A system of equations only has a unique solution if all the equations are linearly independent (this means that we have two or more equations representing the same curve).
Here we have the system:
a*x + 2*y + 4*z = b
2*x + y + 3*z = a + 1
5*x - 2*y + (a + 1)*z = 3
Now we want to show that if a = -5 or a = 2, then the system does not have a unique solution.
To see this, we can just replace a by these values and see that the equations are not linearly independent.
for a = 2 we get:
2*x + 2*y + 4*z = b
2*x + y + 3*z = 2 + 1
5*x - 2*y + (2 + 1)*z = 3
Simplifying this:
2*x + 2*y + 4*z = b
2*x + y + 3*z = 3
5*x - 2*y + 3*z = 3
From the second and third equation we can write:
2*x + y + 3*z = 3 = 5*x - 2*y + 3*z
2*x + y + 3*z = 5*x - 2*y + 3*z
2*x + y = 5*x - 2*y
3*y = 3*x
y = x
now that we know this, we can rewrite the equations as:
4*(x + z) = b
3*(x + z) = 3
Now we can take the quotient of the two equations to get:
(4*(x + z))/3*(x + z) = (4/3) = b/3
b = 4
Then the equations become:
4*(x + z) = 4
3*(x + z) = 3
Here we can divide te above one by 4 and the below one by 3 to get:
(x + z) = 1
(x + z) = 1
So both equations represent the same line, thus, the system does not have a unique solution.
Now for a = -5:
-5*x + 2*y + 4*z = b
2*x + y + 3*z = -5 + 1
5*x - 2*y + (-5 + 1)*z = 3
Simplify it:
-5*x + 2*y + 4*z = b
2*x + y + 3*z = -4
5*x - 2*y - 4*z = 3
Now we can rewrite the first equation as:
-5*x +2*y - b = -4*z
and the third one as:
-4*z = 3 - 5*x + 2*y
Then we can write:
-5*x +2*y - b = -4*z = 3 - 5*x + 2*y
-5*x +2*y - b = 3 - 5*x + 2*y
Now we can remove the -5*x in both sides to get:
+2*y - b = 3 + 2*y
Now we can remove the 2*y in both sides to get:
-b = 3
b = -3
Replacing this in our system we get:
-5*x + 2*y + 4*z = -3
2*x + y + 3*z = -4
5*x - 2*y -4*z = 3
Now we can multiply the first equation by -1:
-1*(-5*x + 2*y + 4*z)= -1*(-3)
5*x - 2*y - 4*z = 3
This is the same as the third equation, so here we can see that the first and third equations represent the same line, thus, the system of equations does not have a unique solution.
If you want to learn more, you can read:
https://brainly.com/question/12895249
