Q1. Determine the value of k for which the given system of equations has unique solution:
a) 2x – 3y = 1 ; k x + 5y = 7 b) 4x – 5y = k ; 2x – 3 y = 12
a)
2x-3y = 1
kx+5y = 7
There is no single value of k which determines a unique solution. In fact, since the slope of
2x-3y=1 is 2/3, any value of k which makes the slope of the other line different from 2/3 will work.
So, if -k/5 = 2/3 then k = -10/3 and the line
-10/3 x + 5y = 7
is parallel to the first line.
So, any k different from -10/3 will produce a single solution.
The other pair of lines is even easier. Since it is clear that the lines have different slopes, they must intersect at a single point, regardless of k's value.
To determine the value of k for which the given system of equations has a unique solution, we need to solve the system of equations and check if there is a unique solution.
a) 2x - 3y = 1
kx + 5y = 7
To solve this system, we can use the method of substitution or elimination. Let's use the method of elimination.
Step 1: Multiply the first equation by 5 and the second equation by 3 to eliminate y.
10x - 15y = 5
3kx + 15y = 21
Step 2: Add the two equations together to eliminate y.
(10x - 15y) + (3kx + 15y) = 5 + 21
10x - 15y + 3kx + 15y = 26
13kx = 26
Step 3: Divide both sides by 13 to isolate kx.
kx = 26/13
kx = 2
Step 4: Divide both sides by k to solve for x.
x = 2/k
Now, let's move on to the second equation.
b) 4x - 5y = k
2x - 3y = 12
Again, we can use the method of elimination.
Step 1: Multiply the second equation by 2 and the first equation by 3 to eliminate y.
4x - 6y = 24
6x - 10y = 3k
Step 2: Multiply the first equation by 5 and the second equation by 2 to eliminate x.
20x - 30y = 60
12x - 20y = 6k
Step 3: Subtract the second equation from the first equation to eliminate x.
(20x - 30y) - (12x - 20y) = 60 - 6k
20x - 12x - 30y + 20y = 60 - 6k
8x - 10y = 60 - 6k
4x - 5y = 30 - 3k
Step 4: Set the equations equal to each other to eliminate x.
4x - 5y = 30 - 3k = k
2x - 3y = 12
Step 5: Subtract the second equation from the first equation to eliminate x.
(4x - 5y) - (2x - 3y) = (k) - (12)
4x - 2x - 5y + 3y = k - 12
2x - 2y = k - 12
Step 6: Combine like terms.
2(x - y) = k - 12
Step 7: Divide both sides by 2 to isolate x - y.
x - y = (k - 12)/2
Since the system has a unique solution, x - y should have the same value in both equations.
Set (k - 12)/2 = 2/k and solve for k.
(k - 12)/2 = 2/k
Cross-multiply:
k(k - 12) = 4(2)
k^2 - 12k = 8
Rearrange the equation:
k^2 - 12k - 8 = 0
Now, you can solve this quadratic equation using the quadratic formula or factoring.
After finding the value(s) for k from the quadratic equation, you can substitute them back into the original equations to verify if they give unique solutions.