To determine the correct answer, let's work through finding the linear equation that goes through the points $(4, 3)$ and $(8, 2)$ in standard form.

Step 1: Find the Slope

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting $(x_1, y_1) = (4, 3)$ and $(x_2, y_2) = (8, 2)$: $m = \frac{2 - 3}{8 - 4} = \frac{-1}{4} = -\frac{1}{4}$

Step 2: Use Point-Slope Form

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using the point $(4, 3)$ and the slope $-\frac{1}{4}$: $y - 3 = -\frac{1}{4}(x - 4)$

Step 3: Convert to Standard Form

We need to rearrange this equation into standard form $Ax + By = C$, where $A$, $B$, and $C$ are integers.

Starting with: $y - 3 = -\frac{1}{4}(x - 4)$ Multiply both sides by 4 to eliminate the fraction: $4(y - 3) = -(x - 4)$ Expanding both sides: $4y - 12 = -x + 4$ Rearrange to get $x + 4y = 16$: $x + 4y = 16$

Conclusion

The correct answer is: C. $x + 4y = 16$

Would you like more details, or do you have any questions?

Here are some related questions for further practice:

1. How would you find the equation if the points were $(2, 5)$ and $(6, 7)$?
2. What is the slope of a line passing through $(3, -1)$ and $(5, 4)$?
3. How can you convert a line in slope-intercept form $y = mx + b$ to standard form?
4. What are the key differences between standard form and point-slope form?
5. How would you find the x-intercept and y-intercept of the equation $x + 4y = 16$?

Tip: Always check the slope first when determining the equation of a line through two points, as it can guide you to the correct form and answer.