# 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)?

Apr 6, 2018

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

#### Explanation:

The rate of change, or the slope, is calculated as $\frac{\Delta y}{\Delta x}$, or $\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$. Plugging in, $m = \frac{7 - 5}{2 - 1} = 2$.

We'll next find the initial value. The slope-intercept equation of a line is $y = m x + b$, 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: $y = 2 x + b$

Plug In:
$y = 2 x + b$
$5 = 2 \left(1\right) + b$
$5 = 2 + b$
$3 = b$

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