To find the equation of a line that is parallel to a given line and passes through a specific point, we need to follow several steps:
1. Identify the slope of the given line:
The given line is [tex]\( x + 2y = 4 \)[/tex].
Let's put this equation in slope-intercept form ( [tex]\( y = mx + b \)[/tex] ), where [tex]\( m \)[/tex] is the slope:
[tex]\[
x + 2y = 4 \implies 2y = -x + 4 \implies y = -\frac{1}{2}x + 2
\][/tex]
So, the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{1}{2} \)[/tex].
2. Use the slope and the given point to find the new line:
A line parallel to the given line will have the same slope, which is [tex]\( -\frac{1}{2} \)[/tex]. We need the equation of the line passing through the point [tex]\( (2, 3) \)[/tex].
3. Point-slope form:
Use the point-slope form of the line equation [tex]\( (y - y_1) = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\( (2, 3) \)[/tex]:
[tex]\[
y - 3 = -\frac{1}{2}(x - 2)
\][/tex]
4. Simplify the equation:
Distribute [tex]\( -\frac{1}{2} \)[/tex]:
[tex]\[
y - 3 = -\frac{1}{2}x + 1
\][/tex]
Add 3 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{2}x + 4
\][/tex]
5. Convert to the standard form:
Multiply both sides by 2 to clear the fraction:
[tex]\[
2y = -x + 8
\][/tex]
Move [tex]\( x \)[/tex] to the left side:
[tex]\[
x + 2y = 8
\][/tex]
So, the equation of the line parallel to [tex]\( x + 2y = 4 \)[/tex] and passing through the point [tex]\( (2, 3) \)[/tex] is:
[tex]\[
x + 2y = 8
\][/tex]
