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NCERT New Syllabus Mathematics
Class: 10
Exercise 3.2
Topic: Pair of Linear Equations in Two Variables
Exploring the Concept of Pair of Linear Equations in Two Variables: NCERT Class 10
Linear equations are everywhere in our daily lives, from calculating expenses to solving practical problems. In this chapter, we dive deeper into understanding how pairs of linear equations work in two variables. Whether finding the meeting point of two straight lines or determining the solution to complex problems, this topic opens doors to various real-world applications.
This chapter explores how two linear equations with two variables can be represented graphically and algebraically and how to find their solutions using different methods. We will explore everything from graphical representations to algebraic solutions such as substitution, elimination, and cross-multiplication.
By the end of this topic, you will understand how to interpret, solve, and apply linear equations in two variables effectively!
EXERCISE 3.2
1. Solve the following pair of linear equations by the substitution method.
(i) x + y = 14, x – y = 4, (ii) s – t = 3, (s/3) + (t/2) = 6, (iii) 3x – y = 3, 9x – 3y = 9
(iv) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3, (v) √2 x + √3 y = 0, √3 x – √8 y = 0
(vi) (3x/2) – (5y/3) = – 2, (x/3) + (y/2) = 13/6
Explanation:
1) Start with a simple linear equation.
2) Solve for one variable (say x or y) using straightforward steps.
3) Substitute the value of the solved variable into the other equation to find the
value of the remaining variable.
Solution:
(i) x + y = 14, x – y = 4
1) x + y = 14 ------------ equation 1
2) x – y = 4 ------------ equation 2
3) Simplify equation 2, we get
x – y = 4
4) Substitute the value of y from equation 3 in equation 1, we gety = x – 4 ------------ equation 3
x + y = 14
x + (x – 4) = 14
2x – 4 = 14
2x = 14 + 4
2x = 18x = 9 ------------ equation 4
5) Put the value of x from equation 4 in equation 3, we get
y = x – 4y = 9 – 4y = 5
6) The value of x = 9 and the value of y = 5.
(ii) s – t = 3, (s/3) + (t/2) = 6
1) s – t = 3 ------------ equation 1
2) (s/3) + (t/2) = 6 ------------ equation 2
3) Simplify equation 1, we get
s – t = 3
s = t + 3 ------------ equation 34) Substitute the value of s from equation 3 in equation 2, we get
(s/3) + (t/2) = 6
((t + 3)/3) + (t/2) = 6
[(2t + 6) + (3t)]/6 = 6
(5t + 6)/6 = 6
(5t + 6) = 36
5t = 305t = 36 – 6
t = 6 ------------ equation 4
5) Put the value of t from equation 4 in equation 3, we get
s = t + 3s = 6 + 3s = 9
6) The value of s = 9 and the value of t = 6.
(iii) 3x – y = 3, 9x – 3y = 9
1) 3x – y = 3 ------------ equation 1
2) 9x – 3y = 9 ------------ equation 2
3) Simplify equation 1, we get
3x – y = 3
y = 3x – 3 ------------ equation 34) Substitute the value of y from equation 3 in equation 2, we get
9x – 3y = 9
9x – 3(3x – 3) = 9
9x – 9x – 9 = 9
– 9 = 9
5) This shows that the lines are coincident with infinitely many solutions.
(iv) 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3
1) 0.2x + 0.3y = 1.3 ------------ equation 1
2) 0.4x + 0.5y = 2.3 ------------ equation 2
3) Multiplying by 10 to equation 1 and equation 2, we get,
2x + 3y = 13 ------------ equation 3
4x + 5y = 23 ------------ equation 4
4) Simplify equation 3, we get
2x + 3y = 13
2x = 13 – 3y
x = (13 – 3y)/2 ------------ equation 5
5) Substitute the value of x from equation 5 in equation 4, we get
4x + 5y = 23
4[(13 – 3y)/2] + 5y = 23
2(13 – 3y) + 5y = 23
26 – 6y + 5y = 23
26 – y = 23
y = 26 – 23
y = 3 ------------ equation 6
5) Put the value of y from equation 6 in equation 5, we get
x = (13 – 3y)/2x = (13 – 3(3))/2x = (13 – 9)/2
x = 4/2
x = 2
6) The value of x = 2 and the value of y = 3.
(v) √2 x + √3 y = 0, √3 x – √8 y = 0
1) √2x + √3y = 0 ------------ equation 1
2) √3x – √8y = 0 ------------ equation 2
3) Simplify equation 2, we get
√3x – √8y = 0
√3x = √8y
x = √(8/3)y ------------ equation 3
4) Substitute the value of x from equation 3 in equation 1, we get
√2x + √3y = 0
√2(8/3)y + √3y = 0
[√2(8/3) + √3]y = 0
[√16/3) + √3]y = 0
y = 0 ------------ equation 4
5) Put the value of y from equation 4 in equation 3, we get
x = √(8/3)yx = √(8/3)x0x = 0
6) The value of x = 0 and the value of y = 0.
(vi) (3x/2) – (5y/3) = – 2, (x/3) + (y/2) = 13/6
1) (3x/2) – (5y/3) = – 2 ------------ equation 1
2) (x/3) + (y/2) = 13/6 ------------ equation 2
3) Simplify equation 1 and equation 2, and we get,
9x – 10y = – 12 ------------ equation 3
2x + 3y = 13 ------------ equation 4
4) Simplify equation 3, we get
9x – 10y = – 12
9x = 10y – 12
x = (10y – 12)/9 ------------ equation 55) Substitute the value of x from equation 5 in equation 4, we get
2x + 3y = 13
2[(10y – 12)/9] + 3y = 13
(20y – 24)/9 + 3y = 13
(20y – 24 + 27y)/9 = 13
(47y – 24)/9 = 13
47y – 24 = 13 x 9
47y = 24 + 117
47y = 141y = 141/47
y = 3 ------------ equation 6
6) Put the value of y from equation 6 in equation 4, we get
2x + 3y = 132x + (3 x 3) = 132x = 13 – 9 = 4
x = 4/2
x = 2
7) The value of x = 2 and the value of y = 3.
Q2. Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which y = mx + 3.
1) 2x + 3y = 11 ------------ equation 1
2) 2x – 4y = – 24 ------------ equation 2
3) Simplify equation 2, we get
2x – 4y = – 24
2x = 4y – 24
x = (4y – 24)/2
x = 2y – 12 ------------ equation 34) Substitute the value of x from equation 3 in equation 1, and we get
2x + 3y = 11
2(2y – 12) + 3y = 11
4y – 24 + 3y = 11
7y – 24 = 11
7y = 24 + 117y = 35
y = 35/7y = 5 ------------ equation 4
5) Put the value of y = 5, from equation 4 in equation 3, and we get
x = 2y – 12x = (2 x 5) – 12
x = 10 – 12
x = – 2
6) The value of x = – 2 and the value of y = 5.
7) Put x = – 2 and y = 5 in the equation y = mx + 3, we get
y = mx + 3
5 = m(– 2) + 3
– 2m = 5 – 3
– 2m = 2
m = 2/(– 2)
m = – 1
8) Therefore, x = – 2, y = 5, and m = – 1.
Q3. Form the pair of linear equations for the following problems and find their
solution by substitution method.
(i) The difference between two numbers is 26 and one number is three times
the other. Find them.
(ii) The larger of two supplementary angles exceeds the smaller by 18
degrees. Find them.
(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she
buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.
(iv) The taxi charges in a city consist of a fixed charge together with the
charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?
(v) A fraction becomes 9/11 if 2 is added to both the numerator and the
denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
(vi) Five years hence, the age of Jacob will be three times that of his son. Five
years ago, Jacob’s age was seven times that of his son. What are their present ages?
Explanation:
1) Let x and y represent the two variables.
2) Based on the given conditions, formulate the corresponding equations.
3) Solve the two equations obtained from the conditions to determine the values of
x and y.
Solution:
(i) The difference between two numbers is 26 and one number is three times
the other. Find them.
1) Let the greater number be x, and the smaller number be y.
2) According to the first condition,
x – y = 26 -------------------------- equation 1.
3) According to the second condition,
x = 3y -------------------------- equation 2.
4) Put the value of x from equation 2 in equation 1, we get,
x – y = 26
3y – y = 26
2y = 26
y = 26/2y = 13 -------------------------- equation 3.
5) Put the value of y from equation 3 in equation 2, we get,
x = 3y
x = 3 x 13
x = 39 -------------------------- equation 4.
6) The numbers are 13 and 39.
7) The equations are x – y = 26, and x = 3y,
where x and y are two numbers (x > y); x = 39, y = 13.
(ii) The larger of two supplementary angles exceeds the smaller by 18
degrees. Find them.
1) Let the greater angle be x, and the smaller angle be y.
2) According to the first condition,
x = y + 18 -------------------------- equation 1.
3) We know that the sum of the supplementary angles is 180 degrees,
x + y = 180 -------------------------- equation 2.
4) Put the value of x = y + 18 from equation 1 in equation 2, we get,
x + y = 180
(y + 18) + y = 180
2y + 18 = 180
2y = 180 – 18
2y = 162
y = 810 -------------------------- equation 3.
5) Put the value of y = 81 from equation 3 in equation 1, we get,
x = y + 18
x = 81 + 18
x = 990 -------------------------- equation 4.
6) The angles are 810 and 990.
7) The equations are x = y + 18, x + y = 180, where x and y are the measures of
the two angles in degrees; x = 990, y = 810.
(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she
buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.
1) Let the cost of the bat be Rs x and the cost of the ball be Rs y.
2) As 7 bats and 6 balls cost Rs 3800, we have,
7x + 6y = 3800
7x = 3800 – 6y
x = (3800 – 6y)/7 ------------ equation 1
3) As 3 bats and 5 balls cost Rs 1750, we have,
3x + 5y = 1750 ------------ equation 2
4) Put the value of x = (3800 – 6y)/7 from equation 1 in equation 2, we get,
3x + 5y = 1750
3(3800 – 6y)/7 + 5y = 1750
[(11400 – 18y) + 35y]/7 = 1750
[(11400 – 18y) + 35y] = 1750 x 7
11400 + 17y = 1750 x 7
11400 + 17y = 12250
17y = 12250 – 11400
17y = 850
y = 850/17
y = 50 -------------------------- equation 3.
5) Put the value of y = 50 from equation 3 in equation 2, we get,
3x + 5y = 1750
3x + 5(50) = 1750
3x + 250 = 1750
3x = 1750 – 250
3x = 1500
x = 1500/3
x = 500 -------------------------- equation 4.
6) The equations are 7x + 6y = 3800, and 3x + 5y = 1750, where x and y are the
costs (in Rs) of one bat and one ball respectively; and the cost of the bat is Rs 500 and the cost of the ball is Rs 50.
(iv) The taxi charges in a city consist of a fixed charge together with the
charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?
1) Let the fixed charges be Rs x and the charges per km be Rs y.
2) According to the first condition, we have,
x + 10y = 105
x = 105 – 10y ------------ equation 1
3) According to the second condition, we have,
x + 15y = 155 ------------ equation 2
4) Put the value of x = 105 – 10y from equation 1 in equation 2, we get,
x + 15y = 155
(105 – 10y) + 15y = 155
105 + 5y = 155
5y = 155 – 105
5y = 50
y = 10 -------------------------- equation 3.
5) Put the value of y = 10 from equation 3 in equation 1, we get,
x = 105 – 10y
x = 105 – 10(10)
x = 105 – 100
x = 5 -------------------------- equation 4.
6) The charges for 25 km travel will be x + 25y.
7) So the total charges for 25 km travel will be,
x + 25y = 5 + 25(10)
= 5 + 250= 255
8) The equations are x + 10y = 105 and x + 15y = 155, where x is the fixed charges (in Rs), y is the charge (in Rs per km), and the fixed charges are x = Rs 5, the charge per kilometer is y = Rs 10, and the total travel cost for 25 km will be Rs 255.
(v) A fraction becomes 9/11 if 2 is added to both the numerator and the
denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6. Find the fraction.
1) Let the numerator be x and the denominator be y.
2) According to the first condition, we have,
(x + 2)/(y + 2) = 9/11
11x + 22 = 9y +18
9y = 11x + 22 – 18
9y = 11x + 4
y = (11x + 4)/9 ------------ equation 1
3) According to the second condition, we have,
(x + 3)/(y + 3) = 5/6
6x + 18 = 5y +15
6x – 5y = 15 – 18
6x – 5y = – 3 ------------ equation 2
4) Put the value of y = (11x + 4)/9 from equation 1 in equation 2, we get,
6x – 5y = – 3
6x – 5(11x + 4)/9 = – 3
[54x – 5(11x + 4)]/9 = – 3
[54x – 5(11x + 4)] = – 3 (9)
[54x – 55x – 20] = – 27
54x – 55x = – 27 + 20
– x = – 7
x = 7 -------------------------- equation 3.
5) Put the value of x = 7 from equation 3 in equation 1, we get,
y = (11x + 4)/9
y = (11(7) + 4)/9
y = (77 + 4)/9
y = 81/9
y = 9 -------------------------- equation 4.
6) The equations are 9y = 11x + 4, and 6x – 5y = – 3, where x and y are the
numerator and denominator of the fraction; the numerator is x = 7 and the denominator is y = 9. So the fraction is 7/9.
(vi) Five years hence, the age of Jacob will be three times that of his son.
Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
1) Let Jacob's present age be x and his son's present age be y.2) 5 years hence their ages will be (x + 5) and (y + 5).3) According to the first relation given in the problem,(x + 5) = 3(y + 5)
(x + 5) = 3y + 15
x = 3y + 15 – 5x = 3y + 10 ---------------------- equation 1
4) 5 years ago, their ages were (x – 5) and (y – 5). 5) According to the relation given in the problem,(x – 5) = 7(y – 5)
(x – 5) = 7y – 35
x = 7y – 35 + 5x = 7y – 30 ---------------------- equation 2
6) Put the value of x = 3y + 10 from equation 1 in equation 2, and we get,x = 7y – 30
3y + 10 = 7y – 30
3y – 7y = – 30 – 10
– 4y = – 40
4y = 40
y = 10 -------------------------- equation 3.
7) Put the value of y = 10 from equation 3 in equation 1, we get,x = 3y + 10
x = 3(10) + 10
x = 30 + 10
x = 40 -------------------------- equation 4.
8) The equations are x = 3y + 10 and x = 7y – 30, where x
and y are the ages inyears of Jacob and his son. So Jacob's present age
is x = 40 years and his son's present age is y = 10 years.
1) Let Jacob's present age be x and his son's present age be y.
2) 5 years hence their ages will be (x + 5) and (y + 5).
3) According to the first relation given in the problem,
(x + 5) = 3(y + 5)
(x + 5) = 3y + 15
x = 3y + 15 – 5x = 3y + 10 ---------------------- equation 1
4) 5 years ago, their ages were (x – 5) and (y – 5).
5) According to the relation given in the problem,
(x – 5) = 7(y – 5)
(x – 5) = 7y – 35
x = 7y – 35 + 5x = 7y – 30 ---------------------- equation 2
6) Put the value of x = 3y + 10 from equation 1 in equation 2, and we get,
x = 7y – 30
3y + 10 = 7y – 30
3y – 7y = – 30 – 10
– 4y = – 40
4y = 40
y = 10 -------------------------- equation 3.
7) Put the value of y = 10 from equation 3 in equation 1, we get,
x = 3y + 10
x = 3(10) + 10
x = 30 + 10
x = 40 -------------------------- equation 4.
8) The equations are x = 3y + 10 and x = 7y – 30, where x
and y are the ages in
years of Jacob and his son. So Jacob's present age
is x = 40 years and his son's present age is y = 10 years.
Conclusion: Mastering Pair of Linear Equations in Two Variables
As we reach the end of this fascinating topic on Pair of Linear Equations in Two Variables, it’s clear how these equations form the foundation for solving real-world problems involving relationships between two variables. By learning how to graph, calculate, and interpret their solutions, you’re not just honing your mathematical skills but also building your problem-solving mindset. Remember, mastering this concept will open doors to more advanced mathematical techniques in the future. So keep practicing, explore different methods like substitution, elimination, and graphical representation, and enjoy the journey of learning!
Stay curious and keep solving!
As we reach the end of this fascinating topic on Pair of Linear Equations in Two Variables, it’s clear how these equations form the foundation for solving real-world problems involving relationships between two variables. By learning how to graph, calculate, and interpret their solutions, you’re not just honing your mathematical skills but also building your problem-solving mindset. Remember, mastering this concept will open doors to more advanced mathematical techniques in the future. So keep practicing, explore different methods like substitution, elimination, and graphical representation, and enjoy the journey of learning!
Stay curious and keep solving!
