9.2.2.2 to solve systems of nonlinear equations with two variables 1

9.2.2.2 to solve systems of nonlinear equations with two variables 1

1. Nonlinear
equations with two
variables and their
systems
9.2.2.1 TO DISTINGUISH BETWEEN LINEAR AND NONLINEAR EQUATIONS
WITH TWO VARIABLES

2. By the end of the lesson, students will be able to:
Define linear and nonlinear equations with two variables
in their own words.
Identify whether a given equation is linear or nonlinear
based on its structure.
Classify at least 8 out of 10 equations correctly as linear
or nonlinear in an exercise.
Explain the difference between linear and nonlinear
equations using examples.

3. 1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
•A linear equation in two variables, x and y, is an equation that can be
written in the form , where m and b are constants (just numbers).
•Likewise, a linear function is a function whose graph is a non-vertical line.
•A linear function has a constant rate of change and can be represented by a
linear equation in two variables.
•A nonlinear function does not have a constant rate of change and its graph is
not a line.

4. 1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the graph represent a linear or nonlinear function? Explain.
The function is nonlinear.
Because the graph is not
a line.
The function is linear.
Because the graph is a
line.

5. 1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
As x increases by 3, y decreases by 6.
The rate of change is constant for both
x and y.
The function is linear.
x 3 6 9 12
y 36 30 24 18
+3 +3 +3
-6 -6 -6

6. 1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
As x increases by 2, y increases by
different amounts.
The rate of change is not constant for
both x and y.
The function is nonlinear.
x 1 3 5 7
y 2 9 20 35
+2 +2 +2
+7 +11 +15

7. You try!!
1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the graph represent a linear or nonlinear function? Explain.
The function is linear.
Because the graph is a
line.
The function is nonlinear.
Because the graph is not
a line.

8. You try!!
1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
As x increases by 1, y increases by 2.
The rate of change is constant for both
x and y.
The function is linear.
x 0 1 2 3
y 3 5 7 9
+1 +1 +1
+2 +2 +2

9. Does the table represent a linear or nonlinear function? Explain.
A linear function has a constant rate of change
As x increases by 1, y decreases by
different amounts.
The rate of change is not constant for
both x and y.
The function is nonlinear.
X 1 2 3 4
y 16 8 4 2
+1 +1 +1
-8 -4 -2
You try!!
1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.

10. Which of the following equations represent linear functions? Explain.
A linear function can be written in the form , where m and b are constants (just
numbers).
You cannot rewrite the equations , , , and in the form .
So these equations cannot represent linear functions.
You can rewrite the equation as
You can rewrite the equation as
So these equations represent linear functions.
1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
, , , , , and

11. Does the equation represent a linear or nonlinear function? Explain.
A linear function can be written in the form .
You can rewrite the
equation as
Equation represents a
linear function.
You try!!
1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.

You can rewrite the
equation as
Equation represents a
linear function.

You cannot rewrite the
equation as
Equation is a nonlinear
function.

12. 1.Students will be able to determine whether functions are linear
functions by using graphs, tables and equations.
In a graph it looks like a ……
Line
In a table we look for what?
A constant rate of change in both x and y values.
In equations we need to be able to do what?
Write the equation in the form of
So let’s review!
What have we covered so far?

13. Quadratic
equation

14. Lesson Objectives – Quadratic
Equations
By the end of the lesson, students will be able to:
Define a quadratic equation and state its standard form
Identify quadratic equations from a list of algebraic equations.
Differentiate quadratic equations from linear and higher-order
equations.
Solve simple quadratic equations using factorization (if solving is part
of the lesson scope).
Apply quadratic equations to model and solve real-life problems
(extension/advanced objective).

17. STANDARD FORM OF A QUADRATIC EQUATION
The standard form of a quadratic equation is
 
ax
2 
+ bx + c = 0
where a,b and c are real numbers and
 a≠0.
‘a’ is the coefficient of
 
x
2
. It is called the quadratic coefficient. ‘b’
is the coefficient of x. It is called the linear coefficient. ‘c’ is the
constant term.

18. Roots of a Quadratic equation:
The values of x for which a quadratic equation is satisfied are called
the roots of the quadratic equation.
If α is a root of the quadratic equation ax2+bx+c=0,then, aα2+bα+c=0.
A quadratic equation can have two distinct roots, two equal roots or
real
roots may not exist.
SOLVING QUADRATIC EQUATION BY FACTORISATION

19. QUADRATIC FORMUL A

21. 1. Check whether the following are quadratic equations:
(i) (x+ 1)
2
=2(x-3)
EX: 4.1 :
Solution:
(i) Given : (x + 1)
2 
= 2(x – 3)
By using the formula for (a+b)
2
 
= a
2+2ab+b
2
 
⇒
x
2 
+ 2x + 1 = 2x – 6
 
⇒
x
2 
+ 7 = 0
Since the above equation is in the form of
ax
2 
+ bx + c = 0.
Therefore, the given equation is
 quadratic
equation.
(ii)Given, x
2 
– 2x = (–2) (3 – x)
By using the formula for (a+b)
2
 
= a
2+2ab+b
2
 
⇒
x
2
 
–
 2x = -6
 + 2x 
 
x⇒
2
 
– 4x
 + 6 = 0
Since the above equation is in the form of
ax
2 
+ bx + c = 0.
Therefore, the given equation is
 quadratic
equation.
(ii) x – 2x = (- 2) (3-x)

22. EX: 4.1 :
1. Check whether the following are quadratic equations:
(iii) (x – 2) (x + 1) = (x – 1) (x + 3) (iv) (x – 3) (2x + 1) = x (x + 5)
Given, (x – 2)(x + 1) = (x – 1)(x + 3)
By using the formula for (a+b)
2
 
= a
2+2ab+b
2
 
⇒
x
2
 
–
 x – 2 = x
2
 
+ 2x
 – 3
3x
 – 1 = 0⇒
Since the above equation is not in the form of
ax
2 
+ bx + c = 0.
Therefore, the given equation is
 not a
quadratic equation.
Given, (x – 3)(2x +1) = x(x + 5)
By using the formula for (a+b)
2=a
2+2ab+b
2

⇒
2x
2
 
– 5x
 – 3 = x
2
 
+ 5x

 x⇒
2
 
– 10x
 – 3 = 0
Since the above equation is in the form of
ax
2 
+ bx + c = 0.
Therefore, the given equation is
 quadratic equation.

23. EX: 4.1 :
1. Check whether the following are quadratic equations:
(v) (2x – 1) (x – 3) = (x + 5) (x – 1) (vi) x
2
+ 3x + 1 = (x – 2)
2
Given, (2x
 – 1)(x – 3) = (x + 5)(x – 1)
By using the formula for (a+b)
2=a
2+2ab+b
2

⇒
2x
2
 
– 7x
 + 3 = x
2
 
+ 4x
 – 5
 
x⇒
2
 
– 11x
 + 8 = 0
Since the above equation is in the form of
ax
2 
+ bx + c = 0.
Therefore, the given equation is
 quadratic
equation.
(vi)Given, x
2 
+ 3x + 1 = (x – 2)
2
By using the formula for (a+b)
2=a
2+2ab+b
2
 
⇒
x
2 
+ 3x + 1 = x
2 
+ 4 – 4x
7x
 – 3 = 0⇒
Since the above equation is not in the form of
ax
2 
+ bx + c = 0.
Therefore, the given equation is
 not a quadratic
equation.

24. (viii) x
3
-4x
2
-x + 1 = (x-2)
3
EX: 4.1 :
1. Check whether the following are quadratic equations:
(vii) (x + 2)
3
= 2x(x
2
– 1)
Given, (x
 + 2)
3 
= 2x(x
2 
– 1)
By using the formula for (a+b)
2
 
= a
2+2ab+b
2
 
⇒
x
3 
+ 8 + x
2 
+ 12x = 2x
3 
– 2x
 
x⇒
3 
+ 14x – 6x
2 
– 8 = 0
Since the above equation is not in the form
of ax
2 
+ bx + c = 0.
Therefore, the given equation is
 not a
quadratic equation.
Given, x
3 –
4x
2 – x +
1 = (x – 2)
3
By
using the formula for (a+b)
2 =
a
2+2ab+b
2
⇒
 
x
3 –
4x
2 – x +
1 = x
3 –
8 – 6x
2  +
12x
⇒
2x
2 –
13x + 9 = 0
Since
the above equation is in the form of
ax
2 + bx + c =
0.
Therefore,
the given equation is quadratic
equation.

25. 2. Represent the following situations in the form of quadratic equations:
The area of a rectangular plot is 528 m
2
. The length of the plot (in metres) is one
more than twice its breadth. We need to find the length and breadth of the plot.
EX: 4.1 :
Solution:
Let us consider,
Breadth of the rectangular plot =
 x m
Thus, the length of the plot = (2x
 + 1) m.
Area of rectangle = length
 × breadth = 528 m
2
Putting the value of length and breadth of the plot in the formula, we get,
(2x
 + 1) × x = 528

⇒
2x
2 
+ x =528

⇒
2x
2 
+ x – 528 = 0
Therefore, the length and breadth of plot, satisfies the quadratic equation, 2x
2 
+ x – 528 = 0,
which is the required representation of the problem mathematically.

26. Let us consider,
The first integer number = x
Thus, the next consecutive positive integer will be = x + 1
Product of two consecutive integers = x × (x +1) = 306  x
⇒
2 
+ x = 306
 
⇒
x
2 
+ x – 306 = 0
Therefore, the two integers x and x+1, satisfies the quadratic
equation, x
2 
+ x – 306 = 0, which is the required representation of
the problem mathematically.
(ii)The product of two consecutive positive integers is 306. We need to find
the integers.
EX: 4.1 :
2. Represent the following situations in the form of quadratic equations:

27. 3) Let us consider,
Age of Rohan’s =
 x  years
Therefore, as per the given question,
Rohan’s mother’s age =
 x + 26
After 3 years,
Age of Rohan’s =
 x + 3
Age of Rohan’s mother will be =
 x + 26 + 3 = x + 29
(iii)Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years
from now will be 360. We would like to find Rohan’s present age.
The product of their ages after 3 years will be equal to 360,
such that
(x
 + 3)(x + 29) = 360
 
⇒
x
2
 
+ 29x + 3x + 87 = 360
 
⇒
x
2
 
+ 32x + 87 – 360 = 0
 
⇒
x
2
 
+ 32x – 273 = 0
Therefore, the age of Rohan and his mother, satisfies the
quadratic equation, x
2
 
+ 32x – 273 = 0, which is the required
representation of the problem mathematically.
EX: 4.1 :
2. Represent the following situations in the form of quadratic equations:

28. Let us consider,
The uniform speed of train =
 
x 
km/h
Time taken to travel 480 km = 480/x
km/hr
As per second condition,
If the speed had beenn8km/hr less, then the
speed of train
= (x 
– 8) km/h
Time taken to travel 480 km = 480/(x-
8) km/hr
Also given, the train will take 3 hours to cover
the same distance.
ATQ
480/(x-8) =480/x +3
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h
less, then it would have taken 3 hours more to cover the same distance. We need to find
the speed of the train.
Therefore,
480 - 480 =3
(x-8) x

⇒
480(x)- 480 (x-8) = 3x(x-8)
480x - 480x + 3840 = 3(x2-8x)
3840 = 3x
2
 
- 24x
3x
2
 
- 24x = 3840
3x
2
 
- 24x -3840 = 0 (divide by 3)
x
2
 
- 8x -1280 = 0
Therefore, the speed of the train, satisfies the
quadratic equation, 3x
2
 
– 8x 
– 1280 = 0, which is the
required representation of the problem mathematically.
EX: 4.1 :
2. Represent the following situations in the form of quadratic equations:

29. EX:4.2

30. 1.Find the roots of the following
quadratic equations by
factorisation:
(i)
 x
2
 
– 3x – 10 = 0
1.(ii) 2x
2
 
+ x – 6 = 0
EX: 4.2 :
1x(-10)=(-10)
Sum = (-5 + 2) = -3
Product=(-5 x 2) = -10
-5 2
2x(-6)=(-12)
Sum = (4 - 3) = 1
Product=(4 x -3 ) = -12
4 -3

31. (iii) √2
 x
2
 
+ 7x + 5√2 = 0
(iv) 2x
2
 
– x +1/8 = 0
√2x5 √2=5x2=10
Sum = (5 + 2) = 7
Product=(5 x 2 ) = 10
5 2
16x1=(16)
Sum = (-4-4) = -8
Product=(-4 x -4 ) = 16
-4 -4
1.Find the roots of the following
quadratic equations by
factorisation:
EX: 4.2 :

32. 100x1=100
Sum = (-10-10) = -20
Product=(-10 x -10 ) = 100
-10 -10
()
(v) 100x
2
 
– 20x + 1 = 0
1.Find the roots of the following
quadratic equations by
factorisation:
EX: 4.2 :

33. 2. Solve the problems given in Example
1.Represent the following situations
mathematically:
(i).John and Jivanti together have 45 marbles.
Both of them lost 5 marbles each, and the
product of the number of marbles they
now have is 124. We would like to find out
how many marbles they had to start with.
Solution:
EX: 4.2 :
∴
(x – 5)(40 –
 
x) = 124
40x-
x
2
-200+5x=124
-x
2
+45x-200-124=0
-
x
2
+45x-324=0
 
⇒
x
2
 
– 45
x 
+ 324 = 0
 
⇒
x
2
 
– 36
x 
– 9
x 
+ 324 = 0
 
⇒
x(x 
– 36) -9(
x 
– 36) = 0
(
⇒
x 
– 36)(
x 
– 9) = 0
Thus, we can say,
x 
– 36 = 0 or 
x 
– 9 = 0
 
⇒
x 
= 36 or 
x 
= 9
Therefore,
If, John’s marbles = 36,
Then, Jivanti’s marbles = 45 – 36 = 9
And if John’s marbles = 9,
Then, Jivanti’s marbles = 45 – 9 = 36
Let us say, the number of marbles John have =
 
x.
Therefore, number of marble Jivanti have = 45 –
 
x
After losing 5 marbles each,
Number of marbles John have =
 
x 
– 5
Number of marble Jivanti have = 45 –
 
x 
– 5 = 40 –
 x
Given that the product of their marbles is 124.

34. (ii).A cottage industry produces a certain number of toys in a day. The cost of production of
each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a
particular day, the total cost of production was Rs.750. We would like to find out the number of
toys produced on that day.
EX: 4.2 :
(ii)Let us say, number of toys produced in a day be
 
x.
Therefore, cost of production of each toy = Rs(55 –
 
x)
Given, total cost of production of the toys = Rs 750
 
∴
x(55 –
 
x) = 750
 
⇒
x
2
 
– 55
x 
+ 750 = 0
 
⇒
x
2
 
– 25
x 
– 30
x 
+ 750 = 0
⇒ x(x 
– 25) -30(
x 
– 25) = 0

⇒
(x 
– 25)(
x 
– 30) = 0
Thus, either
 
x 
-25 = 0 or 
x 
– 30 = 0
 
⇒
x 
= 25 or 
x 
= 30
Hence, the number of toys produced in a day, will be either 25 or 30.

35. 3. Find two numbers whose sum is 27 and product is 182.
Solution: 
Let us say, first number be
 x and the second number is 27 – x.
Therefore, the product of two numbers
x(27 –
 x) = 182
 
x⇒
2
 
– 27x – 182 = 0
 
x⇒
2
 
– 13x – 14x + 182 = 0
 
x(x – 13) -14(x – 13) = 0⇒
(x
 – 13)(x -14) = 0⇒
Thus, either, x
 = -13 = 0 or x – 14 = 0
 
x = 13 or x = 14⇒
Therefore, if first number = 13, then second number = 27 – 13 = 14
And if first number = 14, then second number = 27 – 14 = 13
Hence, the numbers are 13 and 14.
EX: 4.2 :

36. 4. Find two consecutive positive integers, sum of whose squares is 365.
Solution:
 
Let us say, the two consecutive positive integers be
 
x 
and 
x 
+ 1.
Therefore, as per the given questions,
EX: 4.2 :

37. EX: 4.2 :
5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm,
find the other two sides.
Solution;

39. EX:4.3

40. 2. Find the roots of the quadratic equations given in Q.1 above by
applying the quadratic formula.
1) 2x
2
 
– 7
x 
+3 = 0
EX: 4.3:
2)
 2
x
2
 
+ 
x 
– 4 = 0
4)
 2
x
2
 
+ 
x 
+ 4 = 0
3)4x
2
 
+ 4√3
x 
+ 3 = 0
5)
6)
7) The sum of the reciprocals
of Rehman’s ages, (in years) 3
years ago and 5 years from
now is 1/3. Find his present
age.
8) In a class test, the sum of
Shefali’s marks in
Mathematics and English is
30. Had she got 2 marks
more in Mathematics and 3
marks less in English, the
product of their marks would
have been 210. Find her
marks in the two subjects.

41. 6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the
longer side is 30 metres more than the shorter side, find the sides of the field.
EX: 4.3:

42. 7.
 The difference of squares of two numbers is 180. The square of the
smaller number is 8 times the larger number. Find the two numbers.
EX: 4.3:

43. EX: 4.3:
8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would
have taken 1 hour less for the same journey. Find the speed of the train.
Solution

44. 9.
 Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter
takes 10 hours less than the smaller one to fill the tank separately. Find the
time in which each tap can separately fill the tank.
EX: 4.3:
Solution:
 
Let the time taken by the smaller pipe to fill the tank =
x hr.
Time taken by the larger pipe = (x 
– 10) hr
Part of tank filled by smaller pipe in 1 hour = 1/x
Part of tank filled by larger pipe in 1 hour = 1/(x – 10)
As given, the tank can be filled in 9 3/8
 = 75/8 hours
by both the pipes together.

45. 9.
 Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter
takes 10 hours less than the smaller one to fill the tank separately. Find the
time in which each tap can separately fill the tank.
EX: 4.3:

46. 11. Sum of the areas of two squares is 468 m
2
. If the difference of
their perimeters is 24 m, find the sides of the two squares.
EX: 4.3:

47. EX:4.4

48. 1. Find the nature of the roots of the following quadratic equations.
 If the real roots
exist, find them;
(i) 2x
2
 
– 3
x 
+ 5 = 0
(ii) 3x
2
 
– 4√3
x 
+ 4 = 0
(iii) 2x
2
 
– 6
x 
+ 3 = 0
Solutions:
EX: 4.4:
(i) Given: 2x
2
– 3x + 5 = 0
(ii)

Given: 3x
2 
– 4√3
x 
+ 4 = 0

49. 1. Find the nature of the roots of the following quadratic equations.
 If the real
roots exist, find them;
(iii) 2x
2
 
– 6
x 
+ 3 = 0
Solution:
EX: 4.4:

50. 2. Find the values of
 k for each of the
following quadratic equations, so
that they have two equal roots.
(i) 2x
2
 
+ kx + 3 = 0
(ii)
 kx (x – 2) + 6 = 0
Solution:
EX: 4.4:
Hence, k = 6.

51. Let breadth of the rectangular be x m
Then, the length of rectangular will be
2x m.
According to question, we have
EX: 4.4:
3. Is it possible to design a rectangular mango grove whose length is twice its
breadth, and the area is 800 m
2
?
 If so, find its length and breadth.

52. Solution:
 
Let’s say, the age of one friend be x years.
Then, the age of the other friend will be (20 – x) years.
Four years ago,
Age of First friend = (x 
– 4) years
Age of Second friend = (20 –
 
x 
– 4) = (16 –
 x) years
As per the given question, we can write,
(x 
– 4) (16 – 
x) = 48
16x
– x
2
 
– 64 + 4
x 
= 48
 –
x
2
 
+ 20
x
– 
112 = 0
x
2
 
– 20
x
+ 
112 = 0
4. Is the following situation possible? If so, determine their present ages. The sum
of the ages of two friends is 20 years. Four years ago, the product of their ages in
years was 48.
EX: 4.4:

53. Solution:
 Let the length and of the park be 
x.
EX: 4.4:
5.
 Is it possible to design a rectangular park of perimeter 80 and area 400 m2? If
so find its length and breadth.

54. 4.QUADRATIC EQUATIONS
COMPLETED