Midpoint of each diagonal is the same point $\text{\hspace{0.17em}}\left(2,2\right).\text{\hspace{0.17em}}$ Note this is a characteristic of rectangles, but not other quadrilaterals.

Real-world applications

The coordinates on a map for San Francisco are $\text{\hspace{0.17em}}\left(53,17\right)\text{\hspace{0.17em}}$ and those for Sacramento are $\text{\hspace{0.17em}}\left(123,78\right).\text{\hspace{0.17em}}$ Note that coordinates represent miles. Find the distance between the cities to the nearest mile.

If San Jose’s coordinates are $\text{\hspace{0.17em}}\left(76,-12\right),$ where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.

$\text{37}\text{\hspace{0.17em}}$ mi

A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were $\text{\hspace{0.17em}}\left(49,64\right).\text{\hspace{0.17em}}$ One rescue boat is at the coordinates $\text{\hspace{0.17em}}\left(60,82\right)\text{\hspace{0.17em}}$ and a second Coast Guard craft is at coordinates $\text{\hspace{0.17em}}\left(58,47\right).\text{\hspace{0.17em}}$ Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?

A man on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?

54 ft

If we rent a truck and pay a $75/day fee plus$.20 for every mile we travel, write a linear equation that would express the total cost $\text{\hspace{0.17em}}y,$ using $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 mi.

what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice