From the image, the system of equations is:

$\text{Equation 1: } 2x + 3y + 5z = 9$ $\text{Equation 2: } 7x + 3y - 2z = 8$ $\text{Equation 3: } 2x + 3y + 1z = \lambda$

We need to find the values of $\lambda$ and the unknown variables $x$, $y$, and $z$.

Steps:

1. Subtract Equation 3 from Equation 1 to simplify and eliminate $2x + 3y$.
2. Similarly, manipulate other equations to solve for $x$, $y$, and $z$.
3. Finally, substitute the values back to find $\lambda$.

Let me calculate this step by step.The solutions for the variables $x$, $y$, and $z$ in terms of $\lambda$ are:

$x = \frac{59}{20} - \frac{7\lambda}{20}$ $y = \frac{13\lambda}{20} - \frac{163}{60}$ $z = \frac{9}{4} - \frac{\lambda}{4}$

These are the expressions for $x$, $y$, and $z$ in terms of $\lambda$. You can now choose a specific value for $\lambda$ if needed to find exact numerical values for $x$, $y$, and $z$.

Would you like to assign a specific value to $\lambda$ or explore more about this system?

Follow-up questions:

1. What happens if $\lambda = 0$?
2. How do the values of $x$, $y$, and $z$ change as $\lambda$ increases?
3. Can you derive the relationship between $x$ and $y$ independent of $\lambda$?
4. Is there any special case when the system has no solution?
5. What does the value of $z$ represent geometrically in this system?

Tip: You can substitute different values for $\lambda$ to explore how it affects the system.