Let me extract and analyze the mathematical content from the image. One moment, please.The image contains a system of linear equations as follows:

1. $3x - 2y = 6$
2. $9 = 4x + y$
3. $9x - 2z + xy = 17$

Let’s solve this step-by-step:

Step 1: Solve the first two equations to find $x$ and $y$:

Equation 1: $3x - 2y = 6$

Rewriting: $2y = 3x - 6 \implies y = \frac{3x - 6}{2}.$

Equation 2: $9 = 4x + y$

Substitute $y = \frac{3x - 6}{2}$ into $9 = 4x + y$: $9 = 4x + \frac{3x - 6}{2}.$

Simplify: $18 = 8x + 3x - 6 \implies 18 + 6 = 11x \implies 24 = 11x \implies x = \frac{24}{11}.$

Now substitute $x = \frac{24}{11}$ back into $y = \frac{3x - 6}{2}$: $y = \frac{3 \cdot \frac{24}{11} - 6}{2} = \frac{\frac{72}{11} - \frac{66}{11}}{2} = \frac{\frac{6}{11}}{2} = \frac{3}{11}.$

So: $x = \frac{24}{11}, \quad y = \frac{3}{11}.$

Step 2: Use $x$ and $y$ to solve for $z$ in the third equation:

Equation 3: $9x - 2z + xy = 17$

Substitute $x = \frac{24}{11}$ and $y = \frac{3}{11}$: $xy = \frac{24}{11} \cdot \frac{3}{11} = \frac{72}{121}.$

Now substitute into the equation: $9 \cdot \frac{24}{11} - 2z + \frac{72}{121} = 17.$

Simplify: $\frac{216}{11} - 2z + \frac{72}{121} = 17.$

Combine terms with a common denominator. The LCM of 11 and 121 is 121: $\frac{216 \cdot 11}{121} + \frac{72}{121} - 2z = \frac{17 \cdot 121}{121}.$

Simplify: $\frac{2376 + 72}{121} - 2z = \frac{2057}{121}.$

$\frac{2448}{121} - 2z = \frac{2057}{121}.$

Subtract: $\frac{2448}{121} - \frac{2057}{121} = 2z.$

$\frac{391}{121} = 2z.$

Solve for $z$: $z = \frac{391}{242}.$

Final Answer:

$x = \frac{24}{11}, \quad y = \frac{3}{11}, \quad z = \frac{391}{242}.$

Would you like further clarification or a different solving method?

Related Questions:

1. How can substitution be applied effectively in systems of equations?
2. What is the importance of finding a least common denominator in rational equations?
3. Can you explain elimination as an alternative to substitution?
4. How do we interpret solutions of systems of equations geometrically?
5. What happens if a system of equations is inconsistent?

Tip:

Always check your solutions by substituting them back into the original equations to verify accuracy.