Solutions to the system of equations 3x – 5y + z = 10 and x² + y² + z² = 8 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 +