To find the equation of a line that passes through two points, we need to determine the slope and the y-intercept of the line. The equation of the line can be written in the slope-intercept form: [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's go through the steps:
### Step 1: Calculate the slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates of the given points [tex]\((-5, -5)\)[/tex] and [tex]\((1, -2)\)[/tex]:
[tex]\[
m = \frac{-2 - (-5)}{1 - (-5)} = \frac{-2 + 5}{1 + 5} = \frac{3}{6} = 0.5
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( 0.5 \)[/tex].
### Step 2: Calculate the y-intercept
To find the y-intercept [tex]\( b \)[/tex], we use the slope-intercept form of the line equation [tex]\( y = mx + b \)[/tex]. We can substitute the coordinates of one of the points and the slope into this equation. Let's use the point [tex]\((-5, -5)\)[/tex]:
[tex]\[
y = mx + b \implies -5 = 0.5 \cdot (-5) + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
-5 = -2.5 + b \implies b = -5 + 2.5 \implies b = -2.5
\][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\( -2.5 \)[/tex].
### Step 3: Write the equation of the line
Now that we have both the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex], we can write the equation of the line:
[tex]\[
y = 0.5x - 2.5
\][/tex]
So, the equation of the line that passes through the points [tex]\((-5, -5)\)[/tex] and [tex]\((1, -2)\)[/tex] is:
[tex]\[ y = 0.5x - 2.5 \][/tex]
