To solve this problem, follow these steps:
1. Determine the slope of the given line:
The equation of the given line is [tex]\( 3y + 4x = 7 \)[/tex]. We can rewrite this in the slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[
3y + 4x = 7 \implies 3y = -4x + 7 \implies y = -\frac{4}{3}x + \frac{7}{3}
\][/tex]
Therefore, the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{4}{3} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of the line that is perpendicular to another line is the negative reciprocal of the original slope. Thus, the slope of the perpendicular line is:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{-\frac{4}{3}} = \frac{3}{4}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
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.
Given point [tex]\((4, -3)\)[/tex] and slope [tex]\( \frac{3}{4} \)[/tex]:
[tex]\[
y - (-3) = \frac{3}{4}(x - 4) \implies y + 3 = \frac{3}{4}(x - 4)
\][/tex]
4. Simplify the equation:
Distribute the slope on the right-hand side:
[tex]\[
y + 3 = \frac{3}{4}x - \frac{3}{4} \cdot 4
\][/tex]
[tex]\[
y + 3 = \frac{3}{4}x - 3
\][/tex]
Subtract 3 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3}{4}x - 3 - 3
\][/tex]
[tex]\[
y = \frac{3}{4}x - 6
\][/tex]
Thus, the equation of the line passing through [tex]\((4, -3)\)[/tex] and perpendicular to the line [tex]\( 3y + 4x = 7 \)[/tex] is:
[tex]\[
y = \frac{3}{4}x - 6
\][/tex]
