To find the equation of a line that is parallel to another line and passes through a given point, follow these steps:
1. Identify the slope of the given line:
Since the problem involves finding a line parallel to a given line [tex]\( EF \)[/tex], its slope is essential. Parallel lines share the same slope. Let's start by looking at the equation of line [tex]\( EF \)[/tex]:
The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope. For the given line [tex]\( y = \frac{2}{3}x + b \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
2. Use the point-slope form of the equation of the line:
The point-slope form of a line's equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
3. Substitute the known values:
Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{2}{3} \)[/tex], and the point through which the line passes is [tex]\((2, 6)\)[/tex].
[tex]\[
y - 6 = \frac{2}{3}(x - 2)
\][/tex]
4. Solve for [tex]\( y \)[/tex] to get the slope-intercept form:
Expand and simplify the equation to find [tex]\( y \)[/tex].
[tex]\[
y - 6 = \frac{2}{3}(x - 2)
\][/tex]
[tex]\[
y - 6 = \frac{2}{3}x - \frac{4}{3}
\][/tex]
Add 6 to both sides:
[tex]\[
y = \frac{2}{3}x - \frac{4}{3} + 6
\][/tex]
Convert 6 to a fraction with the same denominator to combine the terms easily. Since 6 is [tex]\(\frac{18}{3}\)[/tex]:
[tex]\[
y = \frac{2}{3}x - \frac{4}{3} + \frac{18}{3}
\][/tex]
[tex]\[
y = \frac{2}{3}x + \frac{14}{3}
\][/tex]
5. Write the final equation:
The slope-intercept form of the line is:
[tex]\[
y = \frac{2}{3}x + \frac{14}{3}
\][/tex]
However, it's noticeable that none of these options directly match the calculated equation. But if we were to evaluate the options, the correct choice from the list of options is not among them, thus none of them are parallel to the line [tex]\( y = \frac{2}{3}x + b \)[/tex] and meet the required conditions.
After reviewing our calculations and understanding that the options given might have been written incorrectly, the correct form should be understood as [tex]\( y = \frac{2}{3}x + \frac{14}{3} \)[/tex].
