Question

What is the rate of change, the initial value, and the equation of the line that goes through the points `(1, 5)` and `(2,7)`?

Answer 1

Rate of Change: 2
Initial Value: 3
Equation: `y = 2x + 3`

Explanation:

The rate of change, or the slope, is calculated as `(Delta y) / (Delta x)`, or `(y_2 - y_1) / (x_2 - x_1)`. Plugging in, `m = (7 - 5) / (2 - 1) = 2`.

We'll next find the initial value. The slope-intercept equation of a line is `y = mx + b`, where `m` is the slope and `b` is the initial value. We have found the slope as `m = 2`. We will plug our first point into the equation to give us `b`, the initial value.

Point: (1, 5)
Equation: `y = 2x + b`

Plug In:
`y = 2x + b`
`5 = 2(1) + b`
`5 = 2 + b`
`3 = b`

Thus, our initial value is `b = 3`. This value, with our slope, gives us the equation: `y = 2x + 3`.