To determine the [tex]$y$[/tex]-intercept of a line given its slope and a point it passes through, we use the point-slope form of a linear equation, which can help us identify the y-intercept directly.
The general equation for a line is:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept,
- [tex]\( (x, y) \)[/tex] is any point on the line.
Given:
- The slope [tex]\( m \)[/tex] is [tex]\(-3\)[/tex],
- The line passes through the point [tex]\((0, -7)\)[/tex].
When [tex]\( x = 0 \)[/tex], the term [tex]\( mx \)[/tex] becomes [tex]\( 0 \)[/tex]. This simplifies the equation [tex]\( y = mx + b \)[/tex] to:
[tex]\[ y = b \][/tex]
Since the line passes through the point [tex]\((0, -7)\)[/tex], substituting this into the equation gives:
[tex]\[ -7 = b \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] of the line is:
[tex]\[ b = -7 \][/tex]
Therefore, the [tex]$y$[/tex]-intercept of the line is [tex]\(-7\)[/tex].
From the given choices:
- [tex]\(-7\)[/tex]
- [tex]\(-3\)[/tex]
- [tex]\(0\)[/tex]
- [tex]\(4\)[/tex]
The correct answer is [tex]\(\boxed{-7}\)[/tex].
