CLASS X : CHAPTER - 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

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CLASS X : CHAPTER - 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

IMPORTANT FORMULAS & CONCEPTS

An equation of the form \( ax + by + c = 0 \), where \( a, b \) and \( c \) are real numbers (\( a \neq 0, b \neq 0 \)), is called a linear equation in two variables \( x \) and \( y \).

The numbers \( a \) and \( b \) are called the coefficients of the equation \( ax + by + c = 0 \) and the number \( c \) is called the constant of the equation \( ax + by + c = 0 \).

Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is:

\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)

where \( a_1, a_2, b_1, b_2, c_1, c_2 \) are real numbers, such that \( a_1^2 + b_1^2 \neq 0, a_2^2 + b_2^2 \neq 0 \).

CONSISTENT SYSTEM
A system of simultaneous linear equations is said to be consistent, if it has at least one solution.

INCONSISTENT SYSTEM
A system of simultaneous linear equations is said to be inconsistent, if it has no solution.

METHOD TO SOLVE A PAIR OF LINEAR EQUATION OF TWO VARIABLES

A pair of linear equations in two variables can be represented, and solved, by the:
(i) Graphical method
(ii) Algebraic method

GRAPHICAL METHOD OF SOLUTION OF A PAIR OF LINEAR EQUATIONS

The graph of a pair of linear equations in two variables is represented by two lines.

1. If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.

2. If the lines coincide, then there are infinitely many solutions — each point on the line being a solution. In this case, the pair of equations is dependent (consistent).

3. If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.

Algebraic interpretation of pair of linear equations in two variables

The pair of linear equations represented by these lines \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \):

1. If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) then the pair of linear equations has exactly one solution.
2. If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) then the pair of linear equations has infinitely many solutions.
3. If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) then the pair of linear equations has no solution.

S. No.	Pair of lines	Compare the ratios	Graphical representation	Algebraic interpretation
1	\( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \)	\( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)	Intersecting lines	Unique solution (Exactly one solution)
2	\( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \)	\( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)	Coincident lines	Infinitely many solutions
3	\( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \)	\( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)	Parallel lines	No solution

ALGEBRAIC METHODS OF SOLVING A PAIR OF LINEAR EQUATIONS

Substitution Method
Following are the steps to solve the pair of linear equations by substitution method:
\( a_1x + b_1y + c_1 = 0 \) … (i) and \( a_2x + b_2y + c_2 = 0 \) … (ii)
Step 1: We pick either of the equations and write one variable in terms of the other \[ y = -\frac{a_1x + c_1}{b_1} \dots (\text{iii}) \] Step 2: Substitute the value of \( x \) (or \( y \)) in equation (i) from equation (iii) obtained in step 1.
Step 3: Substituting this value of \( y \) (or \( x \)) in equation (iii) obtained in step 1, we get the values of \( x \) and \( y \).

Elimination Method
Following are the steps to solve the pair of linear equations by elimination method:
Step 1: First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either \( x \) or \( y \)) numerically equal.
Step 2: Then add or subtract one equation from the other so that one variable gets eliminated.
- If you get an equation in one variable, go to Step 3.
- If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.
- If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent.
Step 3: Solve the equation in one variable (\( x \) or \( y \)) so obtained to get its value.
Step 4: Substitute this value of \( x \) (or \( y \)) in either of the original equations to get the value of the other variable.

Cross - Multiplication Method
Let the pair of linear equations be:
\( a_1x + b_1y + c_1 = 0 \) … (1) and \( a_2x + b_2y + c_2 = 0 \) … (2)
\[ \frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1} \dots (3) \] \[ \Rightarrow x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} \quad \text{and} \quad y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1} \]

The arrows between the two numbers indicate that they are to be multiplied and the second product is to be subtracted from the first.
Step 1: Write the given equations in the form (1) and (2).
Step 2: Taking the help of the diagram above, write Equations as given in (3).
Step 3: Find \( x \) and \( y \), provided \( a_1b_2 - a_2b_1 \neq 0 \).

MCQ WORKSHEET-I

1. The pair of equations \( y = 0 \) and \( y = -7 \) has
   1. one solution
   2. two solution
   3. infinitely many solutions
   4. no solution
2. The pair of equations \( x = a \) and \( y = b \) graphically represents the lines which are
   1. parallel
   2. intersecting at (a, b)
   3. coincident
   4. intersecting at (b, a)
3. The value of c for which the pair of equations \( cx - y = 2 \) and \( 6x - 2y = 3 \) will have infinitely many solutions is
   1. 3
   2. -3
   3. -12
   4. no value
4. When lines \( l_1 \) and \( l_2 \) are coincident, then the graphical solution system of linear equation have
   1. infinite number of solutions
   2. unique solution
   3. no solution
   4. one solution
5. When lines \( l_1 \) and \( l_2 \) are parallel, then the graphical solution system of linear equation have
   1. infinite number of solutions
   2. unique solution
   3. no solution
   4. one solution
6. The coordinates of the vertices of triangle formed between the lines and y-axis from the graph is
   
   
   
   Explanation: Graph showing lines intersecting. Vertices appear to be P(0, 5), A(0, 0), and Q(4, 2) or intersection points. The graph shows lines 3x + 4y = 20 and x - 2y = 0.
   1. (0, 5), (0, 0) and (6.5,0)
   2. (4,2), (0, 0) and (6.5,0)
   3. (4,2), (0, 0) and (0,5)
   4. none of these
7. Five years ago Nuri was thrice old as Sonu. Ten years later, Nuri will be twice as old as Sonu. The present age, in years, of Nuri and Sonu are respectively
   1. 50 and 20
   2. 60 and 30
   3. 70 and 40
   4. 40 and 10
8. The pair of equations \( 5x - 15y = 8 \) and \( 3x - 9y = \frac{24}{5} \) has
   1. infinite number of solutions
   2. unique solution
   3. no solution
   4. one solution
9. The pair of equations \( x + 2y + 5 = 0 \) and \( -3x - 6y + 1 = 0 \) have
   1. infinite number of solutions
   2. unique solution
   3. no solution
   4. one solution
10. The sum of the digits of a two digit number is 9. If 27 is added to it, the digits of the numbers get reversed. The number is
    1. 36
    2. 72
    3. 63
    4. 25

MCQ WORKSHEET-II

1. If a pair of equation is consistent, then the lines will be
   1. parallel
   2. always coincident
   3. always intersecting
   4. intersecting or coincident
2. The solution of the equations \( x + y = 14 \) and \( x - y = 4 \) is
   1. x = 9 and y = 5
   2. x = 5 and y = 9
   3. x = 7 and y = 7
   4. x = 10 and y = 4
3. The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2, then the fraction
   1. \( \frac{4}{7} \)
   2. \( \frac{5}{7} \)
   3. \( \frac{6}{7} \)
   4. \( \frac{3}{7} \)
4. The value of k for which the system of equations \( x - 2y = 3 \) and \( 3x + ky = 1 \) has a unique solution is
   1. \( k = -6 \)
   2. \( k \neq -6 \)
   3. \( k = 0 \)
   4. no value
5. If a pair of equation is inconsistent, then the lines will be
   1. parallel
   2. always coincident
   3. always intersecting
   4. intersecting or coincident
6. The value of k for which the system of equations \( 2x + 3y = 5 \) and \( 4x + ky = 10 \) has infinite many solution is
   1. \( k = -3 \)
   2. \( k \neq -3 \)
   3. \( k = 0 \)
   4. none of these
7. The value of k for which the system of equations \( kx - y = 2 \) and \( 6x - 2y = 3 \) has a unique solution is
   1. \( k = -3 \)
   2. \( k \neq -3 \)
   3. \( k = 0 \)
   4. \( k \neq 0 \)
8. Sum of two numbers is 35 and their difference is 13, then the numbers are
   1. 24 and 12
   2. 24 and 11
   3. 12 and 11
   4. none of these
9. The solution of the equations \( 0.4x + 0.3y = 1.7 \) and \( 0.7x - 0.2y = 0.8 \) is
   1. x = 1 and y = 2
   2. x = 2 and y = 3
   3. x = 3 and y = 4
   4. x = 5 and y = 4
10. The solution of the equations \( x + 2y = 1.5 \) and \( 2x + y = 1.5 \) is
    1. x = 1 and y = 1
    2. x = 1.5 and y = 1.5
    3. x = 0.5 and y = 0.5
    4. none of these
11. The value of k for which the system of equations \( x + 2y = 3 \) and \( 5x + ky + 7 = 0 \) has no solution is
    1. 10
    2. 6
    3. 3
    4. 1
12. The value of k for which the system of equations \( 3x + 5y = 0 \) and \( kx + 10y = 0 \) has a non-zero solution is
    1. 0
    2. 2
    3. 6
    4. 8

MCQ WORKSHEET-II (Continued)

1. Sum of two numbers is 50 and their difference is 10, then the numbers are
   1. 30 and 20
   2. 24 and 14
   3. 12 and 2
   4. none of these
2. The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digit exceeds the given number by 18, then the number is
   1. 72
   2. 75
   3. 57
   4. none of these
3. The sum of a two-digit number and the number obtained by interchanging its digit is 99. If the digits differ by 3, then the number is
   1. 36
   2. 33
   3. 66
   4. none of these
4. Seven times a two-digit number is equal to four times the number obtained by reversing the order of its digit. If the difference between the digits is 3, then the number is
   1. 36
   2. 33
   3. 66
   4. none of these
5. A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed, then the number is
   1. 36
   2. 46
   3. 64
   4. none of these
6. The sum of two numbers is 1000 and the difference between their squares is 25600, then the numbers are
   1. 616 and 384
   2. 628 and 372
   3. 564 and 436
   4. none of these
7. Five years ago, A was thrice as old as B and ten years later A shall be twice as old as B, then the present age of A is
   1. 20
   2. 50
   3. 30
   4. none of these
8. The sum of thrice the first and the second is 142 and four times the first exceeds the second by 138, then the numbers are
   1. 40 and 20
   2. 40 and 22
   3. 12 and 22
   4. none of these
9. The sum of twice the first and thrice the second is 92 and four times the first exceeds seven times the second by 2, then the numbers are
   1. 25 and 20
   2. 25 and 14
   3. 14 and 22
   4. none of these
10. The difference between two numbers is 14 and the difference between their squares is 448, then the numbers are
    1. 25 and 9
    2. 22 and 9
    3. 23 and 9
    4. none of these
11. The solution of the system of linear equations \( \frac{x}{a} + \frac{y}{b} = a + b; \frac{x}{a^2} + \frac{y}{b^2} = 2 \) are
    1. x = a and y = b
    2. x = a² and y = b²
    3. x = 1 and y = 1
    4. none of these
12. The solution of the system of linear equations \( 2(ax - by) + (a + 4b) = 0; 2(bx + ay) + (b - 4a) = 0 \) are
    1. x = a and y = b
    2. x = -1 and y = -1
    3. x = 1 and y = 1
    4. none of these

MCQ WORKSHEET-III

1. The pair of equations \( 3x + 4y = 18 \) and \( 4x + \frac{16}{3}y = 24 \) has
   1. infinite number of solutions
   2. unique solution
   3. no solution
   4. cannot say anything
2. If the pair of equations \( 2x + 3y = 7 \) and \( kx + \frac{9}{2}y = 12 \) have no solution, then the value of k is:
   1. \( \frac{2}{3} \)
   2. -3
   3. 3
   4. \( \frac{3}{2} \)
3. The equations \( x - y = 0.9 \) and \( \frac{11}{x + y} = 2 \) have the solution:
   1. x = 5 and y = a
   2. x = 3.2 and y = 2.3
   3. x = 3 and y = 2
   4. none of these
4. If \( bx + ay = a^2 + b^2 \) and \( ax - by = 0 \), then the value of \( x - y \) equals:
   1. a - b
   2. b - a
   3. \( a^2 - b^2 \)
   4. \( b^2 + a^2 \)
5. If \( 2x + 3y = 0 \) and \( 4x - 3y = 0 \), then \( x + y \) equals:
   1. 0
   2. -1
   3. 1
   4. 2
6. If \( \sqrt{a}x - \sqrt{b}y = b - a \) and \( \sqrt{b}x - \sqrt{a}y = 0 \), then the value of x, y is:
   1. a + b
   2. a - b
   3. ab
   4. -ab
7. If \( \frac{2}{x} + \frac{3}{y} = 13 \) and \( \frac{5}{x} - \frac{4}{y} = -2 \), then \( x + y \) equals:
   1. \( \frac{1}{6} \)
   2. \( -\frac{1}{6} \)
   3. \( \frac{5}{6} \)
   4. \( -\frac{5}{6} \)
8. If \( 31x + 43y = 117 \) and \( 43x + 31y = 105 \), then value of \( x - y \) is:
   1. \( \frac{1}{3} \)
   2. -3
   3. 3
   4. \( -\frac{1}{3} \)
9. If \( 19x - 17y = 55 \) and \( 17x - 19y = 53 \), then the value of \( x - y \) is:
   1. \( \frac{1}{3} \)
   2. -3
   3. 3
   4. 5
10. If \( \frac{x}{10} + \frac{y}{5} - 1 = 0 \) and \( \frac{x}{8} + \frac{y}{6} = 15 \), then the value of \( y \) is:
    1. 1
    2. -0.8
    3. 0.6
    4. 0.5
11. If (6, k) is a solution of the equation \( 3x + y - 22 = 0 \), then the value of k is:
    1. 4
    2. -4
    3. 3
    4. -3
12. If \( 3x - 5y = 1 \), \( \frac{2x}{x - y} = 4 \), then the value of \( x + y \) is
    1. \( \frac{1}{3} \)
    2. -3
    3. 3
    4. \( -\frac{1}{3} \)
13. If \( 3x + 2y = 13 \) and \( 3x - 2y = 5 \), then the value of \( x + y \) is:
    1. 5
    2. 3
    3. 7
    4. none of these
14. If the pair of equations \( 2x + 3y = 5 \) and \( 5x + \frac{15}{2}y = k \) represent two coincident lines, then the value of k is:
    1. -5
    2. \( -\frac{25}{2} \)
    3. \( \frac{25}{2} \)
    4. \( -\frac{5}{2} \)
15. Rs. 4900 were divided among 150 children. If each girl gets Rs. 50 and a boy gets Rs. 25, then the number of boys is:
    1. 100
    2. 102
    3. 104
    4. 105

PRACTICE QUESTIONS: SOLVING EQUATIONS

1. Solve for x and y: \( 11x + 15y + 23 = 0; 7x - 2y - 20 = 0 \).
2. Solve for x and y: \( 2x + y = 7; 4x - 3y + 1 = 0 \).
3. Solve for x and y: \( 23x - 29y = 98; 29x - 23y = 110 \).
4. Solve for x and y: \( 2x + 5y = \frac{8}{3}; 3x - 2y = \frac{5}{6} \).
5. Solve for x and y: \( 4x - 3y = 8; 6x - y = \frac{29}{3} \).
6. Solve for x and y: \( 2x - \frac{3}{4}y = 3; 5x = 2y + 7 \).
7. Solve for x and y: \( 2x - 3y = 13; 7x - 2y = 20 \).
8. Solve for x and y: \( 3x - 5y - 19 = 0; -7x + 3y + 1 = 0 \).
9. Solve for x and y: \( 2x - 3y + 8 = 0; x - 4y + 7 = 0 \).
10. Solve for x and y: \( \frac{x}{2} + y = 0.8; \frac{7}{x + \frac{y}{2}} = 10 \).
11. Solve for x and y: \( 152x - 378y = -74; -378x + 152y = -604 \).
12. Solve for x and y: \( 47x + 31y = 63; 31x + 47y = 15 \).
13. Solve for x and y: \( 71x + 37y = 253; 37x + 71y = 287 \).
14. Solve for x and y: \( 37x + 43y = 123; 43x + 37y = 117 \).
15. Solve for x and y: \( 217x + 131y = 913; 131x + 217y = 827 \).
16. Solve for x and y: \( 41x - 17y = 99; 17x - 41y = 75 \).
17. Solve for x and y: \( \frac{5}{x+y} + \frac{1}{x-y} = 2; \frac{15}{x+y} - \frac{5}{x-y} = -2 \).
18. Solve for x and y: \( \frac{2}{x} + \frac{3}{y} = 13; \frac{5}{x} - \frac{4}{y} = -2 \).
19. Solve for x and y: \( \frac{5}{x-1} + \frac{1}{y-2} = 2; \frac{6}{x-1} - \frac{3}{y-2} = 1 \).
20. Solve for x and y: \( \frac{1}{2x} + \frac{1}{3y} = 2; \frac{1}{3x} + \frac{1}{2y} = \frac{13}{6} \).
21. Solve for x and y: \( \frac{3}{x+y} + \frac{2}{x-y} = 2; \frac{9}{x+y} - \frac{4}{x-y} = 1 \).
22. Solve for x and y: \( \frac{x}{a} + \frac{y}{b} = 2; ax - by = a^2 - b^2 \).
23. Solve for x and y: \( ax + by = a - b; bx - ay = a + b \).
24. Solve for x and y: \( (a+2b)x + (2a-b)y = 2; (a-2b)x + (2a+b)y = 3 \).
25. Solve for x and y: \( (a-b)x + (a+b)y = a^2 - 2ab - b^2; (a+b)(x+y) = a^2 + b^2 \).
26. Solve for x and y: \( \frac{x}{a} + \frac{y}{b} = a+b; \frac{x}{a^2} + \frac{y}{b^2} = 2 \).
27. Solve for x and y: \( ax + by = 1; bx + ay = \frac{2ab}{a^2+b^2} \).

PRACTICE QUESTIONS: CONDITIONS FOR SOLVING LINEAR EQUATIONS

1. Find the value of k, so that the following system of equations has no solution: \( 3x - y - 5 = 0; 6x - 2y - k = 0 \).
2. Find the value of k, so that the following system of equations has a non-zero solution: \( 3x + 5y = 0; kx + 10y = 0 \).
3. Find the value of k, so that the system of equations has no solution: \( 3x + y = 1; (2k-1)x + (k-1)y = 2k+1 \).
4. Find the value of k, so that the system of equations has a unique solution: \( kx + 2y = 5; 3x + y = 1 \).
5. For what value of k, the following pair of linear equations has infinite number of solutions: \( kx + 3y = k-3; 12x + ky = k \).
6. Find the value of a and b for which the system of linear equations has infinite number of solutions: \( 2x + 3y = 7; (a-b)x + (a+b)y = 3a + b - 2 \).

PRACTICE QUESTIONS: GRAPHICAL QUESTIONS

1. Solve the following system of linear equations graphically: \( 2x + 3y = 2; x - 2y = 8 \).
2. Solve graphically: \( 2x + 3y = 4; x - y + 3 = 0 \).
3. Solve graphically: \( x + y = 3; 3x - 2y = 4 \).
4. Solve graphically: \( 2x - 3y + 13 = 0; 3x - 2y + 12 = 0 \).
5. Draw the graphs of the equations \( x - y + 1 = 0 \) and \( 3x + 2y - 12 = 0 \). Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
6. Solve graphically: \( 2x + y = 6; 2x - y + 2 = 0 \). Shade the region bounded by these lines and the x-axis. Find the area of the shaded region.
7. Solve graphically: \( 2x + 3y = 12; x - y = 1 \). Shade the region bounded by these lines and the y-axis.

PRACTICE QUESTIONS: WORD PROBLEMS

I. NUMBER BASED QUESTIONS

1. The sum of two numbers is 137 and their difference is 43. Find the numbers.
2. The sum of two numbers is 1000 and the difference between their squares is 25600, then find the numbers.
3. The sum of the digits of a two digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number.
4. The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number.
5. 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.

II. AGE RELATED QUESTIONS

1. Ten years hence, a man’s age will be twice the age of his son. Ten years ago, man was four times as old as his son. Find their present ages.
2. Five years ago Nuri was thrice old as Sonu. Ten years later, Nuri will be twice as old as Sonu. Find the present age of Nuri and Sonu.
3. The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.

III. SPEED, DISTANCE AND TIME RELATED QUESTIONS

1. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still water.
2. Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.

IV. GEOMETRICAL FIGURES RELATED QUESTIONS

1. The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
2. In a \( \Delta ABC \), \( \angle C = 3\angle B = 2(\angle A + \angle B) \). Find the angles.
3. The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

V. TIME AND WORK RELATED QUESTIONS

1. 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

VI. REASONING BASED QUESTIONS

1. Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?
2. A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.
3. 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.

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