Algebraic Representation- Applications of pair of linear equations in two variables

Q.1-Aftab tells his duaghter,"seven years ago,I was seven times as old as you were then.Also three years from now,I shall be three times as old as you will be."Represent this situation algebrically" .

Solution :Let the age of Aftab be x years.and the age of his daughter be y years.

7 years ago... Age of Aftab = x-7 years , Age of his daughter = y-7 years

According to given condition

(x-7) = 7(y-7)

=>x-7y=-42 --------(1)

3 years later....Age of Aftab = x+3 years , Age of his daughter = y+3 years

According to given condition

(x+3) = 3( y+3)

=> x+3 = 3y+9

=>x-3y=6 ---------(2)

Q.2-The coach of a cricket team buys 3 bats & 6 balls for Rs.3900.Later,she buys 1 bat & 2 more balls of the same kind for Rs.1300.Represent this situation algebrically .

Solution:

Let the cost of bats be Rs.x

and the cost of balls be Rs.y.

According to given conditions:-

3x+6y=3900 and

x+3y=1300

Q.3-The cost of 2kg of apples & 1kg of grapes on a day was found to be Rs.160.After a month, cost of 4kg of apples & 2kg of grapes is Rs.300. Represent this situation algebrically.

Solution:

Let the cost of 1 Kg of apples be Rs.x

and the cost of 1 Kg of grapes be Rs.y

According to given conditions:-

2x+y=160 and

4x+2y=300

Solving pair of linear equations by substitution method

Solve for x and y by substitution method

General Method:

Consider the general equations a1x +b1y = c1 and a2x +b2y = c2.
Take either of the two equations and express one of the variable say (x) in terms of the other i.e. y.
Now, sustitute (put) the value of x (in terms of y) obtained in the second equations.
The second equation will become a linear equation in one variable i.e. y . Solve it and find y.
Now, put the value of y in the equation obtained in step 2, to get the value of x.

Example 1

2x + 4y =1 --- (1)
3x + 5y =2 --- (2)
From (1) 2x = 1- 4y
x = (1-4y)/2 ----- (3)
Now put (3) in (1)
2 (1-4y)/2 + 5y = 2
1 - 4y+ 5y = 2
1+y = 2
y=1
Now put (y = 1) in (3)
x = -3/2
y = 1

Example 2

x + 2y = 3 ------ i
2x + 4y = 6 ------ ii
From i,
x = 3 – 2y ------ iii
Now put iii in ii,
2(3 – 2y) + 4y = 6
6 - 4y + 4y = 6
6 = 6
This is always true.
So there are infinitely many solutions.

Example 3
4x + 6y = 7----- i
8x + 12y = 9 ----- ii
From i,
4x = 7 – 6y ------ iii
x = (7 – 6y)/4
Now put iii in ii,
8((7 – 6y)/4) + 12y = 9
14 – 12y +12y = 9
14 = 9
which is a false statement.
So there is no solution.

Solving pair of linear equations-elimination method

Posted a video lesson on solving a pair of linear equations in two variables by elimination method.
Click here!!!

Problems based on pack of cards...

After learning about Pack of cards attempt the following questions:

A card is drawn from a well shuffled deck of cards. Find the probability of drawing

1. a Spade card
2. a Red card
3. a King
4. a Black King
5. neither a King nor a Queen
6. a Face card
7. a Non Face card
8. a jack of Clubs
9. a non Red card
10. a 6 or a 8
11. a Clubs card or a King
12. a Clubs card and a King
13. a Queen of Red suit
14. a Red Face card
15. a Red non Face card

Kings and Queens are removed from a deck of cards. A card is drawn at random . Find the probability of drawing the following

1. a Spade card
2. a Red card
3. a King
4. a Black King
5. neither a King nor a Queen
6. a Face card
7. a Non Face card
8. a jack of Clubs
9. a non Red card
10. a 6 or a 8
11. a Clubs card or a King
12. a Clubs card and a King
13. a Queen of Red suit
14. a Red Face card
15. a Red non Face card

Knowing-Pack/ Deck of Cards

Some students came to me and said that they are not aware of pack of cards. Here is the information on Pack of cards.

1. There are total 52 cards


2. 2 Colours - Red , Black


3. 26 Red Cards , 26 Black Cards


4. 4 Suits - Spade , Diamond , Club , Heart





5. 13 Spade cards (A,1,2,3,....Jack , Queen ,King ).





6. 13 Diamond cards(A,1,2,3,.... Jack, Queen,King).





7. 13 Club cards (A,1,2,3,....Jack, Queen,King).





8. 13 Heart cards (A,1,2,3,....Jack, Queen,King).





9. 12 Face cards - (4 Jacks , 4 Queens , 4 Kings)


10. Total Non Face cards - 40

Activity : Sum of first n odd natural numbers

Aim: To verify that " the sum of first n odd natural numbers is n<sup>2

Material Required: Squared paper, pair of scissors , coloured markers
Procedure:
1.Let n = 10. Cut a square of dimension 10x10 units from a squared paper as shown below.</sup>

^{2.To represent 1, shade top left corner square as shown below.}

^{3.To represent 1+3 , shade 3 more squares adjacent to previous shaded square as shown below (marked with Red arrows)}

^{4. To represent 1+3+5 shade 3 more squares adjacent to previous shaded square as shown below (marked with Blue arrows)}

^{5. Repeat shading squares till you represent 1+3+5+7+9+.....19 (i.e. sum of first 10 odd natural numbers.}

^{6. What do you observe?}

^{Observation table}

^{1 = 1= 12}

^{1+3=4 =22}

^1+3+5=32

^.

^.

^{1+3+5+...19 = ................}

^{Write the Result : ........................................................}

Answer the following questions:

1. What is the sum of first 100 odd natural numbers?
2. What is the sum of first p odd natural numbers?
3. Write the sum of odd natural numbers between 5 and 19.
4. Make a geometric representation for 1+3+5+7.

Rolling a dice experiment

When we roll a dice we get 1,2,3,4,5, or 6. Experiment : Roll a dice 50 times and note down your observations. Answer the following:

1. How many times did you get an even number?


2. How many times did you get an odd number?

Write you Name , Class , Section and Roll No.

Tossing a coin- Experiment

Activity: Take an unbiased coin and toss it 50 times. Note down your observations.

1. How many times out of 50 did you get a Head ?


2. How many times out of 50 did you get a Tail ?
   

Post your observations in the comment box. Do not forget to write your Name , Class , Section and Roll No.

Mathematics in daily life...a class survey

What students of grade 10 said on asking" Where you used Mathematics in the morning?" Aman Bansal : Responded to mother on telling number of sandwiches in breakfast , checked for capacity of water in water bottle. Vaibhav Kohli: Asked for pocket money, paid money to van driver. Aditya Goel: Checked the calendar for date, looked for time in the watch. Charu Chhabra: Purchased a pen from canteen in the school. Ridhima Goel: Paid for buying centre fresh from the shopkeeper nearby house, checked the time left for getting ready to school. Rikita Ghosh: Woke up 10 minutes early than usual time, Counted number of sugar spoons in a glass of milk for breakfast. Prerna Jain: Counted the number of plats in hair before putting a rubber band. Baangi Sharma: Counted the number of biscuits in breakfast, checked the balance. Kanika Sharma: Counted number of Sanskrit periods on that day. Siddharth Jain: Counted number of Honda city cars passing by. Prateek Gupta: Used Math while packing bag according to time schedule in the morning. Oas Jain: Noted how much time is left for Math period to start. Nikhar Sardana: Used Maths while paying money to driver. Isha Ahuja: Applied Maths while adding complan to milk. Raghav Oberoi: While paying school fee at the fee counter. Anshul Grover: While counting number of steps on staircase, checked money in the pocket. Shivam Chawla: Counted the number of buses passing by. Madhur Goel: Counted the money in the pocket before starting from home. Akshit Miglani: Distance from home to school, reached 5 minutes earlier at the bus stop. Vishali Damani: Counted the number of books according to time table while packing school bag. Shubham Yadav: Observed time taken while taking bath in the morning. Vaibhav Sachdeva: Counted the number of times for brushing teeth. Ashish: Used Math for solving a puzzle in the Newspaper. Shivam Bhardwaj: Went to canteen for buying a pen and a register. Paid the bill and used Math. Vinayak Bansal: Calculated number f hours of sleep. Saksham: Purchased a chewing gum and used Math for taking back balance from the shopkeeper. Ankit Aggarwal: Calculated time left to reach the school on time in the morning. Ridhi Gupta: Snoozed alarm for 10 minutes in the morning. Shivam Singhal: Calculated time taken by car to reach the school in the morning. Deepak Jindel: Time in clock Aditya: While counting sugar spoons in milk. Arush Gupta: Counted number of tea spoons of Boost in milk.

Tangram

Aim : Making Figures using tangrams.

Tangram is a Chinese puzzle which consist of 7 pieces which are cut out of a square by the following method.

Material required

Hand made sheet
Pair of scissors
Fevistick
Geometry box.

Method

1. Draw a square ABCD of side 5 inches.
2. Joint BD
3. Mark mid point of BC as E and CD as F.
4. Joint EF.
5. Mark mid point of EF as G and BD as H.
6. Join AHG.
7. Mark mid point of DH as I and HB as J.
8. Join FI and GJ.
9. Cut all 7 pieces and make the figures.


Tangram is a well known Chinees puzzle. The goal is to form various shapes from 7 pieces. This program challenges you to solve a large number of these puzzles. Puzzles range from very simple ones for small children to difficult ones for adults. Also an editor is provided that lets you create your own collections of puzzles. You can even use different sets of pieces. The program is very easy to use and help is provided

http://www.caiman.us/scripts/fw/pageN.php?nr=953&f=tangram.zip

Area of shaded region from graphs

What is the area of shaded region in each of following graphs?