1. Chapter 2 3x + 2.5y = 120 3x + 2.5 (30) = 120 x = 15 x = 0
Linear Equation with 2 variables
2x + 3y = 12
Example 2
3x + 2.5y = 120
3x (30) = 120
x = 15
x = 0
3 (0) y = 120
y = 48

2. Exercise 2.1 Q.24 X = Number of units of product A produced daily
y = Number of units of product B produced daily
3x + 5y = 150
3x + 5 (25) =150
x= 25/ 3 = 8.33 units
c) 3x + 5 (12) =150
x= 90/ 3 = 30 units/ day
30 x 5 = 150 / week

3. Q. 25
8x + 24y + 16 z = 120
If only one type of food is served:
x = 15 ounce
y= 5 ounce
z = 7.5 ounce
Q. 26
a) 120X X2 +250X X 4 = 60000
X1 =500
X2 =200
X3 = 240
X4 = 120

4. Q.29
X = number of buses bought
Y = number of cars bought
Z = number of miles of road repaved
150000x y z = 100,000,000
( 100) ( 200) z = 100,000,000
z = 196 miles
I f spent in only one thing :
X = 666 Buses
y = 555 cars
z = 400 miles

5. Exercise 2.2
X intercept
The point where the graph of an equation crosses the x axis.
Y = 0
Y intercept
The point where the graph of an equation crosses the y axis.
X = 0
Slope of Line
The rate of change in value of y, w.r.t change in value of x
Slope can be positive , negative , zero, undefined.
Slope = ∆ y = y2 – y1
∆ X x2 – x 1
Definition
The change in the value of Y if X increases by 1 unit

6. Q.1
3x- 4y = 24
X intercept = (8, 0)
Y intercept = (0, -6)
Q.9
8x + 5y = -20
X intercept = (2.5, 0)
Y intercept = (0, -4)

7. Q.13
15y- 90 = 0
X intercept = No X intercept
Y intercept = (0, 6)
Q. 37
Solution ??????
Q. 38
Slope = ∆ y = y2 – y1 = = 4 Ans
∆ X x2 – x

8. Exercise 2.3 General Form ax + by = k Slope Intercept Form Y = mx + k
m= Slope
K = y intercept
Q3.
4x – 3y = Slope = 4/ 3
3y = 18 – 4x y intercept = (0, -6)
Y = 18/-3 – 4x/-3
Y = 4x/3 - 6

9. Q. 25
n= t
Graph
Slope = ; y intercept = (0, 29.6)
???????
1995
t = 14
n = 46.4 million women
2000
t = 19
n = 52.4 million women

10. Q.25 C
Slope is the change in “n” w.r.t 1 unit change in t (time), here 1.2 million women would be added in labor force each year.
Y intercept tells that in 1981 there are 29.5 million women in the labor force. ( t= 0)

11. Q. 26
p= t
Graph
Slope = ; y intercept = (0, )
???????
NOTE: Slope = Slope =/= 7500t
Q. 27
C= 5/9 F – 160/9
Slope = 5/9 ; y intercept = (0, – 160/9)
???
C +160/9 = 5/9 F
9/5 C + (9/5)160/9 = F
F= 9/5 C + 32

12. Q. 28
c= 1200 – 12.5p
Slope = by placing one more officer 12.5 crimes are reduced.
c intercept = (0, 1200) if there is no officer there would be 1200 crimes
p intercept = (96, 0) if there are 96 officers there would be no crime.

13. Q. 31 3x+ 4.5y = 150
y = – 2/3 x /3
Slope = – 2/3 by producing one more unit of x, 2/3 units of y would be reduced.
y intercept = (0, 33 1/3) if there is no unit of x is produced 33 1/3 units of y can be produced
x intercept = (50, 0) if there is no unit of y is produced 50 units of x can be produced
Q. 32
2.4x+ 3.5y = 15000
y = – .685x
Slope = – by buying one more share of x, shares of y would be reduced.
y intercept = (0, ) if no share of x is purchased , shares of y can be purchased
x intercept = (6256.5, 0) if no share of y is purchased , shares of x can be purchased

14. Slope & One Point
Y= mX+ c
Point slope formula
y- y1 = m ( x- x1)
If slope is -2 , y intercept (0 , 10)
y- 10 = -2 ( x- 0)
y = -2x + 10 Ans
If slope is 3 , y intercept (2 , 8)
y- 8 = 3 ( x- 2)
y = 3x + 2 Ans

15. Parallel Lines
two lines are parallel if they have same slope
Perpendicular Lines
two lines are perpendicular if the slope of one line is the negative reciprocal of the slope of the other line
m2 = -1/m1

16. Exercise 2.4 Q2. slope is 4 , y intercept (0 , -5) y- 5 = 4( x- 0)
y = 4x + 5 Ans
Q. 9 slope is 3/2 , lies on (-5 , -8)
y + 8 = 3/2 ( x + 5)
y = 3/2x -1/2 Ans
Q.17 slope is undefined , lies on (-3 , -5)
means a vertical line,
x = -3 Ans

17. Exercise 2.4 Q21. (-4, 5) and ( -2 , -3) m= y2 – y1 x2 – x1 m = -3 – 5
-2 + 4
m = -8/ 2
m = -4
Now,
y- 5 = -4( x+4)
y-5 = -4x – 16
y = -4x - 11 Ans

18. Q31. (2, -4) and parallel to the line 3x-4y = 20 -4Y = -3X + 20 Y = 3/4X + 20/ -4 Y= 3/4X - 5 m= ¾ Now, We know that parallel lines have same slope. So, y+ 4 = 3/4( x-2) y+4 = 3/4x –3/2 y = 3/4x –11/2

19. Q35. (-8, -4) and perpendicular to the line 8x-2y = 0 -2Y = -8X + 0 Y = 8/2X + 0 Y= 4X m= 4 ; Y Intercept = (0, 0) Now, We know that perpendicular lines have negative reciprocal slope. So, y+ 4 = -1/4( x + 8) y+4 = -1/4x –2 y= -1/4x –6

20. Q37. Data (0,18000) and (1, 14500) A) m= y2 – y1 x2 – x1 m = (14500 – 18000)/ (1-0) m = so, V = -3500( t -0) V= -3500t B) Slope = , means each year the value of machine would be declined by Rs or annual dep. Is 3500 V intercept ; when machine was purchased ( t=0), the value was

21. V = -3500t
t = -v / /3500
t = -1/3500 V
in 5.14 years or 5 years the value of machine would be 0

22. Q.38 (1,84000 ), (5, 36000) Q.39 (5,41 ), (25, 77) Q.40 (6,11.32 ), (9,12.04) Q. 41