To solve this problem, we need to determine the unit rate (slope) of multiple linear equations given in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
Given the equations:
1. [tex]\( y = \frac{4}{3}x - \frac{5}{3} \)[/tex]
2. [tex]\( y = \frac{5}{4}x - 3 \)[/tex]
3. [tex]\( y = -2x + \frac{17}{3} \)[/tex]
4. [tex]\( y = \frac{7}{4}x - \frac{9}{4} \)[/tex]
5. [tex]\( y = \frac{6}{3}x - \frac{3}{6} \)[/tex]
6. [tex]\( y = \frac{8}{5}x - \frac{1}{6} \)[/tex]
### Step-by-Step Solution:
1. Identify the slopes:
- For the equation [tex]\( y = \frac{4}{3}x - \frac{5}{3} \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{4}{3} \)[/tex].
- For the equation [tex]\( y = \frac{5}{4}x - 3 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{5}{4} \)[/tex].
- For the equation [tex]\( y = -2x + \frac{17}{3} \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( -2 \)[/tex].
- For the equation [tex]\( y = \frac{7}{4}x - \frac{9}{4} \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{7}{4} \)[/tex].
- For the equation [tex]\( y = \frac{6}{3}x - \frac{3}{6} \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{6}{3} \)[/tex], which simplifies to [tex]\( 2 \)[/tex].
- For the equation [tex]\( y = \frac{8}{5}x - \frac{1}{6} \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{8}{5} \)[/tex].
2. List the slopes:
- [tex]\( \frac{4}{3} \)[/tex]
- [tex]\( \frac{5}{4} \)[/tex]
- [tex]\( -2 \)[/tex]
- [tex]\( \frac{7}{4} \)[/tex]
- [tex]\( 2 \)[/tex]
- [tex]\( \frac{8}{5} \)[/tex]
3. Determine the largest unit rate (highest slope):
- Among the values [tex]\( \frac{4}{3} \approx 1.33 \)[/tex], [tex]\( \frac{5}{4} = 1.25 \)[/tex], [tex]\( -2 \)[/tex], [tex]\( \frac{7}{4} = 1.75 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( \frac{8}{5} = 1.6 \)[/tex], the largest value is [tex]\( 2 \)[/tex].
4. Determine the smallest unit rate (lowest slope):
- Among the values [tex]\( \frac{4}{3} \approx 1.33 \)[/tex], [tex]\( \frac{5}{4} = 1.25 \)[/tex], [tex]\( -2 \)[/tex], [tex]\( \frac{7}{4} = 1.75 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( \frac{8}{5} = 1.6 \)[/tex], the smallest value is [tex]\( -2 \)[/tex].
### Conclusion:
- The largest unit rate (slope) is [tex]\( 2 \)[/tex].
- The smallest unit rate (slope) is [tex]\( -2 \)[/tex].
These results can be recorded in the provided table as follows:
[tex]\[
\begin{array}{|l|l|}
\hline
\text{Larger Unit Rate} & \text{Smaller Unit Rate} \\
\hline
2 & -2 \\
\hline
\end{array}
\][/tex]
