02 - Question Bank - Pair Of Linear Equations - Class 10 - Maths Class 10 Notes | EduRev

Crash Course for Class 10 Maths by Let's tute

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Class 10 : 02 - Question Bank - Pair Of Linear Equations - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


 
 
 
 
 
 
 
 
PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 
1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now,  
I shall be two times as old as you will be”. Represent this situation algebraically and graphically. 
Solution: 
Step1: 
Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. 
Step2: 
Five years ago 
Maria’s age was(x – 5) years 
Her Son’s age was (y-5) years 
Step3: 
Two years later 
Maria’s age will be(x + 2) years 
Her son’s age will be (y + 2) years 
Step4: 
Given,   5 years ago Maria’s age = 2 times her son’s age then 
?
) 5 ( 2 5 ? ? ? y x
 
? 0 5 2 ? ? ? y x
 
Step5: 
Given, 2 years hence, Maria’s age = 2 times her son’s age then, 
? ) 2 ( 2 2 ? ? ? y x
 
? 0 2 2 ? ? ? y x
 
Thus, algebraic representation of the given situations is 
0 5 2 ? ? ? y x ………. (I) 
0 2 2 ? ? ? y x ………… (II) 
Step6: 
To obtain equivalent graphical representation we find two points on the line representing each equation. 
That is we find two solutions of each equation. 
Now,     2 2 ? ? y x ……………… FROM (II) 
?
2
2 ?
?
x
y
 
Therefore, 0 ? x ? 1
2
2 0
? ?
?
? y
 
2 ? x ? 0
2
2 2
?
?
? y
 
Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
Page 2


 
 
 
 
 
 
 
 
PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 
1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now,  
I shall be two times as old as you will be”. Represent this situation algebraically and graphically. 
Solution: 
Step1: 
Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. 
Step2: 
Five years ago 
Maria’s age was(x – 5) years 
Her Son’s age was (y-5) years 
Step3: 
Two years later 
Maria’s age will be(x + 2) years 
Her son’s age will be (y + 2) years 
Step4: 
Given,   5 years ago Maria’s age = 2 times her son’s age then 
?
) 5 ( 2 5 ? ? ? y x
 
? 0 5 2 ? ? ? y x
 
Step5: 
Given, 2 years hence, Maria’s age = 2 times her son’s age then, 
? ) 2 ( 2 2 ? ? ? y x
 
? 0 2 2 ? ? ? y x
 
Thus, algebraic representation of the given situations is 
0 5 2 ? ? ? y x ………. (I) 
0 2 2 ? ? ? y x ………… (II) 
Step6: 
To obtain equivalent graphical representation we find two points on the line representing each equation. 
That is we find two solutions of each equation. 
Now,     2 2 ? ? y x ……………… FROM (II) 
?
2
2 ?
?
x
y
 
Therefore, 0 ? x ? 1
2
2 0
? ?
?
? y
 
2 ? x ? 0
2
2 2
?
?
? y
 
Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
  
 
 
 
 
 
 
 
 
 
Step7: 
Plot the points A (0,-1) and B (2, 0) on a graph paper. 
Draw a line passing through the points A and B. 
Then, the line AB represents the equation 0 2 2 ? ? ? y x
 
Step8: 
Also, consider 0 5 2 ? ? ? y x
 
?
2
5 ?
?
x
y
 
Therefore, 1 ? x ? 3
2
5 1
?
?
? y
 
3 ? x ? 4
2
5 3
?
?
? y
 
Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: 
x 1 3 
y 
3 4 
 
Step9: 
Plot the points P(1,3) and Q(3,4) on the same graph paper. 
Draw a line passing through the point P and Q. 
Then the PQ represents the equation 0 5 2 ? ? ? y x
 
We observe that line do not intersect anywhere. 
 
 
x 0 2 
y 
-1 0 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
Page 3


 
 
 
 
 
 
 
 
PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 
1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now,  
I shall be two times as old as you will be”. Represent this situation algebraically and graphically. 
Solution: 
Step1: 
Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. 
Step2: 
Five years ago 
Maria’s age was(x – 5) years 
Her Son’s age was (y-5) years 
Step3: 
Two years later 
Maria’s age will be(x + 2) years 
Her son’s age will be (y + 2) years 
Step4: 
Given,   5 years ago Maria’s age = 2 times her son’s age then 
?
) 5 ( 2 5 ? ? ? y x
 
? 0 5 2 ? ? ? y x
 
Step5: 
Given, 2 years hence, Maria’s age = 2 times her son’s age then, 
? ) 2 ( 2 2 ? ? ? y x
 
? 0 2 2 ? ? ? y x
 
Thus, algebraic representation of the given situations is 
0 5 2 ? ? ? y x ………. (I) 
0 2 2 ? ? ? y x ………… (II) 
Step6: 
To obtain equivalent graphical representation we find two points on the line representing each equation. 
That is we find two solutions of each equation. 
Now,     2 2 ? ? y x ……………… FROM (II) 
?
2
2 ?
?
x
y
 
Therefore, 0 ? x ? 1
2
2 0
? ?
?
? y
 
2 ? x ? 0
2
2 2
?
?
? y
 
Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
  
 
 
 
 
 
 
 
 
 
Step7: 
Plot the points A (0,-1) and B (2, 0) on a graph paper. 
Draw a line passing through the points A and B. 
Then, the line AB represents the equation 0 2 2 ? ? ? y x
 
Step8: 
Also, consider 0 5 2 ? ? ? y x
 
?
2
5 ?
?
x
y
 
Therefore, 1 ? x ? 3
2
5 1
?
?
? y
 
3 ? x ? 4
2
5 3
?
?
? y
 
Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: 
x 1 3 
y 
3 4 
 
Step9: 
Plot the points P(1,3) and Q(3,4) on the same graph paper. 
Draw a line passing through the point P and Q. 
Then the PQ represents the equation 0 5 2 ? ? ? y x
 
We observe that line do not intersect anywhere. 
 
 
x 0 2 
y 
-1 0 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. A shopkeeper sold 4 chocolates and 5 lollipops for Rs 8 to a customer. A little lateranothercustomer 
purchased 5 chocolates and 5 lollipops for Rs 10. Represent the situation algebraically and graphically. 
Solution: 
Step 1: 
Let the price of chocolate be Rsx , and that of lollipop be Rs y . 
Given, 4 chocolates and 5 lollipops were purchased for Rs 8. 
Step2: 8 5 4 ? ? ? y x
 
Given, 5 chocolates and 5 lollipops were purchased for Rs 10. 
Step3: 10 5 5 ? ? ? y x
 
Then the algebraic representation of the given situation is                       
) .( .......... .......... 8 5 4 I y x ? ? 
 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
Page 4


 
 
 
 
 
 
 
 
PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 
1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now,  
I shall be two times as old as you will be”. Represent this situation algebraically and graphically. 
Solution: 
Step1: 
Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. 
Step2: 
Five years ago 
Maria’s age was(x – 5) years 
Her Son’s age was (y-5) years 
Step3: 
Two years later 
Maria’s age will be(x + 2) years 
Her son’s age will be (y + 2) years 
Step4: 
Given,   5 years ago Maria’s age = 2 times her son’s age then 
?
) 5 ( 2 5 ? ? ? y x
 
? 0 5 2 ? ? ? y x
 
Step5: 
Given, 2 years hence, Maria’s age = 2 times her son’s age then, 
? ) 2 ( 2 2 ? ? ? y x
 
? 0 2 2 ? ? ? y x
 
Thus, algebraic representation of the given situations is 
0 5 2 ? ? ? y x ………. (I) 
0 2 2 ? ? ? y x ………… (II) 
Step6: 
To obtain equivalent graphical representation we find two points on the line representing each equation. 
That is we find two solutions of each equation. 
Now,     2 2 ? ? y x ……………… FROM (II) 
?
2
2 ?
?
x
y
 
Therefore, 0 ? x ? 1
2
2 0
? ?
?
? y
 
2 ? x ? 0
2
2 2
?
?
? y
 
Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
  
 
 
 
 
 
 
 
 
 
Step7: 
Plot the points A (0,-1) and B (2, 0) on a graph paper. 
Draw a line passing through the points A and B. 
Then, the line AB represents the equation 0 2 2 ? ? ? y x
 
Step8: 
Also, consider 0 5 2 ? ? ? y x
 
?
2
5 ?
?
x
y
 
Therefore, 1 ? x ? 3
2
5 1
?
?
? y
 
3 ? x ? 4
2
5 3
?
?
? y
 
Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: 
x 1 3 
y 
3 4 
 
Step9: 
Plot the points P(1,3) and Q(3,4) on the same graph paper. 
Draw a line passing through the point P and Q. 
Then the PQ represents the equation 0 5 2 ? ? ? y x
 
We observe that line do not intersect anywhere. 
 
 
x 0 2 
y 
-1 0 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. A shopkeeper sold 4 chocolates and 5 lollipops for Rs 8 to a customer. A little lateranothercustomer 
purchased 5 chocolates and 5 lollipops for Rs 10. Represent the situation algebraically and graphically. 
Solution: 
Step 1: 
Let the price of chocolate be Rsx , and that of lollipop be Rs y . 
Given, 4 chocolates and 5 lollipops were purchased for Rs 8. 
Step2: 8 5 4 ? ? ? y x
 
Given, 5 chocolates and 5 lollipops were purchased for Rs 10. 
Step3: 10 5 5 ? ? ? y x
 
Then the algebraic representation of the given situation is                       
) .( .......... .......... 8 5 4 I y x ? ? 
 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 
 
 
 
 
) ( .......... .......... 10 5 5 II y x ? ?
 
Geometric/ graphical representation 
Step4: 
To obtain the equivalent graphical representation, we find two points on the line representing each equation, 
that is, we find two solution of each equation. 
Now,              
) .( .......... .......... 8 5 4 FromI y x ? ?
 
5
4 8 x
y
?
? ?
 
 
Therefore, 2 ? x 0
5
2 4 8
?
? ?
? ? y
 
2 ? ? x 2 . 3
5
16
5
)] 2 ( 4 [ 8
? ?
? ? ?
? ? y
 
Thus, the two solutions of the equation 8 5 4 ? ? y x are: 
x 2 -2 
y 
0 3.2 
Step5: 
Plot the point A (2, 0) and B (-2, 3.2) on a graph paper. 
Draw a line passing through the points A and B. 
Then, the line AB represents the equation 8 5 4 ? ? y x . 
Step6: 
Now, ) ( .... .......... 10 5 5 II From y x ? ? 
 
5
5 10 x
y
?
? ? 
Therefore, 2 ? x 0
5
2 5 10
?
? ?
? ? y 
2 ? ? x
4
5
20
5
)] 2 ( 5 [ 10
? ?
? ? ?
? ? y
 
Thus, the two solutions of the equation 10 5 5 ? ? y x are: 
x 2 -2 
y 
0 4 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
Page 5


 
 
 
 
 
 
 
 
PAIR OF LINEAR EQUATION ALGEBRAICALLY AND GRAPHICALLY 
1) Maria tells her son “Five years ago, I was two times as old as you were then. Also two years from now,  
I shall be two times as old as you will be”. Represent this situation algebraically and graphically. 
Solution: 
Step1: 
Let Maria’s and her son’s present age be ‘x’ years and ‘y’ years respectively. 
Step2: 
Five years ago 
Maria’s age was(x – 5) years 
Her Son’s age was (y-5) years 
Step3: 
Two years later 
Maria’s age will be(x + 2) years 
Her son’s age will be (y + 2) years 
Step4: 
Given,   5 years ago Maria’s age = 2 times her son’s age then 
?
) 5 ( 2 5 ? ? ? y x
 
? 0 5 2 ? ? ? y x
 
Step5: 
Given, 2 years hence, Maria’s age = 2 times her son’s age then, 
? ) 2 ( 2 2 ? ? ? y x
 
? 0 2 2 ? ? ? y x
 
Thus, algebraic representation of the given situations is 
0 5 2 ? ? ? y x ………. (I) 
0 2 2 ? ? ? y x ………… (II) 
Step6: 
To obtain equivalent graphical representation we find two points on the line representing each equation. 
That is we find two solutions of each equation. 
Now,     2 2 ? ? y x ……………… FROM (II) 
?
2
2 ?
?
x
y
 
Therefore, 0 ? x ? 1
2
2 0
? ?
?
? y
 
2 ? x ? 0
2
2 2
?
?
? y
 
Thus, the two solutions of the equation 0 2 2 ? ? ? y x are: 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
  
 
 
 
 
 
 
 
 
 
Step7: 
Plot the points A (0,-1) and B (2, 0) on a graph paper. 
Draw a line passing through the points A and B. 
Then, the line AB represents the equation 0 2 2 ? ? ? y x
 
Step8: 
Also, consider 0 5 2 ? ? ? y x
 
?
2
5 ?
?
x
y
 
Therefore, 1 ? x ? 3
2
5 1
?
?
? y
 
3 ? x ? 4
2
5 3
?
?
? y
 
Thus, the two solutions of the equation 0 5 2 ? ? ? y x are: 
x 1 3 
y 
3 4 
 
Step9: 
Plot the points P(1,3) and Q(3,4) on the same graph paper. 
Draw a line passing through the point P and Q. 
Then the PQ represents the equation 0 5 2 ? ? ? y x
 
We observe that line do not intersect anywhere. 
 
 
x 0 2 
y 
-1 0 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. A shopkeeper sold 4 chocolates and 5 lollipops for Rs 8 to a customer. A little lateranothercustomer 
purchased 5 chocolates and 5 lollipops for Rs 10. Represent the situation algebraically and graphically. 
Solution: 
Step 1: 
Let the price of chocolate be Rsx , and that of lollipop be Rs y . 
Given, 4 chocolates and 5 lollipops were purchased for Rs 8. 
Step2: 8 5 4 ? ? ? y x
 
Given, 5 chocolates and 5 lollipops were purchased for Rs 10. 
Step3: 10 5 5 ? ? ? y x
 
Then the algebraic representation of the given situation is                       
) .( .......... .......... 8 5 4 I y x ? ? 
 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 
 
 
 
 
) ( .......... .......... 10 5 5 II y x ? ?
 
Geometric/ graphical representation 
Step4: 
To obtain the equivalent graphical representation, we find two points on the line representing each equation, 
that is, we find two solution of each equation. 
Now,              
) .( .......... .......... 8 5 4 FromI y x ? ?
 
5
4 8 x
y
?
? ?
 
 
Therefore, 2 ? x 0
5
2 4 8
?
? ?
? ? y
 
2 ? ? x 2 . 3
5
16
5
)] 2 ( 4 [ 8
? ?
? ? ?
? ? y
 
Thus, the two solutions of the equation 8 5 4 ? ? y x are: 
x 2 -2 
y 
0 3.2 
Step5: 
Plot the point A (2, 0) and B (-2, 3.2) on a graph paper. 
Draw a line passing through the points A and B. 
Then, the line AB represents the equation 8 5 4 ? ? y x . 
Step6: 
Now, ) ( .... .......... 10 5 5 II From y x ? ? 
 
5
5 10 x
y
?
? ? 
Therefore, 2 ? x 0
5
2 5 10
?
? ?
? ? y 
2 ? ? x
4
5
20
5
)] 2 ( 5 [ 10
? ?
? ? ?
? ? y
 
Thus, the two solutions of the equation 10 5 5 ? ? y x are: 
x 2 -2 
y 
0 4 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 
 
 
 
Step7: 
Plot the point P (2, 0) and Q (-2, 4), on the same graph. 
Draw a line passing through the points P and Q. 
 
Then, the line PQ represents the equation 10 5 5 ? ? y x
 
From the graph we observe that, the two lines representing the situation graphically, intersect at(2,0) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
GRAPHICAL METHOD OF SOLVING LINEAR EQUATIONS 
 
   3) 9 students of Class 8 took part in a Science Quiz. If the number of boys is 5 more than 
   the number of girls, find the number of boys and girls who took part in the quiz. 
Solution: 
 
 
X axis- 1cm = 1 unit 
Y axis- 1cm = 1unit 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
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