Integrated Math 2 Unit 1 - Quiz
Itg2 U2Act.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) 5b) 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.
a) Parallel lines are lines that never intersect and always are the same distance from each other.
b) Perpendicularlines are lines that intersect each other at a right angle.
2 What do you know about the slopes of parallel lines?
Slopes of parallel lines are the same.
3. What do you know about the slopes of perpendicular lines?
Slopes of perpendicularlines are the opposite reciprocals of each other.
Opposite means the sign changes, so if one slope sign is positive the other will be negative or vice versa.
The term reciprocal means the numerator and denominator switch so the number that was on top is now on bottom and vice versa.
For each set of equations below, tell whether the two lines are parallel to each other, perpendicular to each other or neither.
4. y = 2x – 5and y = 2x + 8
m1 = 2 m2 = 2
Therefore the slopes are ||slopes to each other
5. y = 1/4 x + 3 and y = - 4x – 3
m1 = 1/4 m2= - 4
Therefore slopes are to each other
6. y = - 2x + 8 and y = - 1/2 x - 10
m1 = -2 m2= - 1/2
Therefore the slopes are neither ||nor
7. y = 12x – 1 and y = 12x + 3
m1 = 12 m2 = 12
Therefore the slopes are ||slopes to each other
8. y = 1/2 (x – 8) + 3 and y = - 1/2 (x + 4) – 1
m1 = 1/2 m2= - 1/2
Therefore the slopes are neither ||nor
9. y = 2/3 (x + 6) – 2 and y = 3/2 (x – 2) + 9
m1 = 2/3 m2= 3/2
Therefore the slopes are neither ||nor
10. y = 2/3 x – 7and y = - 3/2 x + 9
m1 = 2/3 m2= - 3/2
Therefore slopes are to each other
11. y = - 4x and y = 1/4 x – 4
m1 = - 4 m2= 1/4
Therefore slopes are to each other
Itg2 U2 WS2.2
Unit 2 – Slope-Intercept form (Solve for Y)
1. Which of the three forms of equations do you like best? Why?
2 To find the slope which form would you prefer?
I like a line equation that has the slope in it. Thus I prefer Slope-intercept or Modified Point Slope.
Rewrite each equation below in slope-intercept form (in other words, solve for y). Then write the equation of ANY line that is parallel to the given line. And then write the equation of ANY line which is perpendicular. You may write your equations in slope-intercept form.
3. 4x + y = 11
- 4x -4x
y = - 4x + 11
4.x – y = 3
- x - x
- y = - x + 3
y = x – 3
5. 5x – 10y = 9
- 5x - 5x
-10y = - 5x + 9
-10 -10
y = 5/10 – 9/10
y = 1/2 – 9/10
6. -3x + 2y = 8
+3x +3x
2y = 3x + 8
2 2
y = 3/2 x + 4
7. -6x – 3y = 9
+6x +6x
-3y = 6x + 9
-3 -3
y = -2x – 3
8. 9x + 2y = 16
-9x -9x
2y = -9x + 16
2 2
y = - 9/2 x + 8
9. -7x – 2y = 10
+ 7x +7x
-2y = 7x + 10
-2 -2
y = 7/2 x – 5
10. x + 5y = 15
-x -x
5y = -x + 15
5 5
y = - 1/5 x + 3
Itg2 U2WS2.3
Integrated Math 2 – Parallel Lines WS #2-3
For each problem below write the equation of the line in all three forms and graph BOTH lines to make sure you are correct.
1. How can you tell if two lines are parallel before graphing them?
Two lines are parallel if they have the same slope and have different y-intercepts.
For each problem below, write the equation of the line in all three forms and graph BOTH lines to make sure you are correct.
2. A line that passes through the point (-2, 5) and is parallel to the line y = x + 2
A parallel slope means that the slope is the same.
m = 1, (-2, 5)
y = 1(x + 2) + 5
y = x + 2 + 5
y = x + 7
-x + y = 7
x – y = - 7
Graph a)y = x + 2 (Original equation)
Graph b)y = x + 7 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
3. A line that passes through the point (1, 4) and is parallel to the line y = -3x + 1
m = -3, (1, 4)
y = -3(x – 1) + 4
y = -3x + 3 + 4
y = -3x + 7
3x + y = 7
Graph a)y = -3x + 1 (Original equation)
Graph b)y = -3x + 7 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
4. A line that passes through the point (0, 2) and is parallel to the line y = 5x – 6
m = 5, (0, 2)
y = 5(x – 0) + 2
y = 5x – 0 + 2
y = 5x + 2
-5x + y = 2
5x – y = 2
Graph a)y = 5x – 6 (Original equation)
Graph b)y = 5x + 2 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
5. A line that passes through the point (-6, 1) and is parallel to the line y = 1/2x – 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 + 2y = 16
x – 2y = -16
Graph a)y = 1/2 x – 2 (Original equation)
Graph b)y = 1/2 x + 4 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
6. A line that passes through the point (-27, -12) and is parallel to the line y = 2/3 x – 2
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 + 3y = 18
2x – 3y = -18
Graph a)y = 2/3 x – 2 (Original equation)
Graph b)y = 2/3 x + 6 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
7. A line that passes through the point (10, 3) and is parallel to the line y = 3/5 x – 1
m = 3/5, (10, 3)
y = 3/5 (x – 10) + 3
y = 3/5 x – 6 + 3
y = 3/5 x – 3
5y = 3x – 15
-3x + 5y = -15
3x – 5y = 15
Graph a)y = 3/5 x – 1 (Original equation)
Graph b)y = 3/5 x – 3 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
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Itg2 U2WS2.4
Integrated Math 2 – Perpendicular Lines WS #2-4
1. a) What does opposite reciprocal mean?
b) What types of slopes does it refer to?
Slopes that are opposite reciprocals areslopes. Opposite means the sign changes, so if one slope sign is positive the other will be negative or vice versa.
The term reciprocal means the numerator and denominator switch so the number that was on top is now on bottom and vice versa.
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
m = - 1, (-1, 4)
y = -1(x +1) + 4
y = -1x – 1 + 4
y = -1x + 3
+ x + x
x + y = 3
Graph a)y = x + 2 (Original equation)
Graph b)y = -x + 3 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
3. A line that passes through the point (6, 2) and is perpendicular to the line y = -3x + 1
m = 1/3, (-1, 4)
y = 1/3 (x – 6) + 2
y = 1/3 x – 2 + 2
y = 1/3 x
3y = x + 0
– x – x
-x + 3y = 0
x – 3y = 0
Graph a)y = -3x + 1 (Original equation)
Graph b)y = 1/3 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
4. A line that passes through the point (0, 2) and is perpendicular to y = 5x – 6
m = - 1/5, (0, 2)
y = - 1/5 (x – 0) + 2
y = - 1/5 x + 2
5y = - x + 10
+ x + x
x + 5y = 10
Graph a)y = 5x – 6 (Original equation)
Graph b)y = - 1/5 x + 2 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
5. A line that is perpendicular to y = 1/2(x – 2) – 4, and passes through the point (-1, -1)
m = -2, (-1, -1)
y = -2(x + 1) – 1
y = -2x – 2 – 1
y = -2x– 3
+ 2x + 2x
2x +y = 3
Graph a)y = 1/2 x – 3 (Original equation)
Graph b)y = - 2x – 3 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
6. A line that is perpendicular to -2x + 3y = 9, and passes through the point (-12, 7)
-2x + 3y = 9
3y = 2x + 9
y = 2/3 x + 3
m = -3/2, (-12, 7)
y = - 3/2 (x + 12) + 7
y = - 3/2x – 18 + 7
y = - 3/2x – 11
2y = -3x – 22
+ 3x + 3x
3x + 2y = -22
Graph a)y = 2/3 x + 3 (Original equation)
Graph b)y = - 3/2 x – 11 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
7. A line that is perpendicular to 4x + y = – 1, and passes through the point (-8, -3)
4x + y = -1
y = - 4x – 1
m = 1/4, (-8, -3)
y = 1/4 (x + 8) – 3
y = 1/4 x + 2– 3
y = 1/4x – 1
4y = x – 4
– x – x
-x + 4y = - 4
x – 4y = 4
Graph a)y = - 4x – 1 (Original equation)
Graph b)y = 1/4 x – 1 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
Itg2 U2PQz1 - Prep1 for Quiz Fall 2011
Integrated Math 2 – Unit 2 Practice Quiz – ||and
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
m1 = 3 m2 = -5
Therefore slopes are neither ||nor
2. -3x + 5y = 15 and y = - 5/3 x + 6
5y = 3x + 15
y = 3/5 – 3
m1 = 3/5 m2= - 5/3
Therefore slopes are to each other
3. y = 2/3 (x – 6) + 4 and y = 2/3x – 5
m1 = 2/3 m2 = 2/3
Therefore slopes are||slopes to each other
4. -4x + y = 7 and y = 1/4 x – 4
y = 4x + 7
m1 = 4 m2= 1/4
Slopes are neither ||nor (Although the slopes are reciprocals of each other they are not the opposite signs. (The sign did not change)
5. a) What does opposite reciprocal mean? (2 points)
b) What types of slopes does it refer to?
Slopes that are opposite reciprocals areslopes. Opposite means the sign changes, so if one slope sign is positive the other will be negative or vice versa.
The term reciprocal means the numerator and denominator switch so the number that was on top is now on bottom and vice versa.
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
-3x -3x
4y = - 3x + 20
4 4
y = - 3/4x + 5
m1 = - 3/4
|| slope: y =- 3/4 x + #
slope: y = 4/3 x + #
7. 7x – 8y = -80
-7x -7x
- 8y = - 7x – 80
-8 -8
y = 7/8– 10
m1 = 7/8
|| slope: y = 7/8 x + #
slope: y = - 8/7 x + #
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
m = 3, (2, 8)
y = 3(x – 2) + 8
y = 3x – 6 + 8
y = 3x + 2
-3x + y = 2
3x – y = - 2
Graph a)y = 3x – 6 (Original equation)
Graph b)y = 3x + 2 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
9. A line that passes through the point (-9, -5) and is perpendicularto the line y = -3x – 7
m = 1/3 , (-9, -5)
y = 1/3 (x + 9) – 5
y = 1/3 x + 3 – 5
y = 1/3 x – 2
3y = x – 6
-x + 3y = -6
x + 3y = 6
Graph a)y = -3x – 7 (Original equation)
Graph b) y = 1/3 x – 2 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
10. A line that is parallel to the line y = - 1/4 x + 1, and passes through the point (2, 2)
m = - 1/4 , (12, 2)
y = - 1/4 (x – 12) + 2
y = - 1/4 x + 3 + 2
y = - 1/4 x + 5
4y = x + 20
-x + 4y = 20
x – 4y = -20
Graph a)y = - 1/4 x + 1 (Original equation)
Graph b)y = - 1/4 x + 5 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
11. A line that is perpendicular to the line y = 1/3 x – 2, and passes through the point (-4, 8)
m = - 3, (-4, 8)
y = - 3(x + 4) + 8
y = -2x – 12 + 8
y = -2x – 4
2x + y = - 4
Graph a)y = 1/3 x – 2 (Original equation)
Graph b) y = -2x – 4(New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
Itg2 U2PQz1.2 - Prep2 for Quiz Fall 2011
Integrated Math 2 – Unit 2 Practice Quiz – ||and
For questions 1-4 below, determine if the lines are parallel ( ||), perpendicular ( ), or neither and explain why. (3 points each)
1. y = 2/3 (x – 6) + 3and y = 2/3x – 4
m1= 2/3 m2 = 2/3
Therefore slopes are ||slopes to each other
2. y = 4/3x – 2and - 3x + 4y = 20
4y = 3x + 20
y = 3/4x + 5
m1 = 4/3 m2= 3/4
Slopes are neither ||nor (Although the slopes are reciprocals of each other they are not the opposite signs. (The sign did not change)
3. y = - 4(x – 2) – 5and y = 3x – 7
m1 = -4 m2 = 3
Therefore slopes are neither ||nor
4. y = - 3/2x +6 and - 2x + 3y = - 15
3y = 2x – 15
y = 2/3 x – 5
m1 = - 3/2 m2= 2/3
Therefore slopes are to each other
5. a) What does opposite reciprocal mean? (2 points)
b) What types of slopes does it refer to?
Slopes that are opposite reciprocals areslopes. Opposite means the sign changes, so if one slope sign is positive the other will be negative or vice versa.
The term reciprocal means the numerator and denominator switch so the number that was on top is now on bottom and vice versa.
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. 2x + 5y = - 10
-2x -2x
5y = - 2x – 10
5 5
y = - 2/5x – 2
m1 = - 2/5
|| slope: y =- 2/5x + #
slope: y = 5/2 x + #
7. 7x – 4y = 12
-7x -7x
- 4y = - 7x +12
-4 -4
y = 7/4– 3
m1 = 7/8
|| slope: y = 7/4 x + #
slope: y = - 4/7 x + #
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 (8, -2) and is parallel to the line y = - 3/4x – 5
m = - 3/4, (8, -2)
y = - 3/4(x – 8) – 2
y = - 3/4x + 6 – 2
y = - 3/4x + 4
4y = -3x + 16
3x +4y = 16
Graph a)y = - 3/4 x – 5(Original equation)
Graph b)y = - 3/4 x + 4(New equation)
Check to be sure equations lines on graph are parallel as confirmation.
9. A line that passes through the point (6, 4) and is perpendicularto the line y = -3x + 1
m = 1/3 , (6, 4)
y = 1/3(x– 6) + 4
y = 1/3 x – 2 + 4
y = 1/3 x + 2
3y = x + 6
-x + 3y = 6
x – 3y = - 6
Graph a)y = -3x +1 (Original equation)
Graph b) y = 1/3 x + 2 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
10. A line that is parallel to the line y = 2x + 4, and passes through the point (3, 5)
m = 2 , (3, 5)
y = 2(x – 3) + 5
y = 2x – 6 + 5
y = 2x – 1
-2x + y = -1
2x–y = 1
Graph a)y = 2x + 4 (Original equation)
Graph b)y = 2x – 1 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
11. A line that is perpendicular to the line y = 1/2 x – 5, and passes through the point (2, -3)
m = 2, (2, - 3)
y = 2(x – 2) – 3
y = 2x + 4– 3
y = 2x + 1
-2x + y = 1
2x–y = -1
Graph a)y = 1/2 x – 5 (Original equation)
Graph b) y = 2x + 1(New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
Itg2 U2Qz1 - 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
m1= 2 m2 = - 3
Therefore slopes are neither ||nor
2. - 2x + 3y = -12 and y = - 3/2x + 7
3y = 2x – 12
y = 2/3 x – 4
m1 = 2/3 m2= - 3/2
Therefore slopes are to each other
3. y = 3/4 (x – 8) + 1 and y = 3/4x – 1
m1= 3/4 m2 = 3/4
Therefore slopes are ||slopes to each other
4. - 2x + y = 3 and y = 1/2x – 3
y = 2x + 3
m1 = 2 m2= 1/2
Slopes are neither ||nor (Although the slopes are reciprocals of each other they are not the opposite signs. (The sign did not change)
5. a) What does opposite reciprocal mean? (2 points)
b) What types of slopes does it refer to?
Slopes that are opposite reciprocals areslopes. Opposite means the sign changes, so if one slope sign is positive the other will be negative or vice versa.
The term reciprocal means the numerator and denominator switch so the number that was on top is now on bottom and vice versa.
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. 2x + 3y = 15
-2x -2x
3y = - 2x + 15
3 3
y = - 2/3x + 5
m1 = - 2/3
|| slope: y =- 2/3x + #
slope: y = 3/2 x + #
7. 5x – 3y = -9
-5x -5x
- 3y = - 5x – 9
-3 -3
y = 5/3+3
m1 = 5/3
|| slope: y = 5/3x + #
slope: y = - 3/5 x + #
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
m = 2, (3, 7)
y = 2(x – 3) +7
y = - 3/4 x – 6 + 7
y = 2x + 1
-2x + y = -1
2x–y = 1
Graph a)y = 2x – 3 (Original equation)
Graph b)y = 2x + 1 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
9. A line that passes through the point (-2, -4) and is perpendicular to the point y = -2x – 6
m = 1/2 , (-2, - 4)
y = 1/2(x+ 2) – 4
y = 1/2 x +1 – 4
y = 1/2 x – 3
2y = x – 6
-x + 2y = - 6
x –2y = 6
Graph a)y = -2x – 6 (Original equation)
Graph b) y = 1/2 x – 3 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
10. A line that is parallel to the line y = - 1/3x + 1, and passes through the point (9, 3)
m = - 1/3 , (9, 3)
y = - 1/3(x – 9) + 3
y = - 1/3x +3 + 3
y = - 1/3x + 6
3y = - x + 18
x+3y = 18
Graph a)y = - 1/3 x + 1 (Original equation)
Graph b)y = - 1/3x + 6 (New equation)
Check to be sure equations lines on graph are parallel as confirmation.
11. A line that is perpendicular to the line y = 1/2 x + 3, and passes through the point (4, 0)
m = 2, (4, 0)
y = 2(x – 4) – 0
y = 2x – 8
-2x + y = - 8
2x – y = 8
Graph a)y = 1/2 x + 3 (Original equation)
Graph b) y = 2x – 8 (New equation)
Check to be sure equations lines on graph are perpendicular as confirmation.
Itg2 U2WS2.5 Day 1 Graphing
1. What does the solution look like on a graph for a system of equations?
A solution on a graph is the place where both lines intersect.
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
Solution: (1, 1)
3. y = 2x – 6 and y = -2x – 2
Solution: (1, - 4)
4. y = -x + 1and y = 3x + 5
Solution: (- 1, 2 )
5. y = 3(x – 1) + 4 and y = 1/2 x + 1
y = 3x – 3 + 4
y = 3x + 1
Solution: (0, 1)
6. y = 2(x + 3) –5 and y = -3x – 9
y = 2x + 6 – 5
y = 2x + 1
7. y = 2x – 1 and y = -x + 8
Solution: (3, 5)
8. y = (x – 4) + 2 and y = -3x + 2
y = x –2
Solution: ( 1, - 1)
9. y = 1/2 x –3 and y = 3/2 x – 1
Solution: (-2, - 4)
Itg2 U2WS2.6 Day 2 Graphing
1. What does the solution look like on a graph for a system of equations with no solutions?
If linear equations on a graph do notintersect, then there is no solution. This situation only happens when the two equations are parallel.
2. What does the solution look like on a graph for a system of equations with infinite solutions?
If linear equations on a graph are in fact the same equation, then there are an infinite number of solutions. The solution to this system of equations is in fact the equation of the line.
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
– x – x
- 2y = - x + 6
- 2 - 2
y = 1/2 x –3
Infinite Solutions since the lines are the same line
Solution: y = 1/2 x – 3
4. y = 2x – 6 and -2x + y = 3
+ 2x + 2x
y = 2x + 3
No Solution: Lines have the same slope, but have different y-intercepts thus they are parallel
5. y = -x + 1 and y = 1/2 x + 4
Solution: (- 2, 3)
6. y = 2(x – 4) + 3 and -2x + y = 1
y = 2x – 8 + 3 y = 2x + 1
y = 2x – 5
No Solution: Lines have the same slope, but have different y-intercepts thus they are parallel
7. y = 3x + 3 and 6x – 2y = -6
- 2y = - 6x – 6
y = 3x – 3
No Solution: Lines have the same slope, but have different y-intercepts thus they are parallel
8. y = -x + 6 and y = 1/2 x + 3
9. y = (x – 1) – 5 and y = x – 2
y = x – 6
No Solution: Lines have the same slope, but have different y-intercepts thus they are parallel
10. 4x + 3y = 12 and 2x – 3y = 6
3y = - 4x + 12 - 3y = - 2x + 6
3 3 - 3 - 3
y = - 4/3 x + 4 y = 2/3 x– 2
Solution: (3, 0)
Itg2 U2WS2.7 Day 1 Substitution
1. What will indicate to you that you should be using substitution to solve a system of equations?
Whenever one system of the equations has x or y by itself equal to something else then substitution is easy to use.
Solve using substitution.
2. y = 2x – 6 and 3x + 2y = 9
3x + 2(2x – 6) = 9
3x + 4x – 12 = 9
7x – 12 = 9
+ 12 + 12
7x = 21
7 7
x = 3
y = 2x – 6
y = 2(3) – 6
y = 6 – 6
y = 0
(3, 0)
3. 3x + 4y = 8and y = 2x + 13
3x + 4(2x + 13) = 8
3x + 8x + 52 = 8
11x + 52 = 8
– 52 – 52
11x = - 44
11 11
x = - 4
y = 2x + 13
y = 2(- 4) + 13
y = - 8 + 13
y = 5
(- 4, 5)
4. 3y – 2x = 11and y = 9 – 2x
3(9 – 2x) – 2x = 11
27 – 6x – 2x = 11
27 – 8x = 11
– 27 – 27
- 8x = - 16
- 8 - 8
x = 2
y = 9 – 2x
y = 9 – 2(2)
y = 9 – 4
y = 5
(2, 5)
5. x = 3y – 2 and -x + 4y = 4
- (3y – 2) + 4y= 4
- 3y + 2+ 4y= 4
y + 2 = 4
– 2 – 2
y = 2
x = 3(2) – 2
y = 6 – 2
y = 4
(4, 2)
6. x + y = -9 and x = 2y
(2y) + y = - 9
3y= - 9
3 3
y = - 3
x = 2(- 3)
y = - 6
(- 6, - 3)
7. y = 2x – 8 and x = - 3y – 3
y = 2(- 3y – 3) – 8
y = - 6y – 6 – 8
y = - 6y – 14
+ 6y + 6y
7y = - 14
7 7
y = - 2
x = - 3(- 2) – 3
y = 6 – 3
y = 3
(3, - 2)
8. x = - 3y + 16 and y = - 2x + 7
x = - 3(- 2x + 7)+ 16
x = 6x – 21+ 16
x = 6x – 5
– 6x – 6x
-5x = - 5
-5 -5
x = 1
y = - 2(1) + 7
y = - 2 + 7
y = 5
(1, 5)
9. y = - 2x – 14 and y = x + 10
(x + 10) = -2x – 14
+ 2x-10 + 2x -10
3x = - 24
3 3
x = - 8
y = (- 8) + 10
y = 2
(- 8, 2)
Itg2 U2WS2.8 Day 2 Substitution
1. What does it mean if you get a solution of 3=3 when doing substitution? Explain.
Infinite Solution: Lines are the same
2. What does it mean if you get a solution of -2 = 5 when doing substitution? Explain.
No Solution: Lines are parallel
Solve using substitution.
3. y = x + 13 and 2x – y = - 20
2x – (x + 13) = - 20
2x – x – 13 = - 20
x – 13 = - 20
+ 13 + 13
x = -7
y = (-7) + 13
y = 6
(-7, 6)
4. x = 3y – 2 and 6y = 2x + 36
6y = 2(3y – 2) + 36
6y = 6y – 4 + 36
0 = 32
No Solution: Lines are parallel
Proof: Find slope intercept of both equations
3y – 2 = x and 6y = 2x + 36
3y = x + 2 y = 2/6 x + 6
y = 1/3 x + 2 y = 1/3 x + 6
5. y = - 2/3 x + 1and 2x + 3y = 3
2x + 3(- 2/3 x + 1) = 3
2x –2x + 3 = 3
0 = 0
Infinite Solution: Lines are the same
Proof: Find slope intercept of both equations
y = - 2/3 x + 1 2x + 3y = 3
3y = - 2x + 3
y = - 2/3 x + 1
6. x = -2y + 4 and 6y + 3x = 12
6y + 3 (-2y + 4) = 12
6y – 6y + 12 = 12
0 = 0
Infinite Solution: Lines are the same
7. y = x – 9 and x = 2y + 4
x = 2(x – 9)+ 4
x = 2x – 18 + 4
x = 2x – 14
– 2x – 2x
- x = - 14
x = 14
y = (14) – 9
y = 5
(14, 5)
8. 3x = 5 – y and y = - 3x + 1
3x = 5 – (- 3x + 1)
3x = 5 + 3x – 1
0 = 4
No Solution: Lines are parallel
9. x = - 5y – 1 and x = 3y – 33
(- 5y – 1) = 3y – 33
–3y + 1 –3y + 1
- 8y = - 32
- 8 - 8
y = 4
x = - 5(4) – 1
x = -20 – 1
x = -21
(-21, 4)
10. y = - x + 5 and y = 2x – 34
(- x+ 5) = 2x – 34
–2x– 5 –2x – 5
- 3x = - 39
- 3 - 3
x = 13
y = - (13) + 5
y = - 8
(13, - 8)
Itg2 U2WS2.9 Day 1 Elimination
Solve by elimination
1. Describe a problem that would be easiest to solve by elimination. (What form are these problems in?)
When the linear equation is in Standard form then elimination is usually best to solve the system.
2. 3y – 2x = 11 and y + 2x = 9
3y – 2x = 11
y + 2x = 9 (x’s already cancel)
4y = 20
4 4
y = 5
5 + 2x = 9
– 5 – 5
2x = 4
2 2
x = 2 (2, 5)
3. 2x + 3y = -1 and 5x – 3y = 29
2x + 3y = -1
5x – 3y = 29 (y’s already cancel)
7x = 28
7 7
x = 4
2(4) + 3y = -1
8 + 3y = -1
– 8 – 8
3y = -9
3 3
y = -3 (4, -3)
4. 4x + 5y = 33 and - 4x – 3y = -23
4x + 5y = 33
- 4x – 3y = -23 (x’s already cancel)
2y = 10
2 2
y = 5
4x + 5(5) = 33
4x + 25 = 33
– 25 – 25
4x = 8
4 4
x = 2 (2, 5)
5. 5x – 3y = 6 and 2x + 3y = 15
5x – 3y = 6
2x + 3y = 15 (y’s already cancel)
7x = 21
7 7
x = 3
2(3) + 3y = 15
6 + 3y = 15
– 6 – 6
3y = 9
3 3
y = 3 (3, 3)
6. 4x – 3y = - 1 and 3x + y = 9
3(3x + y = 9) (common denominator for y’s = 3)
4x – 3y = - 1
9x +3y = 27
13x = 26
13 13
x = 2
3(2) + y = 9
6 + y = 9
– 6 – 6
y = 3 (2,3)
7. 2x + 5y = 4 and 3x – 10y = -29
2(2x + 5y = 4) (common denominator for y’s = 10)
4x + 10y = 8
3x – 10y = -29
7x = - 21
7 7
x = -3
2(-3) + 5y = 4
- 6 + 5y = 4
+ 6 + 6
5y = 10
5 5
y = 2 (-3, 2)
8. 3x + 4y = 8 and 4x – 2y = -26
2(4x – 2y = -26) (common denominator for y’s = 4)
3x + 4y = 8
8x – 4y = -52
11x = - 44
11 11
x = -4
3(-4) + 4y = 8
-12 + 4y = 8
+ 12 + 12
4y = 20
4 4
y = 5 (-4, 5)
9.2x + 5y = 11 and 3x – 4y = 5
3(2x + 5y = 11)
-2(3x – 4y = 5) (common denominator for x’s = 6)
6x + 15y = 33
- 6x + 8y = - 10
23y = 23
23 23
y = 1
2x + 5(1) = 11
2x + 5 = 11
– 5 – 5
2x = 6
2 2
x = 3 (1, 3)
Itg2 U2WS2.10 Day 2 Elimination
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.
Infinite Solution: Lines are the same
2. When you get a solution of -2 = 2 when solving by elimination, what does the graph of the equations look like? Explain.
No Solution: Lines are parallel
Solve by using substitution.
3. x + y = 4 and 2x + 2y = 4
-2(x + y = 4) (So that x’s will cancel)
2x + 2y = 4
- 2x – 2y = - 8
2x + 2y = 4
0 = - 4
No Solution: Lines are parallel
4. 2x + 14y = 10 and x + 7y = 5
2x + 14y = 10
-2(x + 7y = 5)
0 = - 4
No Solution: Lines are parallel
5. 6x – 5y = 31 and 3x + 2y = 20
6x – 5y = 31
-2(3x + 2y = 20) (common denominator for x’s)
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 (1, 6)
6. 2x – 3y = - 2 and 4x + y = 24
2x – 3y = - 2
3(4x + y = 24) (So that y’s will cancel)
2x – 3y = - 2
12x + 3y = 72
14x = 70
14 14
x = 5
4(5) + y = 24
20 + y = 24
- 20 - 20
y = 4 (5, 4)
7. 2x + y = 8 and 4x + 2y = 12
-2(2x + y = 8) (So that y’s will cancel)
4x + 2y = 12
- 4x – 2y = - 16
4x + 2y = 12
0 = - 4
No Solution: Lines are parallel
8. 2x + 5y = 11 and 3x – 4y = 5
4(2x + 5y = 11) (common denominator for y’s = 20)
5(3x – 4y = 5)
8x + 20y = 44
15x – 20y = 25
23x = 69
23 23
x = 3
2(3) + 5y = 11
6 + 5y = 11
- 6 - 6
5y = 5
5 5
y = 1 (3, 1)
9. 3x + 2y = 12 and 5x – 3y = 1
3(3x + 2y = 12) (common denominator for y’s = 6)
2(5x – 3y = 1)
9x + 6y = 36
10x –6y = 2
19x = 38
19 19
x = 2
3(2) + 2y = 12
6 + 2y = 12
- 6 - 6
2y = 6
2 2
y = 3 (2, 3)
10. 6x – 3y = 12 and 2x – y = 4
6x – 3y = 12 (common denominator for y’s = 3)
-3(2x – y = 4)
6x – 3y = 12
- 6x + 3y = - 12
0 = 0
Infinite Solution: Lines are the same
Itg2 U2 WS2.11
Integrated Math 2 – all 3 methods
1. What does the solution of a system of equation look like on a graph? As an answer?
A solution on a graph is the place where both lines intersect.
Solve each system of equations below by graphing.
2. y = 1/2x – 3 and y = 1/2(x – 8) + 1
y = 1/2x – 4 + 1
y = 1/2x – 3
Infinite Solution: Lines are the same
3. y = 3x - 3 and y = 3x + 2
No Solution: Lines are parallel
Solve each system of equations below by using substitution.
4. y = 4x – 2 and 6x – 2y = -2
6x – 2(4x – 2)= -2
6x – 8x +4 = -2
- 2x +4 = -2
- 4 - 4
- 2x = - 6
- 2 - 2
x = 3
y = 4(3) – 2
y = 12 – 2
y = 10 (3, 10)
5. 5x – y = 8 and x = 10 – 4y
5 (10 – 4y) – y = 8
50 – 20y – y = 8
50 – 21y = 8
- 50 - 50
- 21y = - 42
- 21 - 21
y = 2
x = 10 – 4(2)
x = 10 – 8
x = 2 (2, 2)
Solve by using elimination.
6. 2x – 8y = -18 and x + 8y = 15
2x – 8y = -18 (common denominator for y’s = 8)
x + 8y = 15
3x = - 3
3 3
x = - 1
(- 1) + 8y = 15
+ 1 + 1
8y = 16
8 8
y = 2 (- 1, 2)
7. 2x + 3y = 2 and - 5x + 2y = 33
5(2x + 3y = 2) (common denominator for x’s = 10)
2(- 5x + 2y = 33)
10x + 15y = 10
- 10x + 4y = 66
19x = 76
19 19
x = 4
2(4) + 3y = 2
8 + 3y = 2
- 8 - 8
3y = - 6
3 3
y = - 2 (4, - 2)
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.
The two things I know about parallel lines are:
a) They have the same slope with different y-int.
b) The two lines will never intersect.
c) y = 2x +1 and y = 2x – 5 Lines are parallel
2. What two things do you know about perpendicular lines? Create your own pair of parallel lines and graph them.
The two things I know about perpendicular lines are:
a) The slopes are opposite reciprocals of each other.
b) The two lines will intersect at a right angle (90 0).
c) y = x + 0 and y = - x + 2 Lines are parallel