To determine the equation of the line that passes through the points [tex]\((-2, -2)\)[/tex] and [tex]\( (5, -5) \)[/tex], we follow these steps:
1. Calculate the slope [tex]\( m \)[/tex]:
The formula for the slope [tex]\( m \)[/tex] 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]
For the given points [tex]\((-2, -2)\)[/tex] and [tex]\( (5, -5) \)[/tex]:
[tex]\[
m = \frac{-5 - (-2)}{5 - (-2)} = \frac{-5 + 2}{5 + 2} = \frac{-3}{7} = -\frac{3}{7}
\][/tex]
2. Find the y-intercept [tex]\( b \)[/tex]:
The equation of the line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
To find [tex]\( b \)[/tex], we can use the slope [tex]\( m \)[/tex] and one of the given points. Let's use the point [tex]\( (-2, -2) \)[/tex] for this purpose:
[tex]\[
y_1 = mx_1 + b
\][/tex]
Substituting the values [tex]\((-2, -2)\)[/tex] and [tex]\( m = -\frac{3}{7} \)[/tex]:
[tex]\[
-2 = -\frac{3}{7}(-2) + b
\][/tex]
Simplify the equation:
[tex]\[
-2 = \frac{6}{7} + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = -2 - \frac{6}{7} = -\frac{14}{7} - \frac{6}{7} = -\frac{20}{7}
\][/tex]
3. Form the equation of the line:
Now that we have the slope [tex]\( m = -\frac{3}{7} \)[/tex] and the y-intercept [tex]\( b = -\frac{20}{7} \)[/tex], we can write the equation of the line:
[tex]\[
y = -\frac{3}{7}x - \frac{20}{7}
\][/tex]
From the given options, the one that matches this equation is:
d. [tex]\( y = -\frac{3}{7}x - \frac{20}{7} \)[/tex]
Thus, the equation of the line that passes through the points [tex]\((-2, -2)\)[/tex] and [tex]\( (5, -5) \)[/tex] is:
d. [tex]\( y = -\frac{3}{7} x - \frac{20}{7} \)[/tex].
