CHAPTER – 3
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
ALGEBRAIC INTERPRETATION OF PAIR OF LINEAR EQUATIONS IN TWO
VARIABLES
The pair of linear equations represented by these lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
a b
1. If 1 1 then the pair of linear equations has exactly one solution.
a2 b2
a b c
2. If 1 1 1 then the pair of linear equations has infinitely many solutions.
a2 b2 c2
a b c
3. If 1 1 1 then the pair of linear equations has no solution.
a2 b2 c2
S. No. Pair of lines Compare Graphical Algebraic
the ratios representation interpretation
a1x + b1y + c1 = 0 a1 b1 Intersecting Unique solution (Exactly
1 a2x + b2y + c2 = 0 lines one solution)
a2 b2
a1x + b1y + c1 = 0 a1 b1 c1 Coincident Infinitely many solutions
2 a2x + b2y + c2 = 0 lines
a2 b2 c2
a1x + b1y + c1 = 0 a1 b1 c1 Parallel lines No solution
3 a2x + b2y + c2 = 0
a2 b2 c2
IMPORTANT QUESTIONS
a1 b1 c1
1. On comparing the ratios , find out whether the lines representing the following pairs
a2 b2 c2
of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0 and 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0 and 2x – y + 9 = 0.
a b c
2. On comparing the ratios 1 1 1 , find out whether the following pair of linear equations are
a2 b2 c2
consistent, or inconsistent.
(i) 3x + 2y = 5 ; 2x – 3y = 7 (ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii) 5x – 3y = 11 ; – 10x + 6y = –22
3. Find the number of solutions of the following pair of linear equations:
x + 2y – 8 = 0
2x + 4y = 16
4. Write whether the following pair of linear equations is consistent or not.
x + y = 14, x – y = 4
5. Given the linear equation 3x + 4y – 8 = 0, write another linear equation in two variables such
that the geometrical representation of the pair so formed is parallel lines.
6. Find the value of k so that the following system of equations has no solution:
3x – y – 5 = 0, 6x – 2y + k = 0
7. Find the value of k so that the following system of equation has infinite solutions:
3x – y – 5 = 0, 6x – 2y + k = 0
8. For which values of p, does the pair of equations given below has unique solution?
4x + py + 8 = 0 and 2x + 2y + 2 = 0
, 9. Determine k for which the system of equations has infinite solutions:
4x + y = 3 and 8x + 2y = 5k
10. Find whether the lines representing the following pair of linear equations intersect at a point, are
parallel or coincident:
2x – 3y + 6 = 0; 4x – 5y + 2 = 0
11. Find the value of k for which the system 3x + ky = 7, 2x – 5y = 1 will have infinitely many
solutions.
12. For what value of k, the system of equations 2x – ky + 3 = 0, 4x + 6y – 5 = 0 is consistent?
13. For what value of k, the system of equations kx – 3y + 6 = 0, 4x – 6y + 15 = 0 represents parallel
lines?
14. For what value of p, the pair of linear equations 5x + 7y = 10, 2x + 3y = p has a unique solution.
15. Find the value of m for which the pair of linear equations has infinitely many solutions.
2x + 3y – 7 = 0 and (m – 1)x + (m + 1)y = (3m – 1)
16. For what value of p will the following pair of linear equations have infinitely many solutions?
(p – 3)x + 3y = p; px + py = 12
17. For what value of k will the system of linear equations has infinite number of solutions?
kx + 4y = k – 4, 16x + ky = k
18. Find the values of a and b for which the following system of linear equations has infinite number
of solutions:
2x – 3y = 7, (a + b) x – (a + b – 3) y = 4a + b
19. For what value of k will the equations x + 2y + 7 = 0, 2x + ky + 14 = 0 represent coincident
lines?
20. For what value of k, the following system of equations 2x + ky = 1, 3x – 5y = 7 has (i) a unique
solution (ii) no solution
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).
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
ALGEBRAIC INTERPRETATION OF PAIR OF LINEAR EQUATIONS IN TWO
VARIABLES
The pair of linear equations represented by these lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
a b
1. If 1 1 then the pair of linear equations has exactly one solution.
a2 b2
a b c
2. If 1 1 1 then the pair of linear equations has infinitely many solutions.
a2 b2 c2
a b c
3. If 1 1 1 then the pair of linear equations has no solution.
a2 b2 c2
S. No. Pair of lines Compare Graphical Algebraic
the ratios representation interpretation
a1x + b1y + c1 = 0 a1 b1 Intersecting Unique solution (Exactly
1 a2x + b2y + c2 = 0 lines one solution)
a2 b2
a1x + b1y + c1 = 0 a1 b1 c1 Coincident Infinitely many solutions
2 a2x + b2y + c2 = 0 lines
a2 b2 c2
a1x + b1y + c1 = 0 a1 b1 c1 Parallel lines No solution
3 a2x + b2y + c2 = 0
a2 b2 c2
IMPORTANT QUESTIONS
a1 b1 c1
1. On comparing the ratios , find out whether the lines representing the following pairs
a2 b2 c2
of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0 and 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0 and 2x – y + 9 = 0.
a b c
2. On comparing the ratios 1 1 1 , find out whether the following pair of linear equations are
a2 b2 c2
consistent, or inconsistent.
(i) 3x + 2y = 5 ; 2x – 3y = 7 (ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii) 5x – 3y = 11 ; – 10x + 6y = –22
3. Find the number of solutions of the following pair of linear equations:
x + 2y – 8 = 0
2x + 4y = 16
4. Write whether the following pair of linear equations is consistent or not.
x + y = 14, x – y = 4
5. Given the linear equation 3x + 4y – 8 = 0, write another linear equation in two variables such
that the geometrical representation of the pair so formed is parallel lines.
6. Find the value of k so that the following system of equations has no solution:
3x – y – 5 = 0, 6x – 2y + k = 0
7. Find the value of k so that the following system of equation has infinite solutions:
3x – y – 5 = 0, 6x – 2y + k = 0
8. For which values of p, does the pair of equations given below has unique solution?
4x + py + 8 = 0 and 2x + 2y + 2 = 0
, 9. Determine k for which the system of equations has infinite solutions:
4x + y = 3 and 8x + 2y = 5k
10. Find whether the lines representing the following pair of linear equations intersect at a point, are
parallel or coincident:
2x – 3y + 6 = 0; 4x – 5y + 2 = 0
11. Find the value of k for which the system 3x + ky = 7, 2x – 5y = 1 will have infinitely many
solutions.
12. For what value of k, the system of equations 2x – ky + 3 = 0, 4x + 6y – 5 = 0 is consistent?
13. For what value of k, the system of equations kx – 3y + 6 = 0, 4x – 6y + 15 = 0 represents parallel
lines?
14. For what value of p, the pair of linear equations 5x + 7y = 10, 2x + 3y = p has a unique solution.
15. Find the value of m for which the pair of linear equations has infinitely many solutions.
2x + 3y – 7 = 0 and (m – 1)x + (m + 1)y = (3m – 1)
16. For what value of p will the following pair of linear equations have infinitely many solutions?
(p – 3)x + 3y = p; px + py = 12
17. For what value of k will the system of linear equations has infinite number of solutions?
kx + 4y = k – 4, 16x + ky = k
18. Find the values of a and b for which the following system of linear equations has infinite number
of solutions:
2x – 3y = 7, (a + b) x – (a + b – 3) y = 4a + b
19. For what value of k will the equations x + 2y + 7 = 0, 2x + ky + 14 = 0 represent coincident
lines?
20. For what value of k, the following system of equations 2x + ky = 1, 3x – 5y = 7 has (i) a unique
solution (ii) no solution
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).
