To find the equation of a line that is parallel to the given line [tex]\( y = -\frac{1}{2}x + 3 \)[/tex] and passes through the point [tex]\((2, -3)\)[/tex], follow these steps:
1. Identify the Slope of the Given Line:
- The given line is [tex]\( y = -\frac{1}{2}x + 3 \)[/tex].
- The slope of this line is [tex]\( -\frac{1}{2} \)[/tex].
2. Recall the Property of Parallel Lines:
- Lines that are parallel to each other have identical slopes.
- Therefore, the slope of our desired line is also [tex]\( -\frac{1}{2} \)[/tex].
3. Use the Point-Slope Form Equation:
- 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.
- Here, the slope [tex]\( m = -\frac{1}{2} \)[/tex] and the point [tex]\((x_1, y_1) = (2, -3)\)[/tex].
4. Substitute the Values into the Point-Slope Form:
[tex]\[
y - (-3) = -\frac{1}{2}(x - 2)
\][/tex]
This simplifies to:
[tex]\[
y + 3 = -\frac{1}{2}(x - 2)
\][/tex]
5. Distribute the Slope on the Right Side:
[tex]\[
y + 3 = -\frac{1}{2}x + 1
\][/tex]
6. Isolate [tex]\( y \)[/tex] to Convert into Slope-Intercept Form ( [tex]\(y = mx + b\)[/tex] ):
[tex]\[
y = -\frac{1}{2}x + 1 - 3
\][/tex]
[tex]\[
y = -\frac{1}{2}x - 2
\][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = -\frac{1}{2}x + 3 \)[/tex] and passes through the point [tex]\((2, -3)\)[/tex] is:
[tex]\[
y = -\frac{1}{2}x - 2
\][/tex]
