To determine the equation of a line that passes through the point [tex]\((-4, 1)\)[/tex] and has a slope of [tex]\(-\frac{3}{2}\)[/tex], you can follow these steps:
### Step 1: Start with the Point-Slope Form of the Line Equation
The point-slope form of a line's equation 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.
Given:
- Point [tex]\((x_1, y_1) = (-4, 1)\)[/tex]
- Slope [tex]\( m = -\frac{3}{2} \)[/tex]
### Step 2: Substitute the given values into the Point-Slope Form
Substitute [tex]\( x_1 = -4 \)[/tex], [tex]\( y_1 = 1 \)[/tex], and [tex]\( m = -\frac{3}{2} \)[/tex] into the point-slope form:
[tex]\[ y - 1 = -\frac{3}{2} \left(x - (-4)\right) \][/tex]
### Step 3: Simplify the Equation
First, simplify inside the parentheses:
[tex]\[ y - 1 = -\frac{3}{2} (x + 4) \][/tex]
Next, distribute the slope [tex]\( -\frac{3}{2} \)[/tex]:
[tex]\[ y - 1 = -\frac{3}{2} x - \frac{3}{2} \cdot 4 \][/tex]
Calculate [tex]\( \frac{3}{2} \cdot 4 \)[/tex]:
[tex]\[ \frac{3}{2} \cdot 4 = 6 \][/tex]
So the equation becomes:
[tex]\[ y - 1 = -\frac{3}{2} x - 6 \][/tex]
### Step 4: Convert to Slope-Intercept Form
To convert this into slope-intercept form ([tex]\( y = mx + c \)[/tex]), solve for [tex]\( y \)[/tex]:
Add 1 to both sides of the equation:
[tex]\[ y = -\frac{3}{2} x - 6 + 1 \][/tex]
Combine the constant terms:
[tex]\[ y = -\frac{3}{2} x - 5 \][/tex]
### Final Equation of the Line
The equation of the line that passes through the point [tex]\((-4, 1)\)[/tex] with a slope of [tex]\(-\frac{3}{2}\)[/tex] is:
[tex]\[ y = -\frac{3}{2} x - 5 \][/tex]
