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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?

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    $\begingroup$ How did you get $-5$ in your answer? Please check/show your arithmetic. $\endgroup$ – Michael Burr Sep 12 at 10:19
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    $\begingroup$ Short answer, no, you should always get the same $y$-intercept. You have an arithmetic error. $\endgroup$ – Michael Burr Sep 12 at 10:21
  • $\begingroup$ Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig Sep 12 at 10:58
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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.

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