Correct Answer: C) 5 ### Step-by-Step Solution #### Step 1: Formulate the System of Linear Equations Let the price of an Executive meal package be $x$. Let the price of a Premium meal package be $y$. Let the price of a Standard meal package be $z$. Based on the billing invoices provided for the three conferences, we can construct a system of three linear equations with three variables: $$\begin{aligned} 2x + y + z &= 2400 \quad \text{--- (Equation 1)} \ x + 2y + 3z &= 3200 \quad \text{--- (Equation 2)} \ 5x + 4y + kz &= 8000 \quad \text{--- (Equation 3)} \end{aligned}$$ #### Step 2: Establish Conditions for Infinite Solutions For a system of linear equations to possess an infinite number of solutions, the equations must be linearly dependent. This means that one of the equations can be expressed as a linear combination of the other two. Let Equation 3 be a linear combination of Equation 1 and Equation 2 using the constants $m$ and $n$: $$\text{Equation 3} = m(\text{Equation 1}) + n(\text{Equation 2})$$ Substituting the algebraic expressions: $$5x + 4y + kz = m(2x + y + z) + n(x + 2y + 3z)$$ Expanding and grouping the terms on the right-hand side by their respective variables: $$5x + 4y + kz = (2m + n)x + (m + 2n)y + (m + 3n)z$$ #### Step 3: Solve for the Scaling Coefficients By equating the corresponding coefficients of $x$ and $y$ from both sides of the equation, we obtain a structural sub-system: $$\begin{array}{|c|c|} \hline \text{Variable} & \text{Coefficients Equality Relation} \ \hline x & 2m + n = 5 \ \hline y & m + 2n = 4 \ \hline \end{array}$$ We can solve this two-variable linear system using elimination. Multiply the second relation by 2: $$2m + 4n = 8$$ Now, subtract the first coefficient relation ($2m + n = 5$) from this updated equation: $$(2m + 4n) - (2m + n) = 8 - 5$$ $$3n = 3 \implies n = 1$$ Substitute $n = 1$ back into the second relation to isolate $m$: $$m + 2(1) = 4 \implies m = 2$$ #### Step 4: Verify the Constants against the Total Revenue Vector Before calculating $k$, we must ensure that these scaling factors ($m = 2, n = 1$) remain completely consistent with the absolute constant terms (the invoice values) on the right-hand side of our original system: $$\text{Constant Value} = m(2400) + n(3200)$$ $$\text{Constant Value} = 2(2400) + 1(3200) = 4800 + 3200 = 8000$$ Since $8000 = 8000$, the system is confirmed to be consistent and dependent, validating our values for $m$ and $n$. #### Step 5: Evaluate the Unknown Coordinate $k$ Now, equate the coefficients of the variable $z$ from both sides of our linear combination model: $$k = m + 3n$$ Substitute the known values of $m = 2$ and $n = 1$ into this final expression: $$k = 2 + 3(1)$$ $$k = 2 + 3$$ $$k = 5$$ Therefore, for the corporate catering system to sustain an infinite number of solution sets, the value of $k$ must be exactly 5.