Question

The table shows the number of parts, y, a machine produces each hour, z. Write an equation for this situation, and indicate if this is proportional or nonproportional.

y (number of parts), z (number of hours)
(0,0)
(1, 125)
(2, 250)
(3, 375)
(4, 500)

Answer 1

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.