topic 8 (writing equations of a straight lines)
TRANSCRIPT
![Page 1: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/1.jpg)
PROPERTIES OF STRAIGHT LINES
A. WRITING EQUATION OF A LINE :
1. GIVEN TWO POINTS.
2. GIVEN THE SLOPE AND A POINT B. PARALLEL AND PERPENDICULAR
LINES.
![Page 2: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/2.jpg)
Equations of lines come in several different forms. Two of those are:
Writing Equations of Lines
1. Slope-intercept form
where m is the slope and b is the y-intercept
y mx b
2. General form 0Ax By C Answers will be written in either of these two
forms only
![Page 3: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/3.jpg)
Writing Equations of Lines
A. Given a Point and a Slope
Find the equation of the line that goes through the point (4, 5) and has a slope of 2.
Solution: m = 2 x1 = 4 y1 = 5
Substitute the above given to point- slope form equation a of line.
1 1( )y y m x x
5 2( 4)y x Simplify
![Page 4: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/4.jpg)
Slope-intercept form General Form
5 2( 4)y x
2 8 5y x
2 3y x
5 2( 4)y x
5 2 8y x
2 3 0x y
2 3 0x y
FINAL ANSWER
![Page 5: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/5.jpg)
Find the equation of the line that goes through the point (-3, 2) and has a slope of -4/5.
Solution: m = -4/5 x1 = -3 y1 = 2
1 1( )y y m x x Substitute the above given to point- slope form equation a of line.
42 ( 3)
5y x
![Page 6: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/6.jpg)
Slope-intercept form General Form
42 [ ( 3)]
5y x
42 ( 3)
5y x
4 122
5 5y x
4 2
5 5y x
4 2
5 5y x
5 4 2y x
4 5 2 0x y
FINAL ANSWER
![Page 7: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/7.jpg)
Writing Equations of Lines
B. Given Two Points
Find the equation of the line that passes through the points (-2, 3) and (1, -6).
Solution: x1 = -2 y1 = 3
Substitute the above given to slope formula to find the slope.
2 1
2 1
y ym
x x
x2 = 1 y2 = -6
6 3
1 2m
9
3m
3m
![Page 8: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/8.jpg)
Slope-intercept form General Form
1 1( )y y m x x 3 3[ ( 2)]y x
3 3( 2)y x
3 3y x
3 3 0x y
FINAL ANSWER
3 3 6y x
3 3y x
![Page 9: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/9.jpg)
Definitions
PerpendicularLines
Two lines that makes a 90° angle.The slopes of perpendicular linesare negative reciprocal of each other .
Parallel Lines Lines that never meet .The slopes of parallel lines arethe same.
![Page 10: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/10.jpg)
Writing Equations of Lines C. Given a point and equation of a line parallel to it.
Solution: x1 = 1 y1 = -5
Rewrite the equation to slope- interceptform to get the slope.
4x - 2y =3.
2m
Find the equation of the line that passes through (1, -5) and is parallel to 4x – 2y =3.
- 2y =-4x +3.-4 3
y = x + .-2 -2
3 y =2x - .
2
A
![Page 11: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/11.jpg)
Slope-intercept form General Form
1 1( )y y m x x ( 5) 2( 1)y x
5 2 2y x
2 7y x
2 7 0x y
FINAL ANSWER
2 2 5y x
2 7y x
![Page 12: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/12.jpg)
Writing Equations of Lines D. Given a point and equation of a line perpendicular to it.
Solution: x1 = 1 y1 = -5
Rewrite the equation to slope- interceptform to get the slope.
4x - 2y =3.
m = -½2m
Find the equation of the line that passes through (1, -5) and is perpendicular to 4x – 2y =3.
- 2y =-4x +3.-4 3
y = x + .-2 -2
3 y =2x - .
2
B
![Page 13: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/13.jpg)
Slope-intercept form General Form
1 1( )y y m x x
1( 5) ( 1)
2y x
1 15
2 2y x
1 9
2 2y x
2 9y x
FINAL ANSWER
1 9
2 2y x
2 9 0x y
![Page 14: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/14.jpg)
LINEAR EQUATION WITH TWO VARIABLES
SOLVING SYSTEM OF EQUATION BY:
1.Graph
2.Substitution
3.Elimination
4.Cramer’s Rule
![Page 15: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/15.jpg)
For two-variable systems, there are then three possible types of solutions:
Solving Systems of Linear Equations
A. Independent system:
one solution andone intersection point
1. two distinct non-parallel lines 2. cross at exactly one point3. "independent" system 4. one solution at (x,y )point.
Properties
![Page 16: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/16.jpg)
Solving Systems of Linear Equations
B. Inconsistent system:
no solution andno intersection point. 1. two distinct parallel lines
2. never cross 3. No point of intersection 4. "inconsistent" system5. no solution.
Properties
![Page 17: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/17.jpg)
Solving Systems of Linear Equations
C. Dependent system:
infinitely many solution1. only one line. 2. same line drawn twice. 3. "intersect" at every point4. "dependent" system, 5. Infinitely many solutions.
Properties
![Page 18: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/18.jpg)
Methods of Solving Systems
of Linear Equations
![Page 19: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/19.jpg)
Solve the following system by graphing. 2x – 3y = –24x + y = 24
2x – 3y = –22x + 2 = 3y 4x + y = 24
Solve for y for each equation
y = (2/3)x + (2/3) y = –4x + 24
Systems of Linear Equations: Solving by Graphing
A.
Equation 1 Equation 2
![Page 20: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/20.jpg)
x y = (2/3)x + (2/3) y = –4x + 24
–4 –8/3 + 2/3 = –6/3 = –2 16 + 24 = 40
–1 –2/3 + 2/3 = 0 4 + 24 = 28
2 4/3 + 2/3 = 6/3 = 2 –8 + 24 = 16
5 10/3 + 2/3 = 12/3 = 4 –20 + 24 = 4
8 16/3 + 2/3 = 18/3 = 6 –32 + 24 = –8
Get the ( x, y) values for both equation to facilitate easy graphing. The table below shows it
![Page 21: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/21.jpg)
y = –4x + 24
y = (2/3)x + (2/3)
solution: (x, y) = (5, 4)
Using the table of values we can now graphand look for the intersection:
![Page 22: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/22.jpg)
Systems of Linear Equations: Solving by Substitution
B.
Solve the following system by substitution. 2x – 3y = –24x + y = 24
4x + y = 24y = –4x + 24
Solution:
substitute it for "y" in the first equation
solve the second equation for y:
solve for x
2x – 3(–4x + 24) = –2
x = 5
2x + 12x – 72 = –214x = 70
![Page 23: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/23.jpg)
plug this x-value back into either equation,
and solve for y
4x + y = 24
x = 5
4( 5 ) + y = 24
2x – 3y = –2
x = 5
2( 5 ) – 3y = –2
20 + y = 24 y = 24 - 20
y = 4
10 – 3y = –2- 3y = –2 - 10- 3y = - 12
y = 4
Equation 1 Equation 2
Then the solution is ( x, y ) = (5, 4).
![Page 24: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/24.jpg)
Systems of Linear Equations: Solving by Elimination
C.
Solve the following system using elimination. 2x + y = 93x – y = 16
2x + y = 93x – y = 165x = 25
Solution:
add down, the y's will cancel out
divide through to solve for xx = 5
![Page 25: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/25.jpg)
using either of the original equations, to
find the value of y
x = 5
2( 5 ) + y = 9
x = 5
3( 5 ) – y = 16
10 + y = 9 y = 9 - 10
y = -1
15 – y = 16- y = 16 - 15- y = 1
y = -1
Equation 1 Equation 2
Then the solution is ( x, y ) = (5, -1).
2x + y = 9 3x – y = 16
![Page 26: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/26.jpg)
D. Systems of Linear Equations: Solving by Cramer’s Rule
2x – 3y = –24x + y = 24
Solve the following system using cramer’s rule.
, yxNN
x yD D
Solution:
D - determinant of the coefficient of the variablesNx - determinant taken from D replacing the coefficient of x
and y by their corresponding constant terms leaving all other terms unchanged
Ny -
![Page 27: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/27.jpg)
2 3
4 1D
(2) ( 12) 14D
2 3
24 1xN
2 2
4 24yN
( 2) ( 72) 70xN 70
514
xNx
D
(48) ( 8) 56yN 56
414
yNy
D
FINAL ANSWER
( 5, 4 )
![Page 28: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/28.jpg)
1. (a) Explain why the simultaneous equations 8x – 4y = 20 and y =2x – 3 have no solution . What can you say about the straight lines representing these two equations?
They are parallel
2. The diagram shows the graph of 2y = x - 2. The values of a and b are respectively.
2 and -1
ANSWER THE FOLLOWING PROBLEMS
![Page 29: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/29.jpg)
3. The graphs of x - 2y - 3 = 0 and 6 + 4y - 2x = 0 are identical lines 4. Find the graph of y = -2x - 1?
5. The diagram shows the graph of y = ax + b. Find the values of a and b.
a = 2, b = 2
![Page 30: Topic 8 (Writing Equations Of A Straight Lines)](https://reader033.vdocuments.site/reader033/viewer/2022050818/55895ed4d8b42a4d718b457f/html5/thumbnails/30.jpg)
Find the solution for each system of equationusing any method:
5x – 2y = 0 4x + y = 13
1.
2. 4.
3. y = x + 3 5y + 6x = 15
5y = 6x – 32y = x – 4
1 11
3 3x y
11
2y x
Solution : ( x , y) = ( 2, 5 )
Solution : ( x , y) = ( 0, 3 )
Solution : ( x , y) = ( -2, -3 )
Solution : ( x , y) = ( -1, 4 )