Solve the following pairs or linear equations : 7 x - 2 y x y = 5 ; 8 x + 7 y x y = 15

Pair of equations in two variables which are not linear but can be reduced to linear equations in two variables by making some suitable substitutions are called equations reducible to pair of linear equations. Given, 7 x - 2 y x y = 5 ⇒ - 2 x + 7 y = 5 . . . . . . 1 8 x + 7 y x y = 15 ⇒ 7 x + 8 y = 15 . . . . . . 2 Let 1 x = a & 1 y = b Then equation 1 becomes, - 2 a + 7 b = 5 . . . . . . 3 and equation 2 becomes 7 a + 8 b = 15 . . . . . 4 Now apply substitution method: ⇒ b = 15 - 7 a 8 . . . . 5 Substituting value of b in equation 3 we get - 2 a + 7 15 - 7 a 8 = 5 ⇒ - 16 a + 7 15 - 7 a = 40 ⇒ - 65 a = - 65 ⇒ a = 1 ∴ b = 15 - 7 8 = 8 8 = 1 But a = 1 x ⇒ x = 1 & b = 1 y ⇒ y = 1

Solve the following pairs or linear equations :- 20 x + 1 + 4 y - 1 = 5 ; 10 x + 1 - 4 y - 1 = 1

Pair of equations in two variables which are not linear but can be reduced to linear equations in two variables by making some suitable substitutions are called equations reducible to pair of linear equations. Given, 20 x + 1 + 4 y - 1 = 5 . . . . . 1 and 10 x + 1 - 4 y - 1 = 1 . . . . . . 2 Let 1 x + 1 = a & 1 y - 1 = b Then equation 1 becomes, 20 a + 4 b = 5 and equation 2 becomes 10 a - 4 b = 1 . . . . . 4 Now apply substitution method: ⇒ b = 10 a - 1 4 . . . . 5 Substituting value of b in equation 3 we get 20 a + 4 10 a - 1 4 = 5 ⇒ 2 a + 10 a - 1 = 5 ⇒ 30 a = 6 ⇒ a = 1 5 ∴ b = 10 × 1 5 - 1 4 = 1 4 But a = 1 x + 1 ⇒ x + 1 = 5 ⇒ x = 4 & b = 1 y - 1 ⇒ y - 1 = 4 ⇒ y = 5

Solve the following pairs or linear equations : 3 x + y + 2 x - y = 3 ; 2 x + y + 3 x - y = 11 3

Pair of equations in two variables which are not linear but can be reduced to linear equations in two variables by making some suitable substitutions are called equations reducible to pair of linear equations. Given, 3 x + y + 2 x - y = 3 . . . . . 1 and 2 x + y + 3 x - y = 11 3 . . . . . . 2 Let 1 x + y = a & 1 x - y = b Then equation 1 becomes, 3 a + 2 b = 3 . . . . 3 and equation 2 becomes 2 a + 3 b = 11 3 ⇒ 6 a + 9 b = 11 . . . . . 4 Now apply substitution method: ⇒ b = 11 - 6 a 9 . . . . 5 Substituting value of b in equation 3 we get 3 a + 2 11 - 6 a 9 = 3 ⇒ 27 a + 2 11 - 6 a = 27 ⇒ 27 a + 22 - 12 a = 27 ⇒ 15 a = 5 ⇒ a = 1 3 ∴ b = 11 - 6 × 1 3 9 = 1 But a = 1 x + y ⇒ x + y = 3 . . . . 6 & b = 1 x - y ⇒ x - y = 1 . . . . . 7 Adding equations 6 & 7 we get 2 x = 4 ⇒ x = 2 substituting x = 2 in equation 6 we get 2 + y = 3 ⇒ y = 1 ∴ x = 2 , y = 1

Solve the following pairs or linear equations : 1 2 2 x + 3 y + 12 7 3 x - 2 y = 1 2 ; 7 2 x + 3 y + 4 3 x - 2 y = 2

Pair of equations in two variables which are not linear but can be reduced to linear equations in two variables by making some suitable substitutions are called equations reducible to pair of linear equations. Given, 1 2 2 x + 3 y + 12 7 3 x - 2 y = 1 2 . . . . . 1 and 7 2 x + 3 y + 4 3 x - 2 y = 2 . . . . . . 2 Let 1 2 x + 3 y = a & 1 3 x - 2 y = b Then equation 1 becomes, a 2 + 12 b 7 = 1 2 ⇒ 7 a + 24 b = 7 . . . . . 3 and equation 2 becomes 7 a + 4 b = 2 . . . . . 4 Now apply substitution method: ⇒ b = 2 - 7 a 4 . . . . 5 Substituting value of b in equation 3 we get 7 a + 24 2 - 7 a 4 = 7 ⇒ 7 a + 6 2 - 7 a = 7 ⇒ 7 a + 12 - 42 a = 7 ⇒ 35 a = 5 ⇒ a = 1 7 ∴ b = 2 - 7 × 1 4 4 = 1 4 But a = 1 2 x + 3 y ⇒ 2 x + 3 y = 7 . . . . 6 & b = 1 3 x - 2 y ⇒ 3 x - 2 y = 4 . . . . . 7 Now multiplying equation 6 by 2 we get 4 x + 6 y = 14 . . . . . 8 And multiplying equation 7 by 3 we get 9 x - 6 y = 12 . . . . 9 Adding equation 8 & 9 we get 13 x = 26 ⇒ x = 2 substituting x = 2 in equation 8 we get 8 + 6 y = 14 ⇒ y = 1 ∴ x = 2 , y = 1

Solve the following pairs or linear equations : 1 2 x + 2 y + 5 3 3 x - 2 y = - 3 2 ; 5 4 x + 2 y - 3 5 3 x - 2 y = 61 60

Pair of equations in two variables which are not linear but can be reduced to linear equations in two variables by making some suitable substitutions are called equations reducible to pair of linear equations. Given, 1 2 x + 2 y + 5 3 3 x - 2 y = - 3 2 . . . . . 1 and 5 4 x + 2 y - 3 5 3 x - 2 y = 61 60 . . . . . . 2 Let 1 x + 2 y = a & 1 3 x - 2 y = b Then equation 1 becomes, a 2 + 5 b 3 = - 3 2 ⇒ 3 a + 10 b = - 9 . . . . . 3 and equation 2 becomes 5 a 4 - 3 b 5 = 61 60 75 a - 36 b = 61 . . . . . 4 Now apply substitution method: ⇒ b = 75 a - 61 36 . . . . 5 Substituting value of b in equation 3 we get 3 a + 10 75 a - 61 36 = - 9 ⇒ 108 a + 10 75 a - 61 = - 324 ⇒ 108 a + 750 a - 610 = - 324 ⇒ 858 a = 286 ⇒ a = 1 3 ∴ b = 75 × 1 3 - 61 36 = - 36 36 = - 1 But a = 1 x + 2 y ⇒ x + 2 y = 3 . . . . 6 & b = 1 3 x - 2 y ⇒ 3 x - 2 y = - 1 . . . . . 7 Adding equations 6 & 7 we get 4 x = 2 ⇒ x = 1 2 substituting x = 1 2 in equation 6 we get 1 2 + 2 y = 3 ⇒ y = 5 4 ∴ x = 1 2 , y = 5 4

E M B I B E

Understanding ICSE Mathematics Class 9>Algebra>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4

M L Aggarwal Solutions for Exercise 4: EXERCISE 5.4

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 4: EXERCISE 5.4

Attempt the practice questions from Exercise 4: EXERCISE 5.4 with hints and solutions to strengthen your understanding. Understanding ICSE Mathematics Class 9 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Exercise 4: EXERCISE 5.4 with Hints & Solutions

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 2

Solve the following pairs or linear equations :

$\frac{7 x - 2 y}{x y} = 5; \frac{8 x + 7 y}{x y} = 15$

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 2

Solve the following pairs or linear equations :

$99 x + 101 y = 499 x y; 101 x + 99 y = 501 x y$

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 3

Solve the following pairs or linear equations :

$3 x + 14 y = 5 x y$

$21 y - x = 2 x y$

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 3

Solve the following pairs or linear equations :

$3 x + 5 y = 4 x y$

$2 y - x = x y$

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 4

Solve the following pairs or linear equations :-

$\frac{20}{x + 1} + \frac{4}{y - 1} = 5; \frac{10}{x + 1} - \frac{4}{y - 1} = 1$

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 4

Solve the following pairs or linear equations :

\frac{3}{x + y} + \frac{2}{x - y} = 3; \frac{2}{x + y} + \frac{3}{x - y} = \frac{11}{3}

MEDIUM

9th ICSE

IMPORTANT

Understanding ICSE Mathematics Class 9>Chapter 5 - Simultaneous Linear Equations>EXERCISE 5.4>Q 5

Solve the following pairs or linear equations : $\frac{1}{2 (2 x + 3 y)} + \frac{12}{7 (3 x - 2 y)} = \frac{1}{2}; \frac{7}{2 x + 3 y} + \frac{4}{3 x - 2 y} = 2$