3x – 5y + z = 10
x² + y² + z² = 8
Solutions to the system of equations 3x – 5y + z = 10 and x² + y² + z² = 8 3x – 5y + z = 10 x² + y² + z² = 8
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3x – 5y + z = 10
x² + y² + z² = 8
To solve the system of equations given by:
1. (3x – 5y + z = 10) (Equation 1)
2. (x^2 + y^2 + z^2 = 8) (Equation 2)
we can express (z) from Equation 1 and then substitute it into Equation 2.
### Step 1: Solve Equation 1 for (z)
From Equation 1, we can express (z) as:
[
z = 10 – 3x + 5y
]
### Step 2: Substitute (z) into Equation 2
Now substitute this expression for (z) into Equation 2:
[
x^2 + y^2 + (10 – 3x + 5y)^2 = 8
]
### Step 3: Expand and Simplify
Next, expand ((10 – 3x + 5y)^2):
[
(10 – 3x + 5y)^2 = 100 – 60x + 100y – 9x^2 + 30xy + 25y^2
]
Adding this to the first two terms (x^2) and (y^2):
[
x^2 + y^2 + 100 – 60x + 100y – 9x^2 + 30xy +