Question

Solve the system of equations
X plus Y plus z equals -10
Negative 2X minus Y plus Z equals five
X minus 2Y minus Z equals seven

Answer 1

To solve the system of equations:

1. \( X + Y + Z = -10 \) (Equation 1)
2. \( -2X - Y + Z = 5 \) (Equation 2)
3. \( X - 2Y - Z = 7 \) (Equation 3)

Let's solve these equations step-by-step.

### Step 1: Eliminate Z
We can express \( Z \) in terms of \( X \) and \( Y \) from Equation 1:
\[
Z = -10 - X - Y
\]

Now, substitute this expression for \( Z \) into Equations 2 and 3.

### Substitute Z into Equation 2:
\[
-2X - Y + (-10 - X - Y) = 5
\]
Combine like terms:
\[
-2X - Y - 10 - X - Y = 5
\]
\[
-3X - 2Y - 10 = 5
\]
Now add 10 to both sides:
\[
-3X - 2Y = 15 \quad \text{(Equation 4)}
\]

### Substitute Z into Equation 3:
\[
X - 2Y - (-10 - X - Y) = 7
\]
Combine like terms:
\[
X - 2Y + 10 + X + Y = 7
\]
\[
2X - Y + 10 = 7
\]
Now subtract 10 from both sides:
\[
2X - Y = -3 \quad \text{(Equation 5)}
\]

### Step 2: Solve Equations 4 and 5
Now we have a new system:

1. \( -3X - 2Y = 15 \) (Equation 4)
2. \( 2X - Y = -3 \) (Equation 5)

Let's solve Equation 5 for \( Y \):
\[
Y = 2X + 3 \quad \text{(Equation 6)}
\]

### Substitute Equation 6 into Equation 4:
Now substitute \( Y \) from Equation 6 into Equation 4:
\[
-3X - 2(2X + 3) = 15
\]
Expand and simplify:
\[
-3X - 4X - 6 = 15
\]
Combine like terms:
\[
-7X - 6 = 15
\]
Add 6 to both sides:
\[
-7X = 21
\]
Divide by -7:
\[
X = -3
\]

### Step 3: Find Y and Z
Now that we have \( X = -3 \), substitute this back into Equation 6 to find \( Y \):
\[
Y = 2(-3) + 3 = -6 + 3 = -3
\]

Next, we substitute \( X \) and \( Y \) into Equation 1 to find \( Z \):
\[
Z = -10 - (-3) - (-3)
\]
\[
Z = -10 + 3 + 3 = -4
\]

### Final Solution
The solution to the system of equations is:
\[
X = -3, \quad Y = -3, \quad Z = -4
\]

Thus, the final answer is:
\[
\boxed{(-3, -3, -4)}
\]