Answer:
k = -1/2 or 2
Step-by-step explanation:
For equations ...
ax +by +c = 0
dx +ey +g = 0
Cramer's Rule tells you the solutions are ...
x = (-ce +gb)/(ae -db)
y = (-ga +dc)/(ae -db)
In order for these equations to be consistent, the solution for any pair of them must be the same. It is sufficient for us to find 'k' such that the values of x are the same.
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first and third equations
x = (-1·7 +1·(k+1))/(1·7 -3·(k+1)) = (k -6)/(4-3k)
second and third equations
x = (3·7 +1·5)/(2k·7 -3·5) = 26/(14k -15)
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For these values to be the same, we must have ...
(k -6)/(4-3k) = x = 26/(14k -15)
(k -6)(14k -15) = (4 -3k)(26) . . . . . cross multiply
14k² -99k +90 = -78k +104 . . . . . simplify
14k² -21k -14 = 0 . . . . . . . . . . . . . put in standard form
7(2k +1)(k -2) = 0 . . . . . . . . factor
The solutions are the values of k that make the factors zero: k=2, k=-1/2.
These equations will be consistent if k = -1/2 or k = 2.
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Additional comment
Using our above expression for x, we can find the value of x for the solution x = -1/2 to be ...
x = 26/(14(-1/2) -15) = 26/(-22) = -13/11
The first equation tells us the corresponding value of y will be ...
x +(-1/2 +1)y +1 = 0
y = -2(x+1) = -2(-13/11 +11/11) = 4/11
For k = -1/2, (x, y) = (-13/11, 4/11).
The attached graph shows the solution for k=2.
For k = 2, (x, y) = (2, -1).
