x + y + 3z = 10 ⇒ -3x - 3y - 9z = -30 (1)
-4x + 3y + 5z = 7 (2)
kx + z = 3 (3)
Add (1) and (2) and we get that
-7x - 4z = -23 (4)
Multiply (3) by -4 = -4kx - 4z = -12 (5)
Note that if -4k = -7 ⇒ k = 7/4.....we will have the equations
-7x - 4z = -23
-7x - 4z = -12
And it is impossible for equal values on the left side to produce unequal values on the right
So no solutions if k = 7/4
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