The following equations have infinitely many solutions.33x + 33y + 00z = 33−7−7x + 33y + −6−6z = 252577x + −8−8y + 99z = −41−41Give the right hand side of the vector form of the general solution, using a parameter t. For example, for the equationsx = y + 1, y = z + 1, z = x − 2one correct answer is[x, y, z] = [0, −1, −2] + t [1, 1, 1] = [t, t − 1, t − 2][x, y, z] =
Question
The following equations have infinitely many solutions.33x + 33y + 00z = 33−7−7x + 33y + −6−6z = 252577x + −8−8y + 99z = −41−41Give the right hand side of the vector form of the general solution, using a parameter t. For example, for the equationsx = y + 1, y = z + 1, z = x − 2one correct answer is[x, y, z] = [0, −1, −2] + t [1, 1, 1] = [t, t − 1, t − 2][x, y, z] =
Solution
The given system of equations is:
33x + 33y + 0z = 33 -7x + 33y - 6z = 252 77x - 8y + 99z = -41
First, we simplify these equations:
x + y = 1 -7/33x + y - 6/33z = 252/33 77/33x - 8/33y + 3z = -41/33
Which simplifies to:
x + y = 1 -7/33x + y - 6/33z = 252/33 7/3x - 8/33y + 3z = -41/33
Next, we express x, y, and z in terms of the parameter t:
x = 1 - y y = t z = (252/33 + 7/33*(1-t) - t) / (6/33)
Substituting y = t into the other equations gives:
x = 1 - t z = (252/33 + 7/33*(1-t) - t) / (6/33)
So the general solution in vector form is:
[x, y, z] = [1, 0, 252/33] + t [-1, 1, (7/33 - 6/33
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