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The number of values of k for which the system of equations (k + 1)x + 8y = 4k; kx + (k + 3) y = 3k – 1 has infinitely many solutions is
- a)0
- b)1
- c)2
- d)infinite
Correct answer is option 'B'. Can you explain this answer?
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The number of values of k for which the system of equations (k + 1)x +...
For infinitely many solutions the two equations become identical
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The number of values of k for which the system of equations (k + 1)x +...
To solve this system of equations, we can start by multiplying the first equation by k and the second equation by (k - 3):
(k)(k + 1)x + 8ky = 4k^2
kx + (k - 3)y = 3k(k - 3)
Now we can simplify both equations:
k^2x + kx + 8ky = 4k^2
kx + ky - 3y = 3k^2 - 9k
Next, we can subtract the second equation from the first equation:
k^2x - kx + 8ky - ky + 3y = 4k^2 - 3k^2 + 9k - (-9k)
k^2x - kx + 7ky + 4y = k^2 + 18k
Now we can factor out the common terms:
x(k^2 - k) + y(7k + 4) = k(k + 18)
Since the left-hand side is a linear combination of x and y, and the right-hand side is a product of k and (k + 18), we have two cases to consider:
Case 1: k^2 - k = 0
This equation factors as k(k - 1) = 0, which gives us two solutions: k = 0 and k = 1.
Case 2: 7k + 4 = k + 18
This equation simplifies to 6k = 14, which gives us a single solution: k = 7/3.
Therefore, there are three values of k for which the system of equations is satisfied: k = 0, k = 1, and k = 7/3.
(k)(k + 1)x + 8ky = 4k^2
kx + (k - 3)y = 3k(k - 3)
Now we can simplify both equations:
k^2x + kx + 8ky = 4k^2
kx + ky - 3y = 3k^2 - 9k
Next, we can subtract the second equation from the first equation:
k^2x - kx + 8ky - ky + 3y = 4k^2 - 3k^2 + 9k - (-9k)
k^2x - kx + 7ky + 4y = k^2 + 18k
Now we can factor out the common terms:
x(k^2 - k) + y(7k + 4) = k(k + 18)
Since the left-hand side is a linear combination of x and y, and the right-hand side is a product of k and (k + 18), we have two cases to consider:
Case 1: k^2 - k = 0
This equation factors as k(k - 1) = 0, which gives us two solutions: k = 0 and k = 1.
Case 2: 7k + 4 = k + 18
This equation simplifies to 6k = 14, which gives us a single solution: k = 7/3.
Therefore, there are three values of k for which the system of equations is satisfied: k = 0, k = 1, and k = 7/3.
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The number of values of k for which the system of equations (k + 1)x + 8y = 4k; kx + (k + 3) y = 3k – 1 has infinitely many solutions isa)0b)1c)2d)infiniteCorrect answer is option 'B'. Can you explain this answer?
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