Question: problems involving linear functions and paraller and perpendicular lines
1. **Problem Statement:**
Given a linear function, find the equation of a line parallel or perpendicular to it passing through a specific point.
2. **Key Formulas and Rules:**
- The slope of a line in the form $y=mx+b$ is $m$.
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals: if one slope is $m$, the perpendicular slope is $-\frac{1}{m}$.
- Equation of a line with slope $m$ passing through point $(x_1,y_1)$ is:
$$y - y_1 = m(x - x_1)$$
3. **Example:**
Find the equation of a line parallel to $y=2x+3$ passing through $(1,4)$.
4. **Solution:**
- The slope of the given line is $m=2$.
- Since the new line is parallel, its slope is also $2$.
- Using point-slope form:
$$y - 4 = 2(x - 1)$$
- Simplify:
$$y - 4 = 2x - 2$$
$$y = 2x - 2 + 4$$
$$y = 2x + 2$$
5. **Example:**
Find the equation of a line perpendicular to $y=\frac{1}{2}x - 1$ passing through $(2,3)$.
6. **Solution:**
- The slope of the given line is $m=\frac{1}{2}$.
- The perpendicular slope is:
$$m_{\perp} = -\frac{1}{\frac{1}{2}} = -2$$
- Using point-slope form:
$$y - 3 = -2(x - 2)$$
- Simplify:
$$y - 3 = -2x + 4$$
$$y = -2x + 4 + 3$$
$$y = -2x + 7$$
**Final answers:**
- Parallel line: $y=2x+2$
- Perpendicular line: $y=-2x+7$