To find the equation of a line that is perpendicular to the given line [tex]\( y = 2x + 2 \)[/tex] and passes through the point [tex]\( (9, -2) \)[/tex], follow these steps:
### Step 1: Determine the Slope of the Perpendicular Line
The slope of the given line [tex]\( y = 2x + 2 \)[/tex] is 2. For a line to be perpendicular to another, its slope must be the negative reciprocal of the original line's slope.
- Original slope: [tex]\( m = 2 \)[/tex]
- Perpendicular slope: [tex]\( m_{\perpendicular} = -\frac{1}{2} \)[/tex]
### Step 2: Use the Point-Slope Form of the Equation
The point-slope form of a linear equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Given point: [tex]\( (x_1, y_1) = (9, -2) \)[/tex]
- Perpendicular slope: [tex]\( m_{\perpendicular} = -\frac{1}{2} \)[/tex]
Plug these values into the point-slope form:
[tex]\[ y - (-2) = -\frac{1}{2}(x - 9) \][/tex]
Simplify the equation:
[tex]\[ y + 2 = -\frac{1}{2}(x - 9) \][/tex]
### Step 3: Convert to Slope-Intercept Form
Now, convert this equation to the slope-intercept form [tex]\( y = mx + b \)[/tex].
Start by distributing the slope on the right-hand side:
[tex]\[ y + 2 = -\frac{1}{2}x + \frac{9}{2} \][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{2}x + \frac{9}{2} - 2 \][/tex]
Combine the constants on the right-hand side:
[tex]\[ y = -\frac{1}{2}x + \frac{9}{2} - \frac{4}{2} \][/tex]
[tex]\[ y = -\frac{1}{2}x + \frac{5}{2} \][/tex]
### Final Forms
Point-Slope Form:
[tex]\[ y - (-2) = -\frac{1}{2}(x - 9) \][/tex]
Simplified:
[tex]\[ y + 2 = -\frac{1}{2}(x - 9) \][/tex]
Slope-Intercept Form:
[tex]\[ y = -\frac{1}{2}x + \frac{5}{2} \][/tex]
