To find the equation of the line passing through the points $(4, 3)$ and $(-4, 7)$ in standard form, we can proceed with the following steps:

Step 1: Find the Slope of the Line

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting $(x_1, y_1) = (4, 3)$ and $(x_2, y_2) = (-4, 7)$: $m = \frac{7 - 3}{-4 - 4} = \frac{4}{-8} = -\frac{1}{2}$

Step 2: Use Point-Slope Form to Find the Equation of the Line

With the slope $m = -\frac{1}{2}$ and one of the points (we'll use $(4, 3)$), the point-slope form of the line is: $y - y_1 = m(x - x_1)$ Substituting $m = -\frac{1}{2}$, $x_1 = 4$, and $y_1 = 3$: $y - 3 = -\frac{1}{2}(x - 4)$ Simplify this equation: $y - 3 = -\frac{1}{2}x + 2$ $y = -\frac{1}{2}x + 5$

Step 3: Convert to Standard Form

To write the equation in standard form $Ax + By = C$, we need integer coefficients for $x$ and $y$. Starting from: $y = -\frac{1}{2}x + 5$ Multiply every term by 2 to eliminate the fraction: $2y = -x + 10$ Rearrange terms to get $x + 2y = 10$.

Final Answer

The equation in standard form is: $x + 2y = 10$

Would you like further details on any of these steps?

Here are some related questions to explore:

1. How do you convert point-slope form to slope-intercept form?
2. How do you determine if two lines are parallel?
3. What is the significance of the slope in a line equation?
4. How can you verify if a point lies on a line?
5. What is the difference between standard form and slope-intercept form?

Tip: Remember that standard form $Ax + By = C$ typically has $A$, $B$, and $C$ as integers, and $A$ should be positive.