Let's go through the problem step by step to find the correct equation of Nolan's line.
1. Identify the [tex]$y$[/tex]-intercept:
- Nolan plots the [tex]$y$[/tex]-intercept of the line at [tex]\((0,3)\)[/tex]. This tells us that when [tex]\(x = 0\)[/tex], [tex]\(y = 3\)[/tex].
- In the slope-intercept form of a linear equation, [tex]\(y = mx + b\)[/tex], [tex]\(b\)[/tex] represents the [tex]$y$[/tex]-intercept. Therefore, [tex]\(b = 3\)[/tex].
2. Identify the slope:
- The problem states that Nolan uses a slope of 2. In the slope-intercept form [tex]\(y = mx + b\)[/tex], [tex]\(m\)[/tex] represents the slope. Thus, [tex]\(m = 2\)[/tex].
3. Write the equation:
- Substituting the value of the slope [tex]\(m = 2\)[/tex] and the [tex]$y$[/tex]-intercept [tex]\(b = 3\)[/tex] into the slope-intercept form [tex]\(y = mx + b\)[/tex], we get the equation:
[tex]\[
y = 2x + 3
\][/tex]
So, the equation that represents Nolan's line is:
[tex]\[ y = 2x + 3 \][/tex]
Among the given options:
- [tex]\(y = 2x + 3\)[/tex]
- [tex]\(y = 3x + 2\)[/tex]
- [tex]\(y = 3x + 5\)[/tex]
The correct equation is [tex]\(y = 2x + 3\)[/tex].
