**ALGEBRAIC INTERPRETATION OF PAIR OF LINEAR EQUATIONS IN TWO VARIABLES**

The pair of linear equations represented by these lines a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0

- If a
_{1}/a_{2}≠ b_{1}/b_{2}then the pair of linear equations has*exactly one solution.* - If a
_{1}/a_{2}= b_{1}/b_{2}= c_{1}/c_{2}then the pair of linear equations has*infinitely many solutions.* - If a
_{1}/a_{2}= b_{1}/b_{2}≠ c_{1}/c_{2}then the pair of linear equations has.*no solution*

Q.1. On comparing the ratios a_{1}/a_{2}, b_{1}/b_{2}, c_{1}/c_{2} , find out whether the lines representing the following pairs 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.

Q.2. On comparing the ratios a_{1}/a_{2}, b_{1}/b_{2}, c_{1}/c_{2} , find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5 ; 2x – 3y = 7

(ii) 2x – 3y = 8 ; 4x – 6y = 9

(iii) 5x – 3y = 11 ; – 10x + 6y = –22

Q.3. Find the number of solutions of the following pair of linear equations: x + 2y – 8 = 0; 2x + 4y = 16

Q.4. Write whether the following pair of linear equations is consistent or not: x + y = 14; x – y = 4

Q.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.

Q.6. Find the value of k so that the following system of equations has no solution:

3x – y – 5 = 0, 6x – 2y + k = 0

Q.7. Find the value of k so that the following system of equation has infinite solutions:

3x – y – 5 = 0, 6x – 2y + k = 0

Q.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

Q.9. Determine k for which the system of equations has infinite solutions:

4x + y = 3 and 8x + 2y = 5k

Q.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

Q.11. Find the value of k for which the system 3x + ky = 7, 2x – 5y = 1 will have infinitely many solutions.

Q.12. For what value of k, the system of equations 2x – ky + 3 = 0, 4x + 6y – 5 = 0 is consistent?

Q.13. For what value of k, the system of equations kx – 3y + 6 = 0, 4x – 6y + 15 = 0 represents parallel lines?

Q.14. For what value of p, the pair of linear equations 5x + 7y = 10, 2x + 3y = p has a unique solution.

Q.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)

Q.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

Q.17. For what value of k will the system of linear equations has infinite number of solutions?

kx + 4y = k – 4, 16x + ky = k

Q.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

Q.19. For what value of k will the equations x + 2y + 7 = 0, 2x + ky + 14 = 0 represent coincident lines?

Q.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