Writing an Equation for a Linear Function Given Two Points
If $f$ is a linear function, with $f(3)=−2$, and $f(8)=1$, find an equation for the function in slope-intercept form.
We can write the given points using coordinates. \begin{align*} f(3) & = −2 \to (3,−2)\\ f(8) & = 1 \to (8,1) \end{align*}
We can then use the points to calculate the slope.
\begin{align*} m & = \frac{y_2 - y_1}{x_2 - x_1}\\ & = \frac{1 - (-2)}{8 - 3}\\ & = \frac{3}{5} \end{align*}
Substitute the slope and the coordinates of ONE OF THE POINTS into the point-slope form.
*The book decided to use $(3, -2)$ whereas I decided to use $(8, 1)$.
\begin{align*} y - y_1 & = m(x - x_1)\\ y - (-2) & = \frac{3}{5}(x - 3) \end{align*}
The book goes further in their example.
We can use algebra to rewrite the equation in the slope-intercept form.
\begin{align*} y + 2 & = \frac{3}{5}(x - 3)\\ y + 2 & = \frac{3}{5}x - \frac{9}{5}\\ y & = \frac{3}{5}x - \frac{19}{5} \end{align*}
The points I chose to use $(8, 1)$, which then gave me the answer
\begin{align*} y - 1 & = \frac{3}{5}(x - 8)\\ y & = \frac{3}{5}x - 5 \end{align*}
Because of choosing different points I get what seems a different slope-intercept even when it states "Substitute the slope and the coordinates of ONE OF THE POINTS into the point-slope form.". Is this ok?
It doesn't matter which point one picks, you will get the same solution.
$$y-1=\frac35 \left( x-8\right)$$
$$y=\frac35x - \frac{24}5+1=\frac35x-\frac{24-5}{5}=\frac35x-\frac{19}{5}$$
which is consistent with the other solution.