Advertisements
Advertisements
प्रश्न
For what value of k, the system of equations
x + 2y = 3,
5x + ky + 7 = 0
Have (i) a unique solution, (ii) no solution?
Also, show that there is no value of k for which the given system of equation has infinitely namely solutions
उत्तर
The given system of equations:
x + 2y = 3
⇒ x + 2y - 3 = 0 ….(i)
And, 5x + ky + 7 = 0 …(ii)
These equations are of the following form:
`a_1x+b_1y+c_1 = 0, a_2x+b_2y+c_2 = 0`
where, `a_1 = 1, b_1= 2, c_1= -3 and a_2 = 5, b_2 = k, c_2 = 7`
(i) For a unique solution, we must have:
∴ `(a_1)/(a_2) ≠ (b_1)/(b_2) i.e., 1/5 ≠ 2/k ⇒ k ≠ 10`
Thus for all real values of k other than 10, the given system of equations will have a unique solution.
(ii) In order that the given system of equations has no solution, we must have:
`(a_1)/(a_2) = (b_1)/(b_2 )≠ (c_1)/(c_2)`
`⇒ 1/5 ≠ 2/k ≠ (−3)/7`
`⇒ 1/5 ≠ 2/k and 2/k ≠ (−3)/7`
`⇒k = 10, k ≠ 14/(−3)`
Hence, the required value of k is 10.
There is no value of k for which the given system of equations has an infinite number of solutions.
APPEARS IN
संबंधित प्रश्न
The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another bat and 3 more balls of the same kind for Rs 1300. Represent this situation algebraically and geometrically.
Solve for x and y:
0.4x + 0.3y = 1.7, 0.7x – 0.2y = 0.8.
Solve for x and y:
x + y = 5xy, 3x + 2y = 13xy
Solve for x and y:
23x - 29y = 98, 29x - 23y = 110
Find the value of k for which the system of equations
kx + 3y = 3, 12x + ky = 6 has no solution.
The present age of a man is 2 years more than five times the age of his son. Two years hence, the man’s age will be 8 years more than three times the age of his son. Find their present ages.
Find the value of k for which the system of linear equations has an infinite number of solutions.
2x + 3y – 7 = 0,
(k – 1)x + (k + 2)y=3k
Solve for x:
The condition for the system of linear equations ax + by = c; lx + my = n to have a unique solution is ______.
Read the following passage:
 Lokesh, a production manager in Mumbai, hires a taxi everyday to go to his office. The taxi charges in Mumbai consists of a fixed charges together with the charges for the distance covered. His office is at a distance of 10 km from his home. For a distance of 10 km to his office, Lokesh paid ₹ 105. While coming back home, he took another roµte. He covered a distance of 15 km and the charges paid by him were ₹ 155. |
Based on the above information, answer the following questions:
- What are the fixed charges?
- What are the charges per km?
- If fixed charges are ₹ 20 and charges per km are ₹ 10, then how much Lokesh have to pay for travelling a distance of 10 km?
OR
Find the total amount paid by Lokesh for travelling 10 km from home to office and 25 km from office to home. [Fixed charges and charges per km are as in (i) and (ii).
