To determine the [tex]\( y \)[/tex]-intercept of the line perpendicular to [tex]\( y = \frac{3}{4}x + 3 \)[/tex] that includes the point [tex]\((3, 1)\)[/tex], follow these steps:
1. Find the Slope of the Perpendicular Line:
- The given line has a slope [tex]\( m = \frac{3}{4} \)[/tex].
- The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
- The negative reciprocal of [tex]\( \frac{3}{4} \)[/tex] is [tex]\( -\frac{4}{3} \)[/tex].
2. Use the Point-Slope Form:
- The point-slope form of a line 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 point is [tex]\( (3, 1) \)[/tex] and the slope is [tex]\( -\frac{4}{3} \)[/tex].
- Plugging in the values, we get: [tex]\[ y - 1 = -\frac{4}{3}(x - 3) \][/tex]
3. Convert to Slope-Intercept Form:
- Simplify the equation to find the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]):
[tex]\[
y - 1 = -\frac{4}{3}(x - 3)
\][/tex]
Expand the right-hand side:
[tex]\[
y - 1 = -\frac{4}{3}x + 4
\][/tex]
Adding 1 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{4}{3}x + 4 + 1
\][/tex]
Simplify the constant term:
[tex]\[
y = -\frac{4}{3}x + 5
\][/tex]
4. Identify the [tex]\( y \)[/tex]-Intercept:
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
- From the simplified equation [tex]\( y = -\frac{4}{3}x + 5 \)[/tex], the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is 5.
Hence, the [tex]\( y \)[/tex]-intercept of the line perpendicular to [tex]\( y = \frac{3}{4}x + 3 \)[/tex] that includes the point [tex]\( (3, 1) \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
