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Itg2 U2 Act.1
10/2/2011
Unit 2 – Parallel and Perpendicular Lines Activity
What do you notice about these lines? Part 1
1. The following equations are graphed to the right
Line A: y = 3x + 2
Line B: y = 3x + 4
Identify each line and label it correctly (A or B)
a) What do you notice about the lines?
b) What is the slope of Line A?
c) What is the slope of Line B?
2. Can you make any connection about the lines and their
slopes?
3. Write the equations of two lines that are parallel.
4. The following equations are graphed to the right
Line A: y = -3x + 5
Line B:
a) What do you notice about the lines?
b) What is the slope of Line A?
c) What is the slope of Line B?
5. What conjecture can you make about slopes and
perpendicular lines?
6. Find the negative reciprocal of the following numbers:
a) 5
b)
c) -3
d)
e) 1
7. Write the equation of two sets of lines that are
perpendicular to each other.
Itg2 U2 WS2.1
Unit 2 – Parallel and Perpendicular Lines
1.
a) Define parallel. b) Define perpendicular.
2
What do you know about the slopes of parallel lines?
3.
What do you know about the slopes of perpendicular
lines?
Itg2 U2 WS2.3
Integrated Math 2 – Parallel Lines WS #2-3
1. How can you tell if two lines are parallel before
graphing them?
For each problem below, write the equation of the line in
all three forms and graph BOTH lines to make sure you
are correct.
For each set of equations below, tell whether the two lines
are parallel to each other, perpendicular to each other or
neither.
2. A line that passes through the point (-2, 5) and is
parallel to the line y = x + 2
4.
y = 2x – 5
and
y = 2x + 8
5.
y = 1/4 x + 3
and
y = - 4x – 3
3. A line that passes through the point (1, 4) and is
parallel to the line y = -3x + 1
6.
y = - 2x + 8
and
y = - 1/2 x - 10
7.
y = 12x – 1
and
y = 12x + 3
8.
y = 1/2 (x – 8) + 3
and
y = - 1/2 (x + 4) – 1
9.
2
y = /3 (x + 6) – 2
3
and
y = /2 (x – 2) + 9
10. y = 2/3 x – 7
and
y = - 3/2 x + 9
11. y = - 4x
and
y = 1/4 x – 4
4. A line that passes through the point (0, 2) and is
parallel to the line y = 5x – 6
5. A line that passes through the point (-6, 1) and is
parallel to the line y = 1/2 x – 2
6. A line that passes through the point (-27, -12) and is
parallel to the line y = 2/3 x – 2
Itg2 U2 WS2.2
7. A line that passes through the point (10, 3) and is
Unit 2 – Slope-Intercept form (Solve for Y)
parallel to the line y = 3/5 x – 1
1. Which of the three forms of equations do you like best?
Why?
Itg2 U2 WS2.4
2
To find the slope which form would you prefer?
Integrated Math 2 – Perpendicular Lines WS #2-4
Rewrite each equation below in slope-intercept form (in other
1.
a) What does opposite reciprocal mean?
words, solve for y). Then write the equation of ANY line that is
b) What types of slopes does it refer to?
parallel to the given line. And then write the equation of ANY line
which is perpendicular. You may write your equations in slopeintercept form.
3.
4x + y = 11
4.
x–y=3
5.
5x – 10y = 9
6.
-3x + 2y = 8
7.
-6x – 3y = 9
8.
9x + 2y = 16
9.
-7x – 2y = 10
10. x + 5y = 15
For each problem below, write the equation of the
line in all three forms (You may wish to Graph
BOTH lines to make sure you are doing them are
correct).
2. A line that passes through the point (-1, 4) and is
perpendicular to the line y = x + 2
3. A line that passes through the point (6, 2) and is
perpendicular to the line y = -3x + 1
4. A line that passes through the point (0, 2) and is
perpendicular to y = 5x – 6
5. A line that is perpendicular to y = 1/2 (x – 2) – 4, and
passes through the point (-1, -1)
6. A line that is perpendicular to -2x + 3y = 9, and
passes through the point (-12, 7)
7. A line that is perpendicular to 4x + y = – 1, and
passes through the point (-8, -3)
Itg2 U2 PQz1 - Prep1 for Quiz Fall 2011
Itg2 U2 PQz1.2 - Prep2 for Quiz Fall 2011
Integrated Math 2 – Unit 2 Practice Quiz – || and 
Integrated Math 2 – Unit 2 Practice Quiz 2 – || and 
For questions 1-4 below, determine if the lines are parallel ( || ),
perpendicular (  ), or neither and explain why. (3 points each)
For questions 1-4 below, determine if the lines are parallel ( || ),
perpendicular (  ), or neither and explain why. (3 points each)
1.
y = 3x – 5
and
y = -5(x + 1) – 2
1.
y = 2/3 (x – 6) + 3 and
y = 2/3 x – 4
2.
-3x + 5y = 15
and
y = - 5/3 x + 6
2.
y = 4/3 x – 2
- 3x + 4y = 20
3.
y = 2/3 (x – 6) + 4 and
y = 2/3 x – 5
3.
y = - 4(x – 2) – 5 and
y = 3x – 7
4.
-4x + y = 7
y = 1/4 x – 4
4.
y = - 3/2 x + 6
- 2x + 3y = - 15
5.
a) What does opposite reciprocal mean? (2 points)
5.
a) What does opposite reciprocal mean? (2 points)
and
and
and
b) What types of slopes does it refer to?
b) What types of slopes does it refer to?
Rewrite each equation below in slope-intercept
form. Then write the equation of any line that is
parallel to the given line AND perpendicular to the
given line. (4 points each)
Rewrite each equation below in slope-intercept
form. Then write the equation of any line that is
parallel to the given line AND perpendicular to the
given line. (4 points each)
6.
3x + 4y = 20
6.
2x + 5y = - 10
7.
7x – 8y = -80
7.
7x – 4y = 12
For each problem below, write the equation of the
line in all three forms. GRAPH BOTH LINES to check
your answers. (7 points each)
For each problem below, write the equation of the
line in all three forms. GRAPH BOTH LINES to check
your answers. (7 points each)
8. A line that passes through the point (2, 8) and is
parallel to the line y = 3x – 6
8. A line that passes through the point (8, -2) and is
parallel to the line y = - 3/4 x – 5
9. A line that passes through the point (-9, -5) and is
perpendicular to the line y = -3x – 7
9. A line that passes through the point (6, 4) and is
perpendicular to the line y = -3x + 1
10. A line that is parallel to the line y = - 1/4 x + 1, and
passes through the point (2, 2)
10. A line that is parallel to the line y = 2x + 4, and
passes through the point (3, 5)
11. A line that is perpendicular to the line y = 1/3 x – 2,
and passes through the point (-4, 8)
11. A line that is perpendicular to the line y = 1/2 x – 5,
and passes through the point (2, -3)
Itg2 U2 Qz1 - Quiz 1 Fall 2011
Integrated Math 2 – Unit 2 Quiz – Parallel/Perpendicular
For questions 1-4 below, determine if the lines are parallel ( || ),
perpendicular (  ), or neither and explain why. (3 points each)
1.
y = 2x – 1
and
y = - 3(x – 1) – 4
2.
- 2x + 3y = -12
and
y = - 3/2 x + 7
3
y = /4 (x – 8) + 1 and
y = /4 x – 1
4.
- 2x + y = 3
y=
5.
For each system of equations below, find the solution
by graphing. Write your answer as an ordered pair.
2.
y= x
and
y= -x+2
3.
y = 2x – 6
and
y = -2x – 2
4.
y = -x + 1
and
y = 3x + 5
5.
y = 3(x – 1) + 4
and
y = 1/2 x + 1
6.
y = 2(x + 3) – 5
and
y = - 3x – 9
7.
y = 2x – 1
and
y=-x+ 8
8.
y = (x – 4) + 2
and
y = - 3x + 2
9.
y = 1/2 x – 3
and
y = 3/2 x – 1
3
3.
and
Itg2 U2 WS2.5 Day 1 Graphing
1.
What does the solution look like on a graph for a
system of equations?
1
/2 x – 3
a) What does opposite reciprocal mean? (2 points)
b) What types of slopes does it refer to?
Rewrite each equation below in slope-intercept
form. Then write the equation of any line that is
parallel to the given line AND perpendicular to the
given line. (4 points each)
Itg2 U2 WS2.6 Day 2 Graphing
Integrated Math 2 – Graphing Systems WS #2-6
1. What does the solution look like on a graph for a
system of equations with no solutions?
6. 2x + 3y = 15
7. 5x – 3y = -9
For each problem below, write the equation of the
line in all three forms. GRAPH BOTH LINES to check
your answers. (7 points each)
8.
A line that passes through the point (3, 7) and is
parallel to y = 2x – 3
9.
A line that passes through the point (-2, -4) and is
perpendicular to the point y = -2x – 6
2. What does the solution look like on a graph for a
system of equations with infinite solutions?
For each system of equations below, find the solution
by graphing. Write your answer as an ordered pair.
3.
y = 1/2 x – 3
and
x – 2y = 6
4.
y = 2x – 6
and
-2x + y = 3
5.
y=-x+1
and
y = 1/2 x + 4
6.
y = 2(x – 4) + 3
and
-2x + y = 1
7.
y = 3x + 3
and
6x – 2y = - 6
8.
y=-x+6
and
y = 1/2 x + 3
9.
y = (x – 1) – 5
and
y= x–2
10.
4x + 3y = 12
and
2x – 3y = 6
1
10. A line that is parallel to the line y = - /3 x + 1, and
passes through the point (9, 3)
11. A line that is perpendicular to the line y = 1/2 x + 3,
and passes through the point (4, 0)
Itg2 U2 WS2.7 Day 1 Substitution
1. What will indicate to you that you should be using
substitution to solve a system of equations?
Solve each system of equations below by using substitution.
2.
y = 2x – 6
and
3x + 2y = 9
3.
2x + 4y = 8
and
y = 2x + 13
4.
3y – 2x = 11
and
y = 9 – 2x
5.
x = 3y – 2
and
-x + 4y = 4
6.
x+y=-9
and
x = 2y
7.
y = 2x – 8
and
x = - 3y – 3
8.
x = - 3y + 16
and
y = - 2x + 7
9.
y = - 2x – 14
and
y = x + 10
Itg2 U2 WS2.9 Day 1 Elimination
Solve by elimination
1. Describe a problem that would be easiest to solve by
elimination. (What form are these problems in?)
2. 3y – 2x = 11
and
y + 2x = 9
3.
2x + 3y = -1
and
5x – 3y = 29
4.
4x + 5y = 33
and
- 4x – 3y = -23
5.
5x – 3y = 6
and
2x + 3y = 15
and
3x + y = 9
6.
4x – 3y = - 1
7. 2x + 5y = 4
and
3x – 10y = -29
8.
and
4x – 2y = -26
and
3x – 4y = 5
3x + 4y = 8
9. 2x + 5y = 11
Itg2 U2 WS2.10 Day 2 Elimination
Itg2 U2 WS2.8 Day 2 Substitution
1. What does it mean if you get a solution of 3 = 3
when doing substitution? Explain.
2.
What does it mean if you get a solution of -2 = 5
when doing substitution? Explain.
Solve by elimination
1.
When you get a solution of 4=4 when solving by
elimination, what does the graph of the equations look
like? Explain.
2.
When you get a solution of -2 = 2 when solving by
elimination, what does the graph of the equations look
like? Explain.
Solve each system of equations below by using substitution.
3.
y = x + 13
and
2x – y = - 20
4.
x = 3y – 2
and
6y = 2x + 36
5.
y = - 2/3 x + 1
and
2x + 3y = 3
6.
x = -2y + 4
and
6y + 3x = 12
7.
y= x–9
and
x = 2y + 4
8.
3x = 5 – y
and
y = - 3x + 1
9.
x = - 5y – 1
and
x = 3y – 33
10.
y= -x+5
and
y = 2x – 34
Solve by using substitution.
3. x + y = 4
and
2x + 2y = 4
4.
2x + 14y = 10 and
x + 7y = 5
5.
6x – 5y = 31 and
3x + 2y = 20
6.
2x – 3y = - 2 and
4x + y = 24
7. 2x + y = 8
8. 2x + 5y = 11
9. 3x + 2y = 12
10. 6x – 3y = 12
and
and
and
and
4x + 2y = 12
3x – 4y = 5
5x – 3y = 1
2x – y = 4
Itg2 U2 WS2.11
Use the data below to create a scatterplot. Then draw a
line of best fit. Make sure to label your graph.
Integrated Math 2 – all 3 methods
1. What does the solution of a system of equation look
like on a graph? As an answer?
Solve each system of equations below by graphing.
2. y = 1/2 x – 3
and
y = 1/2 (x – 8) + 1
3. y = 3x - 3
and
y = 3x + 2
Solve each system of equations below by using substitution.
4. y = 4x – 2
and
6x – 2y = - 2
5. 5x – y = 8
and
x = 10 – 4y
Solve by using elimination.
6. 2x – 8y = -18
and
x + 8y = 15
7. 2x + 3y = 2
and
- 5x + 2y = 33
Itg2 U2 Unit 2 Review for Test
1. What two things do you know about parallel lines?
Create your own pair of parallel lines and graph them.
2. What two things do you know about perpendicular lines?
Create your own pair of parallel lines and graph them.
Put the following in Slope-Intercept form (Solve for y)
3. - 3x + y = - 2
4.
4x – 6y = 12
5.
2x + 2y = 4
Solve each system of equations below by graphing.
6. y = 3x – 6
and
y = 1/2 x + 4
7. y = 1/2 x + 5
and
y = - 2x – 5
8. y = 2x – 4
and
y = 2x + 3
Solve each of the systems of equations by substitution.
9. x = 5y – 2
and
2x – 2y = - 12
10. 3x – y = 18
and
y = 2x – 13
Solve by using elimination.
11. - 4x + 2y = 22
and
4x + 3y = 13
12. 3x – 5y = 23
2x + 3y = - 10
and
13. Pick two points from your line (not your data) and write
the equation of the line in all three forms.
14. Use the equation of the line that you just created to
answer the following questions.
a) At what distance would you predict if there were 32
goals made?
b) How many field goals would you predict were made
from a distance of 38 yards??
Itg2 U2 WS2.1 ½ (Class-Work)
Parallel and Perpendicular Lines
Indicate whether the following are || , Neither
||
1.
y = 5x – 2
and y = 5x + 1
2.
y = 1/2 x – 4
and y = 1/2 x + 3
||
3.
y = - 5x – 6
and y = 1/5 x + 5

4.
y = - 1/2 x – 8
and y = - 1/2 x + 7
||
5.
y = 2x – 2
and y = 5x + 1
6.
y = 5x – 2
and y = - 1/5 x + 1

7.
y = 2/5 x – 4
and y = 2/5 x + 3
||
8.
y = - 5/6 x – 6
and y = 6/5 x + 5

9.
y = - 1/2 x – 8
and y = 1/2 x + 7
Neither
10.
y = 2x – 2
and y = -2x + 1
Neither
11.
y = 5(x – 2) + 1
and y = 5(x + 3) + 6
12.
y = 2/5(x + 2) + 7 and y = - 5/2 (x – 3) + 2

13.
y = - 3/4(x – 2) + 3 and y = - 3/4 (x + 3) – 3
||
14.
y = 2(x – 2) + 2
and y = - 1/2 (x – 2) + 4

15.
y = 3(x + 2) + 14 and y = 3(x + 2) – 3
||
16.
y = 5(x – 2) + 7
||
17.
y = 2/5 (x + 2) – 5 and y = 2/5 (x – 2) + 8
18.
y = -6/5 (x – 2) + 4 and y = - 5/6 (x – 2) + 9 Neither
19.
y = - 1/2 (x – 2) – 3 and y = - 1/2 (x + 2) + 1
||
20.
y = 2(x – 2) + 2
and y = - 1/2 (x – 2) – 2

and y = 5(x – 2) + 5
Changing equations into S-I form (Solve for y).
Next write two equations: (a) || and (b) 
Ex1. -2y = 5x – 2
2y = - 5x + 2
y = - 5/2 x + 1
||

y = - 5/2 x + 1
y = 2/5 x + 3
Neither
||
Ex2. 4x + 2y = 8
2y = - 4x + 8
y = -2x + 4
||

y = -2x + 4
y = 1/2 x – 5
Ex3. - 9x + 3y = 12
3y = 9x + 12
y = 3x + 4
||

Ex4.
y = 3x – 2
y = - 1/3 x + 7
7x – 3y = 12
-3y = - 7x + 12
-3
-3
7
y = /3 x – 4
||
y = 7/3 x + 9

y = - 3/7 x + 2
-------------------------------------------------1)
x + 6y = 12
-x
-x
6y = -x + 12
6
6
1
y = - /6 x + 2
2)
-2x + 3y = 9
+2x
+2x
3y = 2x + 9
3
3
2
y = /3 x + 3
3)
4x – 8y = 16
-4x
-4x
-8y = -4x + 16
-8
-8
y = 1/2 x - 2
4)
2x + 3y = 15
3y = -2x + 15
y = - 2/3 x + 5
5)
-x + 4y = 12
4y = x + 12
y = 1/4 x + 3
6)
-6x – 2y = 10
-2y = 6x + 10
-2
-2
y = -3x - 5
||
Itg2 U2 WS2.3 1/2
Integrated Math 2 – Parallel Lines WS #2-3
Find the equation of the line that passes through the given
point and is parallel to the given line.
Write the equation of the line in all three forms
1. (3, 7) y = 2x – 3
A parallel slope means that the slope is the same.
m = 2 , (3, 7)
y = 2(x – 3) + 7
y = 2x – 6 + 7
y = 2x + 1
-2x + y = 1
2x – y = -1
2. (-2, 5) y = x + 2
m = 1 , (-2, 5)
y = 1(x + 2) + 5
y=x+2+5
y= x+ 7
-x
-x
-x + y = 7
x–y=-7
3. (1, 4) y = -3x + 1
m = - 3 , (1, 4)
y = -3(x – 1) + 4
y = -3x + 3 + 4
y = - 3x + 7
+3x
+3x
3x + y = 7
4. (0, 2) y = 5x – 6
m = 5 , (0, 2)
y = 5(x – 0) + 2
y = 5x + 2
y = 5x + 2
-5x
-5x
-5x + y = 2
5x – y = -2
5.
(-6, 1) y = 1/2 x – 2
m = 1/2 , (-6, 1)
y = 1/2 (x + 6) + 1
y = 1/2 x + 3 + 1
y = 1/2 x + 4
2y = x + 8
-x
-x
-x + 2y = 8
x – 2y = -8
6.
(9, 3) y = - 1/3 x + 1
m = - 1/3 , (9, 3)
y = - 1/3(x – 9) + 3
y = -3x + 3 + 3
y = 1/3 x + 6
3y = x + 18
-x
-x
-x + 3y = 18
x – 3y = -18
7.
(-27, -12) y = 2/3 x
m = 2/3 , (-27, -12)
y = 2/3 (x + 27) – 12
y = 2/3 x + 18 – 12
y = 2/3 x + 6
3y = 2x + 18
-2x
-2x
-2x + 3y = 18
2x – 3y = -18
8.
(10, 3) y = 3/5 x – 1
m = 3/5 , (9, 3)
y = 3/5 (x – 10) + 3
y = 3/5 x – 6 + 3
y = 3/5 x – 3
5y = 3x – 15
-3x
-3x
-3x + 5y = -15
3x – 5y = 15
Itg2 U2 WS2. B EXTRA PRACTISE (Needs to be fixed up)
1) A line contains the points (3, -1) and (-1, 2). Another
line graphed in the same coordinate plane contains
the points (2, 0) and (-2, 3).
a) How do these two lines relate to each other?
b) Write the equation of each line in any of the three
forms.
m1 = (- 1) – (2)
m2 = (0) – (3)
(3) – (-1)
(2) – (-2)
= - 3/ 4
4 5/5 = 1 1/5 + b
– 1 1/5 – 1 1/5
3 4/5 = b
y = - 3/5 x + 3 4/5
2. A line contains the points (7, 3) and (-1, -4).
Find the equation of the line.
= ( 3) – (- 4)
( 7) – (- 1)
= 3+ 4
7+ 1
= 7/8
= - 3/ 4
y1 = - 3/4(x – 3) – 1
y = - 3/4 x + 9/4 – 4/4
y = - 3/4 x + 5/4
y = - 3/4 x + 5/4
y2 = - 3/4(x – 2) + 0
y = - 3/4 x + 6/4
y = - 3/4 x + 3/2
y = mx + b
y = (7/8) x + b
(3) = (7/8)(7) + b
3 = 49/8 + b
Therefore the two lines are parallel,
since they have the same slope and different
y-intercepts
2) A line contains the points (0, -2) and (1, 3). Another
line graphed in the same coordinate plane contains
the points (3, -1) and (-2, 0). How do these two lines
relate to each other?
m1 = ( - 2) – (3) m2 = ( - 1) – (0)
(0) – ( 1)
(3) – (- 2)
=
- 5/
-1
=
3 = 7 1/8 + b
– 7 1/8 – 7 1/8
- 4 1/8 = b
y = 7/8 x – 4 1/8
3. A line contains the points (4, -3) and (8, -1).
Find the equation of the line.
= (- 3) – (- 1)
( 4) – ( 8)
- 1/
5
= -3+ 1
-4
= -2
-4
= 1/2
= 5
Therefore the two lines are perpendicular,
since the slopes are the opposite reciprocal of
each other.
y = mx + b
y = (1/2) x + b
(- 3) = (1/2)(8) + b
3) A line contains the points (0, -1) and (-1, 2). Another
line graphed in the same coordinate plane contains
the points (2, 0) and (-2, 3). How do these two lines
relate to each other?
m1 = ( - 1) – (2) m2 = ( 0) – (3)
(0) – (- 1)
(2) – (- 2)
=
- 3/
1
=
1. A line contains the points (-2, 5) and (3, 2).
Find the equation of the line.
m =(5)–(2)
(- 2) – (3)
y = mx + b
y = (- 3/5) x + b
(5) = (- 3/5)(- 2) + b
5 = 6/5 + b
-7 = b
y = 1/2 x – 7
- 3/
4
= -3
Therefore the two lines will intersect, but they
are neither perpendicular nor parallel.
= - 3/5
-3 =4+ b
-4 -4
2)
Write the equation of a line in slope-intercept form:
(a) parallel and (b) perpendicular to the line:
4x - 2y = 8 and contains the point (-6, 2).
Solve both: (a) mathematically and (b) graphically.
(a) Mathematical Method
- 4x + 2y = - 8 (Make y positive by multiplying by -1)
+ 4x
+ 4x (Eventually get in the form: y = mx + b)
2y = 4x – 8
2
2
y = 2x – 4
Parallel Slope Perpendicular Slope
y = 2x + b
y = - ½ x + b [Substitute in (–6,2)]
(2) = 2(-6) + b (2) = - ½ (-6) + b
2 = - 12 + b
2 = 3 + b
(4) = - 2/3(6) + b
+ 12 + 12 _ -3
-3
_
14 = b
-1 = b
y = 2x + 14
y =-½x – 1
4= -4 + b
+ 4 + 4 _
8 = b
(b) Graphic Method
(4) = 3/2 (6) + b
4 = 9 + b
-9 -9
_
-5 = b
y = - 2/3 x + 8
y = 3/2 x – 5
Itg2 U2 WS2. B EXTRA PRACTISE
Integrated Math 2 – Solve by elimination WS #2-9b
Solve the following system of equations by elimination
1.
6x – 3y = 12
2x + 2y = 4
(-3)
6x – 3y = 12
-6x – 9y = -12
- 12y = 0
-12 -12
y =0
(1 Step)
2x + 2(0) = 4
2x = 4
2 2
x= 2
2.
3) Write the equation of a line in slope-intercept form:
(a) parallel and (b) perpendicular to the line:
3x - y = - 2 and contains the point (1, 3).
Solve mathematically only
- 3x + y = 2 (Make y positive by multiplying by -1)
+ 3x
+ 3x (Eventually get in the form: y = mx + b)
y = 3x + 2
Parallel Slope Perpendicular Slope
y = 3x + b
y = - 1/3 x + b [Substitute in (1,3)]
(3) = 3(1) + b
3= 3 + b
-3 -3 _
0 = b
y = 3x + 0
y = 3x
(3) = - 1/3(1) + b
3 =
+ 1/3
- 1/3 + b
+ 1/3 _
3 1/3 = b
y = - 1/3 x + 3 1/3
4) Write the equation of a line in slope-intercept form:
(a) parallel and (b) perpendicular to the line:
2x + 3y = 9 and contains the point (6, 4).
Solve mathematically only
2x + 3y = 9 (Make y positive by multiplying by -1)
- 2x
- 2x (Eventually get in the form: y = mx + b)
3y = - 2x + 9
3
3
y =
- 2/3 x + 3
Parallel Slope Perpendicular Slope
y = - 2/3 x + b
y = 3/2 x + b [Substitute in (6,4)]
3x + 2y = 13
4x + 4y = 24
-6x – 4y = -26
4x + 4y = 24
- 2x = -2
-2
-2
x = 1
3(1) + 2y = 13
3 + 2y = 13
–3
–3
2y = 10
2 2
y= 5
(2,0)
(-2)
(1 Step)
(1,5)
3.
5x + 2y = - 8
x + 7y = 5
5x + 2y = - 8
-5x – 35y = -25
-33y = -33
-33
-33
y = 1
7.
(-5)
(1 Step)
x + 7(1) = 5
x + 7 = 5
–7 –7
x = -2
4.
6x – 5y = 31
3x + 2y = 20
6x – 5y = 31
-6x – 4y = -40
- 9y = - 9
-9
-9
y = 1
3x + 2(1) = 20
3x + 2 = 20
–2 –2
3x = 18
3
3
x =6
5.
2x + 3y = -13
5x – 6y = 8
4x + 6y = -26
5x – 6y = 8
9x = - 18
9
9
x = -2
2(-2) + 3y = -13
- 4 + 3y = -13
+4
+ 4
3y = - 9
3
3
y =-3
6.
4(5) + y = 24
20 + y = 24
– 20
– 20
y = 4
(2)
4x – 2y = 2
3x + 2y = 12
7x = 14
7
7
x = 2
(1 Step)
3(2) + 2y = 12
6 + 2y = 12
–6
–6
2y = 6
2
2
y = 3
(-2,1)
(-2)
(2, 3)
(1 Step)
8.
3x + 2y = 12 (3)
5x – 3y = 1 (2)
9x + 6y = 36
10x – 6y = 2
19x
= 38
19
19
x= 2
(2 Steps)
3(2) + 2y = 12
6 + 2y = 12
–6
–6
2y = 6
2 2
y= 3
(6, 1)
(2)
(2, 3)
(1 Step)
9.
4x – 3y = 25
-3x + 8y = 10
12x – 9y = 75
-12x + 32y = 40
23y = 115
23 23
y=5
(-2, -3)
2x – 3y = -2
4x + y = 24 (3)
2x – 3y = - 2
12x + 3y = 72
14x
= 70 35
14
14 7
x= 5
2x – y = 1
3x + 2y = 12
(1 Step)
(5,4)
4x – 3(5) = 25
4x – 15 = 25
+ 15 + 15
4x = 40
4
4
x = 10
(3)
(4)
(2 Steps)
(5, 10)