To write an equation for the relationship between the number of parts \( y \) produced by the machine and the number of hours \( z \), we can observe the data points given:
- When \( z = 0 \), \( y = 0 \)
- When \( z = 1 \), \( y = 125 \)
- When \( z = 2 \), \( y = 250 \)
- When \( z = 3 \), \( y = 375 \)
- When \( z = 4 \), \( y = 500 \)
From this information, we can determine that for every hour \( z \), the machine produces 125 parts. This shows the relationship is linear.
To formulate the equation, we'll use the slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
Where:
- \( m \) is the slope (the rate of change)
- \( b \) is the y-intercept (the value of \( y \) when \( z = 0 \))
In this case:
- The slope \( m = 125 \) (the number of parts produced per hour).
- The y-intercept \( b = 0 \) (when no hours have passed, no parts are produced).
Thus, the equation relating \( y \) and \( z \) is:
\[ y = 125z \]
Now, let's determine if the relationship is proportional or nonproportional. A relationship is proportional if it passes through the origin (0,0) and can be written in the form \( y = kx \) (where \( k \) is a constant).
Since our equation is \( y = 125z \) and it passes through the origin, this indicates that the relationship is indeed proportional.
In summary:
Equation: \( y = 125z \)
Type: Proportional.
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