Let's solve the problem step by step.

1. Identify the coordinates of the two points:

   • The first point (left) has the coordinates $(7, 4)$.
   • The second point (right) has the coordinates $(14, 9)$.

2. Calculate the slope (m) of the line passing through these two points: The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the coordinates of the two points $(7, 4)$ and $(14, 9)$: $m = \frac{9 - 4}{14 - 7} = \frac{5}{7}$

3. Write the equation of the line: The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using the slope $\frac{5}{7}$ and the point $(7, 4)$: $y - 4 = \frac{5}{7}(x - 7)$

4. Compare this equation with the given options: The first three options are in different forms of the point-slope and slope-intercept format:

   • $y - 6 = \frac{5}{7}(x - 7)$ → incorrect, since the point $(7, 6)$ does not match the given points.
   • $y = \frac{5}{7}x + 1$ → correct, after expanding the line equation into slope-intercept form.
   • $y - 4 = \frac{5}{7}(x - 7)$ → correct, this matches the point-slope form with the point $(7, 4)$.
   • $y = \frac{5}{7}x - 1$ → incorrect, this does not match the calculated y-intercept.
   • $y + 4 = \frac{5}{7}(x + 7)$ → incorrect, this is not in the correct form based on the given points.

Correct Answers:

• $y = \frac{5}{7}x + 1$
• $y - 4 = \frac{5}{7}(x - 7)$

Would you like any further clarification or a detailed breakdown of the solution? Here are 5 related questions to think about:

1. How do you derive the y-intercept form from the point-slope form?
2. Can you find the slope if the coordinates were reversed?
3. How does the slope impact the steepness of the line on the graph?
4. What happens if both points have the same x-coordinate?
5. Can you write the equation in standard form?

Tip: Always double-check which form of a line equation you are being asked to identify (point-slope, slope-intercept, or standard form).