Let's solve this system of linear equations:

1. $x + y + z = -5$
2. $x + 2y + 2z = -15$
3. $2x - y + 3z = -4$

We will use either substitution or elimination to find the values of $x$, $y$, and $z$. Let's proceed step by step using the elimination method for efficiency.

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Step 1: Eliminate one variable

We subtract Equation (1) from Equation (2) to eliminate $x$.

$(x + 2y + 2z) - (x + y + z) = -15 - (-5)$

y + z = -10 \tag{4}

Now, subtract Equation (1) from Equation (3):

$(2x - y + 3z) - (x + y + z) = -4 - (-5)$

x - 2y + 2z = 1 \tag{5}

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Step 2: Solve for two variables

From Equation (4), express one variable in terms of the other:

y = -10 - z \tag{6}

Now, substitute Equation (6) into Equation (5):

$x - 2(-10 - z) + 2z = 1$

$x + 20 + 2z + 2z = 1$

x + 4z = -19 \tag{7}

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Step 3: Express $x$ in terms of $z$

From Equation (7):

x = -19 - 4z \tag{8}

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Step 4: Substitute into Equation (6)

We now substitute Equation (8) and Equation (6) into Equation (1) to solve for $z$.

$(-19 - 4z) + (-10 - z) + z = -5$

Simplify:

$-19 - 4z - 10 = -5$

$-29 - 4z = -5$

$-4z = 24 \quad \Rightarrow \quad z = -6$

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Step 5: Find $x$ and $y$

Using $z = -6$ in Equation (8):

$x = -19 - 4(-6) = -19 + 24 = 5$

Using $z = -6$ in Equation (6):

$y = -10 - (-6) = -10 + 6 = -4$

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Final Solution

The solutions are:

$x = 5, \quad y = -4, \quad z = -6$

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Do you have any questions or need further clarification?

5 Related Questions

1. How do you solve systems of equations using substitution?
2. Can this system be solved using matrix methods like Gaussian elimination?
3. How do we interpret inconsistent or dependent systems of equations?
4. What are the applications of systems of linear equations in real life?
5. How can you use software tools (e.g., Python or MATLAB) to solve systems of equations?

Tip: Always double-check your final solution by substituting the values back into the original equations.