ALGEBRAIC INTERPRETATION OF PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
The pair of linear equations represented by these lines a1x + b1y + c1Â = 0 and a2x + b2y + c2Â = 0
- If a1/a2 ≠b1/b2 then the pair of linear equations has exactly one solution.
- If a1/a2Â = b1/b2Â = c1/c2Â then the pair of linear equations has infinitely many solutions.
- If a1/a2 = b1/b2 ≠c1/c2 then the pair of linear equations has no solution.
Q.1. On comparing the ratios a1/a2, b1/b2, c1/c2Â , 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 a1/a2, b1/b2, c1/c2 , 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