To find the equation of a line parallel to the given line [tex]\( y = 2x + 3 \)[/tex] that passes through the point [tex]\( (2, 6) \)[/tex], we need to follow these steps:
1. Identify the slope: Since we need the new line to be parallel to [tex]\( y = 2x + 3 \)[/tex], it must have the same slope. The slope of the given line [tex]\( y = 2x + 3 \)[/tex] is 2.
2. Use the point-slope form of the equation of a line: The point-slope form of the equation of a line is given by:
[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. Here, the point [tex]\((2, 6)\)[/tex] and the slope [tex]\( m = 2 \)[/tex] will be used.
3. Substitute the given point and slope into the point-slope form:
[tex]\[
y - 6 = 2(x - 2)
\][/tex]
4. Simplify the equation to get it into slope-intercept form (y = mx + b):
[tex]\[
y - 6 = 2(x - 2)
\][/tex]
[tex]\[
y - 6 = 2x - 4
\][/tex]
[tex]\[
y = 2x - 4 + 6
\][/tex]
[tex]\[
y = 2x + 2
\][/tex]
Therefore, the equation of the line parallel to [tex]\( y = 2x + 3 \)[/tex] that passes through the point [tex]\( (2, 6) \)[/tex] is:
[tex]\[
y = 2x + 2
\][/tex]
This matches the slope-intercept form [tex]\( y = mx + b \)[/tex] with [tex]\( m = 2 \)[/tex] and [tex]\( b = 2 \)[/tex], which means the correct equation from the given options is:
[tex]\[
y = 2x + 2
\][/tex]
