Determine the value of k for which the system of linear equations
3x − y = 2
6x + ky = 2
has no solution.
k =?
To determine the value of k for which the system of linear equations has no solution, we need to examine the coefficients of the variables.
The given system of linear equations is:
1) 3x - y = 2
2) 6x + ky = 2
We can start by eliminating the variable y by multiplying equation 1 by 2 and equation 2 by 1:
1) 6x - 2y = 4
2) 6x + ky = 2
Next, we can subtract equation 2 from equation 1 to eliminate the variable x:
(6x - 2y) - (6x + ky) = 4 - 2
-2y - ky = 2
Now, we factor out y from the left side of the equation:
y(-2 - k) = 2
For the system to have no solution, the coefficient of y on the left side of the equation must be zero.
Therefore, -2 - k = 0
Solving for k:
-2 - k = 0
k = -2
Hence, the value of k for which the system of linear equations has no solution is k = -2.
To determine the value of k for which the system of linear equations has no solution, we need to find the condition under which the two equations are parallel lines.
To do this, we can start by rearranging both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Equation 1: 3x - y = 2
Rearranging, we get:
-y = -3x + 2
y = 3x - 2
Equation 2: 6x + ky = 2
Rearranging, we get:
ky = -6x + 2
y = (-6/k)x + 2/k
Now, we compare the slopes of the two equations. If the slopes are equal, the lines are parallel and there is no solution.
Comparing the slopes, we have:
Equation 1: m1 = 3
Equation 2: m2 = -6/k
For the lines to be parallel, the slopes should be equal, so we set m1 = m2:
3 = -6/k
Now, we solve this equation for k:
Multiply both sides by k:
3k = -6
Divide both sides by 3:
k = -2
Therefore, the value of k for which the system of linear equations has no solution is k = -2.
you want them to be parallel, so
3/6 = -1/k
3k = -6
k = -2
the system of linear equations has no solution
... if the lines are parallel ... have the same slope ... -6/k = 3