To determine the equation of a line that passes through the point (3, -2) and is perpendicular to the line [tex]\( y = \frac{3}{4} x + 6 \)[/tex], let's go through the solution step by step:
### Step 1: Identify the slope of the given line
The given line is [tex]\( y = \frac{3}{4} x + 6 \)[/tex]. From this equation, we see that the slope [tex]\( m \)[/tex] of the line is [tex]\( \frac{3}{4} \)[/tex].
### Step 2: Find the slope of the perpendicular line
A line that is perpendicular to another line will have a slope that 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].
### Step 3: Use the point-slope form to write the equation of the perpendicular line
We know the slope [tex]\( m_{\text{perpendicular}} = -\frac{4}{3} \)[/tex] and the line passes through the point [tex]\( (3, -2) \)[/tex].
The point-slope form of a line is given by:
[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.
Substituting the known values:
[tex]\[ y - (-2) = -\frac{4}{3} (x - 3) \][/tex]
### Step 4: Simplify the equation
Simplify the equation to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y + 2 = -\frac{4}{3} x + 3(-\frac{4}{3}) \][/tex]
[tex]\[ y + 2 = -\frac{4}{3} x - 4 \][/tex]
[tex]\[ y = -\frac{4}{3} x - 4 - 2 \][/tex]
[tex]\[ y = -\frac{4}{3} x - 6 \][/tex]
Thus, the equation of the line passing through [tex]\( (3, -2) \)[/tex] and perpendicular to the line [tex]\( y = \frac{3}{4} x + 6 \)[/tex] is:
[tex]\[ y = -\frac{4}{3} x + 2 \][/tex]
The coefficients of the line in the form [tex]\( y = mx + b \)[/tex] are:
[tex]\[ m = -1.3333333333333333 \][/tex]
[tex]\[ b = 2 \][/tex]
Thus, the equation is:
[tex]\[ y = -1.3333333333333333 x + 2 \][/tex]
