To find the equation of a line perpendicular to the given line [tex]\( y = 5x + 4 \)[/tex] and passing through the point (-3, 2), we'll follow these steps:
### Step 1: Identify the Slope of the Given Line
The given line's equation is [tex]\( y = 5x + 4 \)[/tex]. The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope of the given line [tex]\( m \)[/tex] is 5.
### Step 2: Find the Slope of the Perpendicular Line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of 5 is:
[tex]\[ \text{slope\_perpendicular} = -\frac{1}{5} \][/tex]
### Step 3: Use the Point-Slope Form of the Line Equation
We need to use the point-slope form of the line equation to find the equation of the perpendicular line. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((-3, 2)\)[/tex] and [tex]\( m \)[/tex] is the slope of the perpendicular line [tex]\(-\frac{1}{5}\)[/tex].
Substitute the point and the slope into the point-slope form:
[tex]\[ y - 2 = -\frac{1}{5}(x + 3) \][/tex]
### Step 4: Simplify to Slope-Intercept Form
To convert this equation to slope-intercept form ([tex]\( y = mx + b \)[/tex]):
[tex]\[ y - 2 = -\frac{1}{5}x - \frac{1}{5} \cdot 3 \][/tex]
[tex]\[ y - 2 = -\frac{1}{5}x - \frac{3}{5} \][/tex]
Add 2 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{5}x - \frac{3}{5} + 2 \][/tex]
Express 2 as a fraction with a denominator of 5:
[tex]\[ y = -\frac{1}{5}x - \frac{3}{5} + \frac{10}{5} \][/tex]
Combine the fractions:
[tex]\[ y = -\frac{1}{5}x + \frac{7}{5} \][/tex]
So the equation of the line perpendicular to [tex]\( y = 5x + 4 \)[/tex] and passing through the point (-3, 2) is:
[tex]\[ y = -\frac{1}{5}x + \frac{7}{5} \][/tex]
From the given options, the correct one is:
[tex]\[ y = -\frac{1}{5}x + \frac{7}{5} \][/tex]
