To find the [tex]\( y \)[/tex]-intercept of the line perpendicular to the given line [tex]\( y = \frac{3}{4}x + 3 \)[/tex] and passing through the point [tex]\((3, 1)\)[/tex], we need to follow these steps:
1. Determine the slope of the given line:
The equation of the line is [tex]\( y = \frac{3}{4}x + 3 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{3}{4} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of the line perpendicular to the given line would be the negative reciprocal of [tex]\( \frac{3}{4} \)[/tex]. Therefore, the slope of the perpendicular line is [tex]\( -\frac{4}{3} \)[/tex].
3. Use the point-slope form of the line equation:
The point-slope form of the line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
We know the slope [tex]\( m = -\frac{4}{3} \)[/tex] and the line passes through the point [tex]\((3, 1)\)[/tex]. Substituting these values into the equation gives:
[tex]\[ y - 1 = -\frac{4}{3}(x - 3) \][/tex]
4. Solve for the [tex]\( y \)[/tex]-intercept:
We rearrange the equation into the slope-intercept form [tex]\( y = mx + b \)[/tex] to find the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]. Distribute the slope on the right-hand side:
[tex]\[ y - 1 = -\frac{4}{3}x + 4 \][/tex]
Add 1 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{4}{3}x + 5 \][/tex]
Here, the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is 5.
Thus, the [tex]\( y \)[/tex]-intercept of the line perpendicular to [tex]\( y = \frac{3}{4}x + 3 \)[/tex] and passing through the point [tex]\((3, 1)\)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
