Question: Answer the following questions. 1) Choose if the linear representation is proportional or…

Answer the following questions.

1. Choose if the linear representation is proportional or nonproportional.

\[ \begin{array}{c|ccccc} x & -16 & -12 & -8 & -4 & 0 \\ \hline y & 4 & 3 & 2 & 1 & 0 \end{array} \]

• nonproportional
• proportional

2. Choose if the linear representation is proportional or nonproportional.

\[ y = \frac{7}{9}x - 18 \]

• proportional
• nonproportional

Solution

1. To determine if the linear representation is proportional, the ratio \( \frac{y}{x} \) must be constant for all pairs. Calculate the ratio for each pair of \((x, y)\): When \( x = -16 \), \( y = 4 \): \[ \frac{4}{-16} = -\frac{1}{4} \] When \( x = -12 \), \( y = 3 \): \[ \frac{3}{-12} = -\frac{1}{4} \] When \( x = -8 \), \( y = 2 \): \[ \frac{2}{-8} = -\frac{1}{4} \] When \( x = -4 \), \( y = 1 \): \[ \frac{1}{-4} = -\frac{1}{4} \] When \( x = 0 \), \( y = 0 \): \[ \frac{0}{0} \quad \text{(undefined but zero matches zero)} \] Since the ratios are consistent and include zero for both, the relationship is proportional. The correct choice is: proportional. 2) For the equation \( y = \frac{7}{9}x - 18 \), a linear representation is proportional if it is in the form \( y = kx \) with a zero constant term. Here, the constant term is \(-18\). Thus, the relationship is nonproportional. The correct choice is: nonproportional.