To determine the equation of the line that Nolan drew, we need to use the information provided: the [tex]\( y \)[/tex]-intercept and the slope of the line.
1. Identify the [tex]\( y \)[/tex]-intercept:
- The [tex]\( y \)[/tex]-intercept is the point where the line crosses the [tex]\( y \)[/tex]-axis.
- According to the problem, the [tex]\( y \)[/tex]-intercept is at [tex]\((0, 3)\)[/tex].
- This means when [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex].
2. Determine the slope:
- The slope ([tex]\( m \)[/tex]) of the line is given as 2.
- The slope tells us how steep the line is and the direction it goes. A slope of 2 means that for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units.
3. Writing the equation:
- The general form of the equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
- Substituting the given slope and [tex]\( y \)[/tex]-intercept into the equation:
[tex]\[
y = 2x + 3
\][/tex]
Hence, the equation that represents Nolan's line is:
[tex]\[
y = 2x + 3
\][/tex]
Among the given choices:
1. [tex]\( y = 2x + 1 \)[/tex]
2. [tex]\( y = 2x + 3 \)[/tex]
3. [tex]\( y = 3x + 2 \)[/tex]
4. [tex]\( y = 3x + 5 \)[/tex]
The correct answer is:
[tex]\[
\boxed{y = 2x + 3}
\][/tex]
