The image shows the graph of a linear function $g$ that passes through the points $(-7, -4)$ and $(7, 6)$. We can determine the equation of the line by calculating the slope and using the point-slope form.

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:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Here, the points are:

• $(x_1, y_1) = (-7, -4)$
• $(x_2, y_2) = (7, 6)$

Substitute these values:

$m = \frac{6 - (-4)}{7 - (-7)} = \frac{6 + 4}{7 + 7} = \frac{10}{14} = \frac{5}{7}$

So, the slope of the line is $\frac{5}{7}$.

Step 2: Use point-slope form to find the equation

The point-slope form of a line is given by:

$y - y_1 = m(x - x_1)$

Substituting one of the points, say $(-7, -4)$, and the slope $m = \frac{5}{7}$:

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

Simplify:

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

Distribute the slope:

$y + 4 = \frac{5}{7}x + 5$

Now, subtract 4 from both sides:

$y = \frac{5}{7}x + 5 - 4$

$y = \frac{5}{7}x + 1$

Final Equation of the Line:

The equation of the line is:

$y = \frac{5}{7}x + 1$

Let me know if you would like further clarification on any part of the solution.

Here are some related questions for practice:

1. How do you find the equation of a line when given only its slope and a point on the line?
2. Can you find the y-intercept of the line by just looking at the graph?
3. What happens to the graph of the line if the slope is negative instead of positive?
4. How do you calculate the slope if the coordinates of the points are fractions?
5. What is the meaning of the slope and y-intercept in a real-world context?

Tip: Always check your work by substituting the points back into the equation to ensure they satisfy the equation of the line.