To determine the [tex]\( y \)[/tex]-intercept of the line, we need to find the equation of the line in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the [tex]\( y \)[/tex]-intercept.
First, let's calculate the slope [tex]\( m \)[/tex]. The slope [tex]\( m \)[/tex] of the line that passes 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]
Let's use the points [tex]\((-28, -54)\)[/tex] and [tex]\((-21, -40)\)[/tex] from the table to find the slope.
[tex]\[
m = \frac{-40 - (-54)}{-21 - (-28)} = \frac{-40 + 54}{-21 + 28} = \frac{14}{7} = 2
\][/tex]
Now we know the slope [tex]\( m \)[/tex] is 2.
Next, we need to find the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]. We can use any of the given points and plug them into the line equation [tex]\( y = mx + b \)[/tex] to solve for [tex]\( b \)[/tex]. Let's use the point [tex]\((-28, -54)\)[/tex].
Starting with the slope-intercept form:
[tex]\[
y = mx + b
\][/tex]
Substitute [tex]\( y = -54 \)[/tex], [tex]\( m = 2 \)[/tex], and [tex]\( x = -28 \)[/tex]:
[tex]\[
-54 = 2(-28) + b
\][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[
-54 = -56 + b
\][/tex]
[tex]\[
b = -54 + 56
\][/tex]
[tex]\[
b = 2
\][/tex]
So, the [tex]\( y \)[/tex]-intercept of the line is [tex]\( 2 \)[/tex].
Thus, the [tex]\( y \)[/tex]-intercept is
[tex]\[
\boxed{2}
\][/tex]
