Question

The number y of miles you run after x weeks is represented by the equation y = 8x. Graph the equation and interpret the slope.

A) The slope represents the total number of miles you've run over the x weeks.
B) The slope represents the number of weeks it takes to run 8 miles.
C) The slope represents the rate at which the number of miles you run increases per week.
D) The slope represents the initial number of miles you start with.

Answer 1

The correct interpretation of the slope in the equation
`\( y = 8x \)` is:

C) The slope represents the rate at which the number of miles you run increases per week.

The equation
`\( y = 8x \)` describes a linear relationship between the number of miles you run
`\( y \)` and the number of weeks
`\( x \)`. The slope of this line is the coefficient of
`\( x \)`, which is 8 in this case.

Now, let's interpret the slope:

- The slope in a linear equation
`\( y = mx + b \)` represents the rate of change of
`\( y \)` with respect to
`\( x \)`. In this context, it indicates how many miles you run increase for each additional week.

- Option A ("The slope represents the total number of miles you've run over the x weeks.") is not correct because the slope represents the rate of increase per week, not the total number of miles run.

- Option B ("The slope represents the number of weeks it takes to run 8 miles.") is incorrect. It's the reverse; the slope represents the number of miles run in one week.

- Option C ("The slope represents the rate at which the number of miles you run increases per week.") is correct. It means that for each additional week, you run 8 more miles.

- Option D ("The slope represents the initial number of miles you start with.") is incorrect. The initial number of miles would be the y-intercept in a linear equation (the value of
`\( y \)` when
`\( x = 0 \)`), not the slope.

Therefore, the correct interpretation of the slope in the equation
`\( y = 8x \)` is:

C) The slope represents the rate at which the number of miles you run increases per week.

Regarding the graph, the line would pass through the origin (0,0) and would have a slope of 8, meaning it rises 8 units vertically for every 1 unit it moves horizontally.