To determine the equation of the line $R$ in the form $y = mx + c$, we need to find:

1. The slope ($m$).
2. The $y$-intercept ($c$).

Step 1: Determine the slope ($m$)

The slope of a line is calculated as:

$m = \frac{\Delta y}{\Delta x}$

From the graph:

• The line passes through two clear points: $(-2, -3)$ and $(2, 3)$.

Using these points: $\Delta y = 3 - (-3) = 6, \quad \Delta x = 2 - (-2) = 4$

So the slope is: $m = \frac{\Delta y}{\Delta x} = \frac{6}{4} = \frac{3}{2}$

Step 2: Find the $y$-intercept ($c$)

The $y$-intercept is the point where the line crosses the $y$-axis. From the graph, the line crosses the $y$-axis at $(0, -1)$.

Thus: $c = -1$

Final Equation

Substitute $m = \frac{3}{2}$ and $c = -1$ into $y = mx + c$:

$y = \frac{3}{2}x - 1$

Final Answer:

$m = \frac{3}{2}, \quad c = -1$

Would you like me to explain any steps further or explore related problems?

Related Questions:

1. How do you find the slope if no points are explicitly given?
2. What is the general method to verify a line equation using a point?
3. Can you calculate the slope if one of the points is fractional or a decimal?
4. What is the meaning of the $y$-intercept in real-world applications?
5. How would the equation change if the line was vertical?

Tip:

For any line, the slope can also be visualized as "rise over run" on a graph—count the vertical and horizontal steps between two points.