To find the equation relating $x$ and $y$ for the points on this line, we can use the given points on the line to determine the slope and then form the linear equation.

Step-by-Step Solution

1. Identify Points on the Line: We have two specific points on the line:

   • $(2, -4)$
   • $(10, 3)$ (point $C$)

2. Calculate the Slope: The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using the points $(2, -4)$ and $(10, 3)$: $m = \frac{3 - (-4)}{10 - 2} = \frac{3 + 4}{8} = \frac{7}{8}$

3. Form the Equation: We can use the point-slope form of a line: $y - y_1 = m(x - x_1)$ Using the point $(2, -4)$ and the slope $\frac{7}{8}$: $y - (-4) = \frac{7}{8}(x - 2)$ Simplify this to get: $y + 4 = \frac{7}{8}x - \frac{7}{4}$ Subtract 4 from both sides to isolate $y$: $y = \frac{7}{8}x - \frac{7}{4} - 4$ Converting $4$ to a fraction with denominator 4: $y = \frac{7}{8}x - \frac{7}{4} - \frac{16}{4}$ $y = \frac{7}{8}x - \frac{23}{4}$

Final Equation

The equation of the line relating $x$ and $y$ is: $y = \frac{7}{8}x - \frac{23}{4}$

Would you like a detailed explanation on any part of this solution or have further questions?

Related Questions

1. How can you convert the slope-intercept form to the standard form of a line?
2. What are some ways to check if a point lies on a line given its equation?
3. How do you interpret the slope in real-world scenarios?
4. What other forms of line equations exist besides slope-intercept and point-slope?
5. Can you find an equation of a line perpendicular to this one?

Tip:

Always double-check your calculations, especially when converting between fraction forms, as small errors can lead to incorrect equations.