Kaliyuga Ekalavya

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 1(ii) Question: Solve the following pair of linear equations by the elimination method and the substitution method: 3x + 4y = 10 and 2x – 2y = 2 Given: 3x + 4y = 10 2x – 2y = 2 To Find: The values of x and y using elimination and substitution methods. Formula: No specific formula, but the elimination and substitution methods are used. Solution: Elimination Method: 1. Multiply the second equation by 2 to make the coefficients of y opposites: 4x - 4y = 4 2. Add the first equation and the modified second equation: (3x + 4y) + (4x - 4y) = 10 + 4 ⇒ 7x = 14 ⇒ x = 2 3. Substitute x = 2 into either original equation to solve for y. Using the second equation: 2(2) - 2y = 2 ⇒ 4 - 2y = 2 ⇒ 2y = 2 ⇒ y = 1 Substitution Method: 1. Solve the second equation for x: 2x - 2y = 2 ⇒ 2x = 2y + 2 ⇒ x = y + 1 2. Substitute this expression for x into the first equation: 3(y + 1) + 4y = 10 ⇒ 3...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 1(i) Question: Solve the following pair of linear equations by the elimination method and the substitution method: \( x + y = 5 \) and \( 2x - 3y = 4 \) Given: The two linear equations are: \( x + y = 5 \)     (1) \( 2x - 3y = 4 \)     (2) To Find: The values of \( x \) and \( y \) using: Elimination method Substitution method Formula: Elimination Method: Eliminate one variable by making the coefficients of that variable equal (or opposites) and adding or subtracting the equations. Substitution Method: Express one variable in terms of the other using one equation, then substitute this value into the second equation to solve. Solution: Step 1: Write the given equations. $$ \begin{align*} x + y &= 5 \quad \text{(1)} \\ 2x - 3y &= 4 \quad \text{(2)} \end{align*} $$ Step 2: Multiply equation (1) by 3 to make...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 1(iii) Question: Solve the following pair of linear equations by the elimination method and the substitution method: 3x – 5y – 4 = 0 and 9x = 2y + 7 Given: 3x – 5y – 4 = 0 9x = 2y + 7 To Find: Solve the given pair of linear equations using elimination and substitution methods. Formula: No specific formula, but we'll use the principles of elimination and substitution. Solution: Elimination Method Rewrite the equations as: 3x – 5y = 4 (1) 9x – 2y = 7 (2) Multiply (1) by 3: 9x – 15y = 12 (3) Subtract (2) from (3): (9x – 15y) – (9x – 2y) = 12 – 7 ⇒ -13y = 5 ⇒ y = -5 13 Substitute y = -5 13 in (1): 3x – 5( -5 13 ) = 4 ⇒ 3x + 25 13 = 4 ⇒ 3x = 4 - 25 13 = -7 13 ⇒ x = -7 39 Substitution Method From (1), 3x = 5y + 4 ⇒ x = 5y + 4 3 (3) Substitute (3) in (2): 9( 5y + 4 3 ) = 2y + 7 ⇒ 15y + 12 = 2y + 7 ⇒ 13y = -5 ⇒ y = -5 13 Substitute y = -5 13 in (3): x = 5( ...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 1(iv) Question: Solve the following pair of linear equations by the elimination method and the substitution method: x 2 + 2y 3 = -1 and x - y 3 = 3 Given: The given pair of linear equations are: x 2 + 2y 3 = -1 ---(1) x - y 3 = 3 ---(2) To Find: We need to solve the given pair of linear equations using the elimination method and the substitution method. Formula: No specific formula, we will use the elimination and substitution methods. Solution: Elimination Method: Multiply equation (2) by 2: 2x - 2y 3 = 6 ---(3) Add equations (1) and (3): x 2 + 2x + 2y 3 - 2y 3 = -1 + 6 ⇒ 5x 2 = 5 ⇒ x = 2 Substitute x = 2 in equation (2): 2 - y 3 = 3 ⇒ y = -3 Substitution Method: From equation (2), x = 3 + y 3 ---(4) Substitute (4) in equation (1): 3 + y/3 2 + 2y 3 = -1 ⇒ 9 + y 6 + 4y 6 = -1 ⇒ 9 + 5y = -6 ⇒ 5y = -15 ⇒ y = -3 Substitute y = -3 in equation (4): x ...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 2(i) Question: Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method: If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction? Given: Let the fraction be x y . Adding 1 to the numerator and subtracting 1 from the denominator gives 1. Adding 1 to the denominator gives 1 2 . To Find: The fraction x y . Formula: Elimination method for solving linear equations. Solution: From the given information, we can form two linear equations: x+1 y-1 = 1 ⇒ x + 1 = y - 1 ⇒ x - y = -2 ...(1) And, x y+1 = 1 2 ⇒ 2x = y + 1 ⇒ 2x - y = 1 ...(2) Subtracting (1) from (2): (2x - y) - (x - y) = 1 - (-2) ⇒ x = 3 Substituting x = 3 in (1): 3 - y = -2 ⇒ y =...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 2(ii) Question: Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu? Given: Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. To Find: The present ages of Nuri and Sonu. Formula: Let Nuri's present age be N and Sonu's present age be S. We will form two linear equations based on the given information and solve them using the elimination method. Solution: Five years ago: N - 5 = 3(S - 5) ⇒ N - 5 = 3S - 15 ⇒ N - 3S = -10 ---(1) Ten years later: N + 10 = 2(S + 10) ⇒ N + 10 = 2S + 20 ⇒ N - 2S = 10 ---(2) Subtracting equation (1) from equation (2): (N - 2S) - (N - 3S) = 10 - (-10) ⇒ N - 2S - N + 3S = 20 ⇒ S =...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 2(iii) Question: Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number. Given: Let the two-digit number be 10a + b, where a and b are the digits. The sum of the digits is 9, and nine times the number is twice the number obtained by reversing the order of the digits. To Find: The value of the two-digit number. Formula: Elimination method for solving linear equations. Solution: We have the following equations: a + b = 9 ---(1) 9(10a + b) = 2(10b + a) ---(2) From equation (1), we get a = 9 - b. Substituting this into equation (2): 9(10(9 - b) + b) = 2(10b + (9 - b)) ⇒ 9(90 - 10b + b) = 2(10b + 9 - b) ⇒ 9(90 - 9b) = 2(9b + 9) ⇒ 810 -...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 2(iv) Question: Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method: Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many notes of Rs 50 and Rs 100 she received. Given: Total amount withdrawn = Rs 2000 Total number of notes = 25 Notes are of Rs 50 and Rs 100 only. To Find: Number of Rs 50 notes and number of Rs 100 notes. Formula: Let x be the number of Rs 50 notes and y be the number of Rs 100 notes. Solution: Forming the equations: x + y = 25 (Total number of notes) 50x + 100y = 2000 (Total amount) Solving by elimination method: Multiply the first equation by 50: 50x + 50y = 1250 Subtract this from the second equation: (50x + 100y) - (50x + 50y) = 2000 - 1250 ⇒ 50y = 750 ⇒ y = 750 ...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 2(v) Question: Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day. Given: Saritha paid Rs 27 for a book kept for 7 days. Susy paid Rs 21 for a book kept for 5 days. Fixed charge for first 3 days and additional charge for each day thereafter. To Find: Fixed charge and charge for each extra day. Formula: Elimination method for solving linear equations. Solution: Let the fixed charge for the first three days be Rs x and the additional charge for each day thereafter be Rs y. For Saritha: x + 4y = 27 ---(1) For Susy: x...