Let’s analyze your system of equations step by step based on the image:

The System of Equations:

1. $x - 2y + z = 8$ $(1)$
2. $3x + 4y + 3z = -6$ $(2)$
3. $-x - 4y - 3z = 14$ $(3)$

Step 1 Analysis:

• You correctly combined equations (1) and (3) to eliminate $x$:
  • Adding $(1)$ and $(3)$:
    $(x - 2y + z) + (-x - 4y - 3z) = 8 + 14$ Simplifies to: -6y - 2z = 22 \quad \text{(labeled as a)}.

Now for Step 2, you need to eliminate another variable. Here’s what to do:

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Step 2 Instructions:

1. Use equations (2) and (3) to eliminate $x$:
   • Add $3 \times (1)$ to $(2)$:
     $(3x + 4y + 3z) + (-3x - 12y - 9z) = -6 + (-42)$ Simplifies to: $-8y - 6z = -48.$ Divide the entire equation by 2: -4y - 3z = -24 \quad \text{(labeled as b)}.

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Step 3:

Now you have two equations in two variables $y$ and $z$:

1. $-6y - 2z = 22$ $(a)$
2. $-4y - 3z = -24$ $(b)$

Solve this system by substitution or elimination:

• Multiply $(a)$ by 3 and $(b)$ by 2 to eliminate $z$:
  • $3(-6y - 2z) = 3(22) \rightarrow -18y - 6z = 66$,
  • $2(-4y - 3z) = 2(-24) \rightarrow -8y - 6z = -48$.

Subtract the second equation from the first: $-18y - 6z - (-8y - 6z) = 66 - (-48).$ Simplify: $-10y = 114 \quad \Rightarrow \quad y = -11.4.$

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Would you like me to finish solving for $z$ and $x$? Let me know!

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5 Follow-Up Questions:

1. Do you want me to continue solving for $z$ and $x$?
2. Would you like a detailed breakdown of the elimination steps again?
3. Should I verify your earlier step for consistency?
4. Would you like tips on solving systems using substitution instead?
5. Need help graphing or visualizing these equations?

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Tip:

When working with systems of equations, always double-check your signs to avoid small errors.