Question

Solve the following system of equations for all three variables
-5x + 7y +2z =5
2x + 7y + 6z+10
3x - 7y - 4z =-3


x = 3; y = -4; z = -4
x = 3; y = -4; z = -4

x = 3; y = 4; z = -4
x = 3; y = 4; z = -4

x = 4; y = 3; z = -4
x = 4; y = 3; z = -4

x = 3; y = -4; z = 4

Answer 1

To solve the system of equations, we begin with the equations you've provided:

1. \(-5x + 7y + 2z = 5\) (Equation 1)
2. \(2x + 7y + 6z = -10\) (Equation 2)
3. \(3x - 7y - 4z = -3\) (Equation 3)

It seems there was a slight formatting issue with Equation 2. I will correct it to \(2x + 7y + 6z = -10\).

Let's solve the system step by step by using the method of elimination or substitution.

### Step 1: Express one variable in terms of others
From Equation 1:

\[
-5x + 7y + 2z = 5 \quad \Rightarrow \quad 2z = 5 + 5x - 7y \quad \Rightarrow \quad z = \frac{5 + 5x - 7y}{2}
\]

### Step 2: Substitute into other equations
Substitute \(z\) from above into Equation 2:

\[
2x + 7y + 6\left(\frac{5 + 5x - 7y}{2}\right) = -10
\]

This simplifies to:

\[
2x + 7y + 3(5 + 5x - 7y) = -10
\]
\[
2x + 7y + 15 + 15x - 21y = -10
\]
\[
17x - 14y + 15 = -10
\]
\[
17x - 14y = -25 \quad \Rightarrow \quad 17x = 14y - 25 \quad \Rightarrow \quad x = \frac{14y - 25}{17} \quad \text{(Equation 5)}
\]

Next, substitute \(z\) from Equation 1 into Equation 3:

\[
3x - 7y - 4\left(\frac{5 + 5x - 7y}{2}\right) = -3
\]
\[
3x - 7y - 2(5 + 5x - 7y) = -3
\]
This simplifies to:
\[
3x - 7y - 10 - 10x + 14y = -3
\]
\[
-7x + 7y - 10 = -3
\]
\[
-7x + 7y = 7 \quad \Rightarrow \quad -x + y = 1 \quad \Rightarrow \quad y = x + 1 \quad \text{(Equation 6)}
\]

### Step 3: Combine Equations 5 and 6
Substitute Equation 6 into Equation 5:

\[
x = \frac{14(x + 1) - 25}{17}
\]
\[
x = \frac{14x + 14 - 25}{17}
\]
\[
17x = 14x - 11
\]
\[
3x = -11 \quad \Rightarrow \quad x = -\frac{11}{3}
\]

### Step 4: Find y and z
Substituting \(x\) back into Equation 6:

\[
y = -\frac{11}{3} + 1 = -\frac{11}{3} + \frac{3}{3} = -\frac{8}{3}
\]

And now we find \(z\) using either of the equations. Let's use Equation 1:

\[
-5\left(-\frac{11}{3}\right) + 7\left(-\frac{8}{3}\right) + 2z = 5
\]
This gives:
\[
\frac{55}{3} - \frac{56}{3} + 2z = 5 \Rightarrow -\frac{1}{3} + 2z = 5
\]
\[
2z = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \Rightarrow z = \frac{8}{3}
\]

### Final Solution
The values we found are:

\[
x = -\frac{11}{3}, \quad y = -\frac{8}{3}, \quad z = \frac{8}{3}
\]

None of the provided solutions \( (3, -4, -4); (3, 4, -4); (4, 3, -4); (3, -4, 4) \) match our computed values. Thus, it's possible that the provided solutions are incorrect or not derived from the given equations.

To validate, you can plug in these values back into the original equations to see if they satisfy all three equations.