To find the equation of a line in slope-intercept form given a slope and a point on the line, we use the following approach:
1. Identify the Slope (m) and the Point (x1, y1):
- The slope [tex]\( m \)[/tex] is given as [tex]\( \frac{1}{2} \)[/tex].
- The line passes through the point [tex]\((4, -1)\)[/tex].
2. Use the Point-Slope Form of the Equation of a Line:
- The point-slope form of a line is given by:
[tex]\[
y - y1 = m(x - x1)
\][/tex]
- Substituting the given values into this form:
[tex]\[
y - (-1) = \frac{1}{2}(x - 4)
\][/tex]
- Simplify the equation:
[tex]\[
y + 1 = \frac{1}{2}(x - 4)
\][/tex]
3. Distribute the Slope:
- Distribute [tex]\(\frac{1}{2}\)[/tex] across [tex]\((x - 4)\)[/tex]:
[tex]\[
y + 1 = \frac{1}{2}x - 2
\][/tex]
4. Isolate y:
- To express the equation in y = mx + b form, isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{2}x - 2 - 1
\][/tex]
[tex]\[
y = \frac{1}{2}x - 3
\][/tex]
So, the equation of the line in slope-intercept form is:
[tex]\[
y = \frac{1}{2}x - 3
\][/tex]
