Linear Lines y = mx + b y − y1 = m( x − x1 ) Ax + By = C Slopes of Lines €
y 2 − y1
x 2 − x1 €
€
Parallel Lines Perpendicular Lines €
1 Lines, Lines, Lines Partner Sheets _______________________ y = m x + b _______________________ y – y = m ( x – x) _______________________ ax + by = c _______________________ parallel _______________________ perpendicular 2 1.1 A Review of Linear Equations
Part 1: Making Sense 1) What does it mean for a pattern to be linear? 2) Linear equations in slope/intercept form look like y = mx + b. Which part of that equation is the slope? y-­‐intercept? 3) When the line goes up (increases) the slope is________________________. 4) When the line goes down (decreases) the slope is ________________________. 5) How can I tell if the y-­‐intercept is greater than or less than zero? 6) Why do you think that y = mx + b is called slope/intercept form? Writing linear equations. Use the slope and y-­‐intercept given to write a linear equation. 1) m = -­‐2 4) m = 42 b = 3 b = -­‐6 5) m = -­‐12 2) m = 1 b = 5 2
b = 3 1
6) m
=
€
5
2
3) m = − 4
b= 0 b = -­‐4
€
€
3 €
€
Graphing linear equations written in slope-­intercept form. 2
Example 1: y = x + 2 3
• Plot the y-­intercept. • Locate other points using the slope. • Connect the points with a line. €
5
Example 2: y = x − 3 2
€
Graph line on the graph provided. Be sure to use a different color for each line. Graphs 1-­3 Graphs 4-­6 4
1
1. y = − x + 8 4. y = − x + −6 5
3
5. y = 5x −1 2. y = −3x + 5 €
7
6. y = x
4
2
3. y = x − 6 €
3
4 €
€
Part 2: Calculating slope. Slope Formula: Calculate the slope for each set of coordinates. 1) (6, -­‐10) and (-­‐15, 15) 5) (17, -­‐13) and (0, 1) 2) (19, -­‐16) and (-­‐7. -­‐15) 6) (3, 0) and (-­‐11, -­‐15) 3) (-­‐4, 7) and (-­‐6, -­‐4) 7) (-­‐8, 15) and (-­‐5, 30) 4) (3, -­‐20) and (5, 8) 8) (0, -­‐3) and (12, 21) 5 Part 3: Calculating the y-­intercept Calculate the y-­‐intercept using the given slope and point on the line. 1) m = 3, (-­‐7, 4) 2) slope = 7; (1, 2) 3) m = -­‐9, (2, -­‐3) 4) slope = -­‐4; (-­‐2, 5) 5) m = 4, (-­‐7, 9) 6) m= -­‐1; (2, -­‐4) 1
7) m = , (9, -­‐1) 2
€
1
8) m = ; (3, 1) 2
€
6 Part 4: Putting it all Together Write the equation for a line passing through the points listed. 1) (1, 5) and (3, 2) 4) (-­‐1, -­‐6) and (3, 2) 2) (8, -­‐7) and (10, 2) 5) (-­‐3, -­‐5) and (-­‐1, -­‐5) 3) (3, 4) and (-­‐3, 16) 6) (-­‐2, -­‐7) and (1, -­‐1) 7 1.2 What’s the point, anyway?
Use the line on each graph to… 1) identify two points on the line 2) calculate the slope 3) calculate the y-­‐intercept 4) write an equations for the line #1) Two Points on the line: (____, ____) #2) m = b = Two Points on the line: (____, ____) #3) m = b = Two Points on the line: (____, ____) m = b = (____, ____) equation: (____, ____) equation: (____, ____) equation: 8 The graph below shows how the number of visitors to a local beech varies depending on the temperature for the day. Use the information to… a) Draw a line of best fit for the data displayed below b) identify two points on the line c) calculate the slope d) calculate the y-­‐intercept e) write an equations for the line #4) Two Points on the line: (____, ____) (____, ____) m = b = equation: 9 1.3 Shake it Up
Actuaries use information about the magnitude and severity of earthquakes in a particular
geographic area to help insurance companies determine how to serve their customers.
The magnitude describes the size of the earthquake at the source. The table below
describes the earthquake’s intensity – the effects of the earth’s shaking – at different
locations away from Parkfield, CA, a town famous for its seismic activity.
Date of the Earthquake: Sept. 28, 2004
Epicenter; 9 miles south of Parkfield, CA
Magnitude at Source: 6.0
Distance from Intensity (Modified
Epicenter (km)
Mercalli Scale)
Armona, CA
89
4
Avenal, CA
35
5
Boise, ID
934
1
Coalinga, CA
47
5
Denver, CO
1,431
1
Fellows, CA
99
4
Henderson, NV
497
2
Hutchinson, KS
2,019
1
Las Vegas, NV
483
3
Lithchfield Park, AZ
778
3
Pahump, NV
401
3
Rancho Palos Verdes, CA
294
4
Salt Lake City, UT
930
2
Sevier, UT
781
2
Strathmore, CA
129
5
Town
1. On a separate sheet of graph paper, graph the (Distance from, Intensity) data.
2. Draw a line of best fit for your data.
3. Pick two points on your line to calculate the slope. What is the meaning of the
slope?
4. Use your slope and one point to calculate the y-intercept. What is the meaning of
the y-intercept?
5. Write the equation for your line of best fit.
10 6. Do you think it would cost more or less to insure customers in the Parkfield region
against earthquake damage than it would customers in places where there are no
recorded earthquakes? Why?
7. Use your equation to answer the following questions:
a) What would be your distance from the epicenter if the intensity were a 6?
b) If your city was 500 miles from the epicenter, what is the likely intensity?
11 1.4 Bike Weight and Jump Heights
In BMX dirt-bike racing, jumping high or “getting air” depends on many factors; the
rider’s skill, the angle of the jump, and the weight of the bike. Here are data
about the maximum height for various bike weights.
WEIGHT HEIGHT
(pounds) (inches)
19
10.35
19.5
10.3
20
10.25
20.5
10.2
21
10.1
22
9.85
22.5
9.8
23
9.79
23.5
9.7
24
9.6
1. Graph the data on graph paper and draw a line of best
fit to model the trend in the data.
2. Is there a positive, negative, or no relationship between
the data?
3. Calculate the slope. What is the meaning of the slope?
4. Calculate the y-intercept. What is the meaning of the y-intercept?
5. Write the equation for your line of best fit.
6. Use your equation to predict the maximum height for a bike the weighs…
a) 21.5 pounds.
b) 17 pounds.
12 1.5 Nathan’s Great Hotdog Eating Contest
This data set represents the winning number of hot dogs eaten at the annual
Nathan’s Great Hot Dog Eating Contest held in Coney Island, NY every July 4th to
determine who is the most patriotic. Your task is to create the graph, with a line of
best fit, and the equation that goes along with this data set. Then answer the
questions that go along with this problem. x represents the year the contest was
held with 1972 being year 0 and y being the number of hot dogs eaten.
5.
1.
On a separate sheet of graph paper, create a graph
of the data to the left.
2.
Draw a line of best fit and explain how you decided
to draw this particular line.
3.
What does the data tell us about the variables?
4.
Pick 2 points of the line and use those points to
write the equation for your line of best fit.
According to your equation, how many hotdogs were eaten in 1972? How do
you know?
13 6.
Look up this information on the web. How does your prediction compared to
the actual result?
7.
According to your equation, how many hot dogs were eaten in year 10? 25?
8.
Look up this data on the web. How do your predictions compare to the actual
results? What could account for these differences?
9.
If a winner had eaten 48.5 hotdogs, which year did they most likely win?
14 1.6 Global Warming
This table shows how the earth’s average normal temperature has changed sine
1880.
Year
1880
1900
1920
1940
1960
1980
2000
2010
Change in
Normal
Temperature
-0.2oC
-0.16oC
-0.27oC
0.08oC
0.03oC
0.17oC
0.55oC
0.64oC
1. On a separate sheet of graph paper, create a
scatterplot of the data.
2. Draw a line of best fit for your data.
3. Write the equation for your line of best fit.
4. Explain what your m and b mean for your line.
5. According to your equation, how much will the temperature have changed by
2020? 2050?
6. After studying the data, do you believe that global warming is real? Why or
why not?
15 1.7 The Toyland Bungee Jump
Here’s your chance to take part in an extreme sport without the risk! In this
activity you’ll set up a bungee jump and collect data relating the distance a
“jumper” falls and the number of rubber bands in the bungee cord. Then you’ll use
your model to find the number of bands you’d need in the cord for a near miss from
a specific height.
Step 1: Make a bungee cord by attaching two rubber bands to your “jumper”. (You
may need to make a harness by twisting a rubber band around the toy.)
Step 2: Place your jumper on the edge of a table or another surface while holding
the end of the bungee cord. Then let your jumper fall from the table. Use your
tape measure to measure the maximum distance the jumper falls on the first
plunge.
Step 3: Do this several times to make sure you get an accurate reading. Then
record the data in a table.
Step 4: Repeat the jump adding another rubber band to your bungee cord. Record
your data.
Step 5: Continue to add rubber bands until your jumper almost hits the floor.
Step 6: Graph your data (Number of rubber bands, distance fallen) on a sheet of
graph paper.
Step 7: Draw a line of best fit that models your data.
Step 8: Write an equation for your line of best fit.
Step 9: Enter your equation into your graphing calculator.
Step 10: Use your equation to make the following predictions;
a) How many rubber bands would be needed to jump 150 feet?
b) How many feet would you fall with 25 rubber bands?
16 2.1 Linear Equations in Point/Slope Form
Another form of a linear equation is called point-slope form. It looks like this…
y − y1 = m(x − x1 )
•
•
m still represents the slope of the line, but y1 and x1 represent one point on the line.
We will learn how to write linear equations in this form and then use our equation
solving skills to change it into y = mx + b.
€
Equivalent Equations: Look at the 6 equations written in point/slope form.
a) y – 3 = -2(x – 1)
b) y – 5 = -2(x – 5)
c) y – 9 = -2(x + 2)
d) y – 0 = -2(x – 2.5)
e) y – 7 = -2(x + 1)
f) y – 9 = -2(x – 7)
1. Is it possible that all six equations represent the same line? Explain.
2. Divide the equations among the members of your table and use the distributive
property to re-write each equation in slope/intercept form. Star the problems that
you are responsible for. Make sure to record the work from the other people at
your table.
a) y – 3 = -2(x – 1)
d) y – 0 = -2(x – 2.5)
b) y – 5 = -2(x – 5)
e) y – 7 = -2(x + 1)
c) y – 9 = -2(x + 2)
f) y – 9 = -2(x – 3)
3. What do you notice about the equations above.
17 4. Here are 15 different equations. They represent only 4 different lines. With your
group, calculate the slope/intercept form for each equation. Then, sort the equations
into their groups. Point/slope form
Point/slope form
y – 0 = 2(x – 2.5)
y – 8 = -6(x + 2)
y – (-19) = -2(x – 7)
y – 0 = 6(x + 0.5)
y – 9 = 6 (x – 1)
y - 7 = 2(x – 6)
y – (-13) = 2(x + 4)
y – 15 = 2(-10 – x)
y – 20 = -6(x + 4)
y – 3 = 2(x – 4)
y – 26 = -6(x + 5)
y – 3 = 6(x – 0)
y – 5 = -2(x + 5)
y – 11 = -2(x + 8)
y – (-4) = -6(x + 0)
18 2.2 Other Forms of linear equations
Ax + By = C
This is the “standard form” of a line, where neither A or B are equal to zero.
Standard Form is useful for finding the x and y intercepts. The x and y intercepts give us a
different way to make a graph of the line.
Example 1: What are the x and y intercepts for a line with the equation 3x + 4y = 12?
Substitute and solve.
Now that you have the x and y intercepts, you can plot
them on a coordinate grid, connect them, and you have
the line for the equation.
Example 2: Let’s try it again. Give the x and yintercepts for the equation. Plot them on the graph
provided and then draw a line to connect them
showing the graph for the equation.
2x – 3y = 5
19 Practice:
• Give the x and y intercepts for each linear equation.
• Plot the points on the coordinate grid.
• Connect the points with a line.
1 2x + 5y = 10
2 3x + 8y = 24
3 -2x + 7y = 14
4 -10x + 5y = 45
20 2.3 Converting between Forms
As we saw on the very first page, it is always easiest to graph a line when it is in slopeintercept form. You can convert (change) any linear equation written in standard form or
point-slope form to slope-intercept form.
Example 1: Convert −3x + 2y = −14 to slopeintercept form; give the slope and y intercept and
then graph the line.
€
Example 2: Convert 8x + 2y = 14 to slope-intercept
form; give the slope and y intercept and then sketch
a graph.
€
Practice:
• Convert the equation into slope intercept form.
• Give the slope and y intercepts.
1 4 x − 3y = −9
slope & y-­‐intercept €
21 2 6y − 5x = −42 slope & y-­‐intercept €
3 −3x + y = 10 slope & y-­‐intercept €
4 4 x
+16y = 32 slope & y-­‐intercept €
5 −9x − 3y = −18 slope & y-­‐ intercept €
6 −6x + 7y = −28 slope & y-­‐ intercept €
22 2.4 Mixing Up Forms
Convert all the equations to the same form, then group them.
y −11 = −5( x +1)
y = −5x + 6
€
€
€
30x + 6y = 36
y = −5x −1
€
€
y − 9 = −5( x + 2)
2
x −3
3
y=
€
€
2
( x + 3)
3
−24 x + 36y = 144
€
€
2
( x + 6)
3
y −2 =
€
y=
€
y+7=
8x −12y = 36
2
x+4
3
−5x − y = 1
€
Line 1 Line 2 Line 3 Line 4 23 3.1: Parallel and Perpendicular Lines
Part 1: On the grid below plot the two line described below.
Line 1: starts at (-5, -1) and ends at (5, 4)
Line 2: starts at (-2, -3) ands at (10, 3)
1. What type of lines are formed?
2. Calculate the slope of each line.
Line 1 - >
Line 2 - >
3. What do you notice about the slope of the lines?
Part 2: On the grid below plot the two line described below.
Line 1: starts at (-6, 1) and ends at (0, -7)
Line 2: starts at (-7, -6) ands at (1,0)
1. What type of lines are formed?
1. Calculate the slope of each line.
Line 1 - >
Line 2 - >
2. What do you notice about the slope of the lines?
24 Part 3: On the grid below plot the points, label them with their letters and connect each
one to the next. Lastly, connect point D to point A.
A (-5, -1)
B (7, 1)
C (6, 7)
D (-6, 5)
1. What shape is formed when
you connect the points?
2. Use the coordinates for each point to calculate the slope of each side of the figure.
AB->
BC->
CD->
DA->
3. What do you notice about the slopes of the lines?
25 3.2: Focus on Parallel Lines
Warm Up:
1) What are parallel lines?
2) How do the slopes of parallel lines compare?
3) Are the lines y = 2x + 3 and y = 2x + 5 parallel?
Explain your reasoning.
Part 1: Model Problems
1) Write the equation of a line that is parallel to y = 2x + 3 that passes through the
point (6,2).
2) Write the equation of a line that is parallel to y =
point (2,6).
1
x + 5 that passes through the
2
€
Part 2: Practice Problems
1. Write the equation of a line that is parallel to y = - 6x + 2 and that has a y-intercept of
6.
2. Write the equation of a line that is parallel to y = 2x + 3 and that has a y-intercept of
12.
3. Write the equation of a line that is parallel to y = point (12,4).
1
x – 6 that passes through the
4
€
26 4. Write the equation of a line that is parallel to y = - 6x + 2 that passes through the
point (-2, -3).
5. Calculate the equation of a line parallel to y = 3x + 1 that goes through the point (2,8).
6. Calculate the equation of a line parallel to y = 2x +7 and that goes through the
point (4, 12).
7. Calculate the equation of a line parallel to y = 4x + 12 that goes through the point (1,9).
8. Calculate the equation of a line parallel to y = 4x + 12 that goes through the
point (-2, 3).
Part 3: Bonus Problems
1. Calculate the equation of a line parallel to y = 5 that goes through the point (-2, -3).
2. Calculate the equation of a line parallel to x = 5 that goes through the point (6, -3).
27 3.3 Focus on Perpendicular Lines
Warm Up:
1) What are perpendicular lines?
2) How do the slopes of perpendicular
lines compare?
3) What is the opposite reciprocal of the following numbers?
3
1
A)
B) 5
C) −
2
2
E)
€
€
11
7
F) 4
1
3
€
G) −3
€
D) -2
1
2
€
Part 1: Model Problems
1. Write the equation of a line that is perpendicular to y = -5x + 2 that passes through the
point (10,6).
2. Write the equation of a line that is perpendicular to y = x – 6 that passes through the
point (6,4)
1
3. Write the equation of a line that is perpendicular to y = − x + 2 that passes through the
8
point (-4, 2).
€
28 Part 2: Practice
1. Write the equation of a line perpendicular to y = −
(0, 5).
11
x + 12 and whose y-intercept is
12
€
3
2. Write the equation of a line parallel to y = − x + 12 and whose y-intercept is (0, 5).
4
€
3. Write the equation of a line perpendicular to y = −2 x – 3 and whose y-intercept is
(0,11).
€
5
4. Write the equation of a line perpendicular to y = − x – 3 and whose y-intercept is
6
(0, 11).
€
5. Write the equation of a line that is perpendicular to y = −
point (6,7).
2
x – 3 that passes through the
3
€
6. Write the equation of a line that is perpendicular to y = -2x + 4 that passes through the
point (8, 8).
7. Write the equation of a line that is perpendicular to y =
point (5, -10)
3
x – 3 that passes through the
8
€
29 8. The two lines below are not parallel. Explain why.
y = 2x – 3 and y = -2x + 3
9. Are the two lines above perpendicular? Explain why.
10. Write the equation for a line that passes through the points (1,1) and (3,5).
11. Write an equation for a line that is perpendicular to the line in #10.
12. Write the equation for a line that passes through the points (1,3) and (2,4).
13. Write an equation for a line that is perpendicular to the line in #12.
14. What is the equation of a line with a slope of 4 that goes through the point (1,9)?
15. Write an equation for a line that will be perpendicular to the line in #14.
16. Write an equation for a line that will be parallel to the line in #14.
17. Test your equations for #14-16 by entering them into your graphing calculators. Make
a sketch of what you see below.
30 3.4: Parallel and Perpendicular Lines Definitions
What are parallel lines?
Parallel lines have slopes that are ___________________.
What are perpendicular lines?
Perpendicular lines have slopes that are _______________________.
Below is a list of linear equations. Make pairs of equations so that each pair will either make parallel or perpendicular lines. y = 0.5x − 3
Parallel Perpendicular 1 x +5
y
=
2
y = − 32 x − 6
y = 310 x + 3
y = − 1 5 x − 2.5
y = 5.5x − 4
2
y = 3x +5
y = −1.5x + 2
11
y
=
x +2
2
y = 5x
1
y = −3 x + 3
3
4
y = x + (−3)
6
y = 10 2 x + 4
€
31 Calculate the slope for each set of coordinates and then tell if the slopes would create lines that are parallel, perpendicular, or neither. 1. (3, 9) and (7, 17) (2, -­‐4) and (-­‐4, -­‐1) slope: slope: parallel, perpendicular, or neither 2. (2, 3) and (10, 7) (7, 0.5) and (-­‐2, 5) slope: slope: parallel, perpendicular, or neither 3. ( -­‐6, -­‐11) and (2, 13) (1, -­‐17) and (8, 4) slope: slope: parallel, perpendicular, or neither 4. ( -­‐2, 3.5) and (4, 5) (0, -­‐6) and (7, -­‐34) slope: slope: parallel, perpendicular, or neither 32 3.5 Equations for Lines that Create Figures
Part 1: What shapes are composed of…
Part 2: Creating figures with parallel and perpendicular sides.
1. On the grid below, draw the lines AB and PQ with the coordinates listed
below.
A (1, 2)
B (10, 6)
P (1, 9)
Q (5, 0)
a) Are the lines parallel?
b) Calculate the slope of line AB.
c) Calculate the slope of line PQ.
d) According to the slopes of each line, what type of lines are they?
33 2. The points P (-3, 1), Q (1, 2), R (0, -1) and S (-4, -2) are the vertices
(corners) of a quadrilateral.
a) Calculate the slope of a line going from P
to Q.
b) Calculate the slope of a line going from Q
to R.
c) Calculate the slope of a line going from R
to S.
d) Calculate the slope of a line going from S to P.
e) Look at the slopes you calculated. They can tell you what shape you have
without plotting the points. Is PQRS a rectangle or a parallelogram?
f) How do you know?
3. One side of a rectangle has a slope of
−2
. List the slopes of the other
3
three sides. (Draw a picture if you think it would help)
€
4. One side of a rectangle has a slope of 5. List the slopes of the other 3
sides.
34 5. The points A (3, 2), B (6, 0), C (5, 4) and D (2, 6) are the vertices of a
quadrilateral. Calculate the slopes of each side. Is quadrilateral ABCD a
rectangle or parallelogram?
6. The lines AB and PQ are perpendicular. The coordinates of the points are
A (3, 2)
B (7, 4)
P (3, 7)
Q (6, q)
What is the value of q?
*7. A triangle has vertices (corners) A (3, 1), B (7, 5) and C (1, 3). Calculate
the slopes of each side. Is this triangle a right triangle?
*8. A triangle has vertices (corners) A (4, 7), B (8, 2) and C (7, 3). Calculate
the slopes of each side. Is this triangle a right triangle?
35 9. On the grid below, draw a quadrilateral.
Calculate the slopes of each side.
Use the slopes to prove that your quadrilateral is a square, rectangle,
parallelogram, or none of these.
36 3.6 Putting it All Together!
Now that we recognize parallel and perpendicular lines and can easily convert between
all the different forms of linear, we are going to combine these skills.
Example 1: Change 3x + 5y = 15 into slope-intercept
form.
• Give an equation with a line parallel to it with
a y-intercept of -8.
• Give an equation with a line perpendicular to
it with a y-intercept of 9.
Example 2: Change 4x – 6y = 14 into slope-intercept
form.
• Give an equation with a line parallel to it with
a y-intercept of 4.
• Give an equation with a line perpendicular to
it with a y-intercept of -13.
Practice:
• Convert each equation into slope-intercept form.
• Give the equation for a line parallel to it with a y-intercept of 5.
• Give the equation for a line perpendicular to the original equation with a
y-intercept of -3.
8x + 7y = 49
7x – 3y = 21
1 2 Parallel Perpendicular Parallel Perpendicular 37 3 -6x + 9y = 54
4 Parallel Parallel Perpendicular 5 5x – 7y = -42
6 9x – 10y = 90
Parallel Parallel Perpendicular 9x + 11y = -88
Perpendicular 8 -6x – 2y = 18
Parallel Parallel Perpendicular 9 Perpendicular 7 -8x – 4y = 40
15x + 5y = 144
Parallel Perpendicular Perpendicular 1
0 10x – 6y = -35
Parallel Perpendicular 38 €
€
Example 3: What is the slope in the equation
2
( y − 5) = ( x + 6) ?
3
• Give an equation with a line parallel to it
with a y-intercept of -8.
• Give an equation with a line perpendicular
to it with a y-intercept of 9.
Example 4: What is the slope in the equation
( y + 9) = −5( x + 2) ?
• Give an equation with a line parallel to it
with a y-intercept of 4.
• Give an equation with a line perpendicular
to it with a y-intercept of -13.
39 Practice:
• For each equation written in point-slope form: Identify the slope.
• Give the equation for a line parallel to it with a y-intercept of ______.
• Give the equation for a line perpendicular to the original equation with a
y-intercept of _____.
1
1 2 ( y + 3) = −3( x −10) ( y −19) = − ( x −1)
4
€
€
€
Parallel €
Perpendicular 5
3 ( y + 4 ) = ( x + 7) 8
Parallel 4 €
Perpendicular 6
5 ( y − 8) = − ( x −15)
5
Parallel €
Parallel Perpendicular 9 1
0 ( y − 4 ) = −.25( x + 5) €
Parallel Perpendicular €
Perpendicular 8
( y −11) = − ( x +14 ) 7
Parallel €
Perpendicular ( y + 59) = −10( x − 23) Parallel 8 Perpendicular ( y −1) = 14( x − 6) Parallel 6 Perpendicular 7 ( y − 2) = −8( x +12)
€
Parallel Perpendicular 1
( y + 7) = ( x +12)
6
Parallel Perpendicular 40 MCA Practice
1. A trapezoid is drawn on a coordinate grid. The equation for 1 side of the trapezoid is
3x − 2y = 12 . Which could be an equation for another side of the trapezoid?
a. y =
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3
x +5
2
3
c. y = − x −12
2
b. y = 3x + 8
d. y = 2x − 5
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2. Which is the equation of the same line as y = 3x − 8 ?
a. 3x − 2y = 8
b. −3x − 2y = −8
c. 6x − y = 16
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d. 6x − 2y = 16
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3. A rectangle is drawn on a coordinate grid. The equation for 1 side of the rectangle is
4
( y + 5) = − ( x − 2) . Which could be an equation for another side of the rectangle?
3
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a. 4 x −12y = 5
c. 6x − 8y = 17
b. −9x −15y = −9
d. 4 x − 3y = 4
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4. Which is the equation of the same line as ( y − 3) = 2( x + 3) ?
1
a. y = − x + 9
2
b. y = 2x − 5
c. 6x − 3y = 9
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d. 6x − 3y = −27
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