To determine the equation of the line that passes through the points [tex]\((-5, -1)\)[/tex] and [tex]\( (5, 5) \)[/tex], we can follow these steps:
1. Calculate the slope (m):
The formula for the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates of our points:
[tex]\[
m = \frac{5 - (-1)}{5 - (-5)} = \frac{5 + 1}{5 + 5} = \frac{6}{10} = \frac{3}{5}
\][/tex]
2. Find the y-intercept (b):
The slope-intercept form of a line is defined as:
[tex]\[
y = mx + b
\][/tex]
To find [tex]\( b \)[/tex], we use one of the points and the calculated slope. We will use the point [tex]\((-5, -1)\)[/tex]:
[tex]\[
y = mx + b \implies -1 = \frac{3}{5} \cdot (-5) + b
\][/tex]
Then, solve for [tex]\( b \)[/tex]:
[tex]\[
-1 = \frac{3}{5} \cdot (-5) + b \implies -1 = -3 + b
\][/tex]
[tex]\[
-1 + 3 = b \implies b = 2
\][/tex]
3. Write the equation of the line:
Using the slope [tex]\( \frac{3}{5} \)[/tex] and the y-intercept [tex]\( 2 \)[/tex], the equation of the line is:
[tex]\[
y = \frac{3}{5} x + 2
\][/tex]
Now we compare this equation with the given options and find that:
[tex]\[
\boxed{y = \frac{3}{5} x + 2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
