Determine the Equation of a Line Through Points (8, 3) and (2, 7)

To find the equation of the line passing through the points $(8,3)$ and $(2,7)$, follow these steps:

• Step 1: Calculate the slope $m$.
• Step 2: Use the slope and one of the points to solve for the y-intercept $b$.
• Step 3: Write the equation of the line in the form $y = mx + b$.

Step 1: Calculate the slope $m$. The slope $m$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{2 - 8} = \frac{4}{-6} = -\frac{2}{3}$.

Step 2: With the slope known, use one point (for example, $(8,3)$) to find $b$ in the slope-intercept form $y = mx + b$. Substitute the values:

$3 = -\frac{2}{3}(8) + b$.

This simplifies to:

$3 = -\frac{16}{3} + b$.

Add $\frac{16}{3}$ to both sides to solve for $b$:

$b = 3 + \frac{16}{3} = \frac{9}{3} + \frac{16}{3} = \frac{25}{3}$.

Step 3: Write the equation using the calculated slope and y-intercept:

$y = -\frac{2}{3}x + \frac{25}{3}$.

To express it as a mixed number, $\frac{25}{3}$ is $8\frac{1}{3}$, so:

$y = -\frac{2}{3}x + 8\frac{1}{3}$.

Thus, the correct equation of the line is $y = -\frac{2}{3}x + 8\frac{1}{3}$, which corresponds to choice 3.

The final answer is: $y = -\frac{2}{3}x + 8\frac{1}{3}$.