To determine the point-slope form of a line that has a slope of -4 and passes through the point [tex]\((-3, 1)\)[/tex], we use the point-slope form equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where [tex]\( m \)[/tex] is the slope of the line, and [tex]\((x_1, y_1)\)[/tex] is a point on the line. Given:
- Slope (m) = -4
- Point [tex]\((x_1, y_1) = (-3, 1)\)[/tex]
Substituting the given values into the point-slope form equation:
1. Identify the correct placement of the point and slope in the equation:
[tex]\[ y - 1 = -4(x - (-3)) \][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[ y - 1 = -4(x + 3) \][/tex]
Among the given options, let's match it:
- Option 1: [tex]\( y-(-3) = -4(x-1) \)[/tex]
- This simplifies to [tex]\( y + 3 = -4(x - 1) \)[/tex], which is not correct.
- Option 2: [tex]\( y - 1 = -4[x - (-3)] \)[/tex]
- This simplifies to [tex]\( y - 1 = -4(x + 3) \)[/tex], which matches.
- Option 3: [tex]\( -1 - y_1 = -4(-3 - x_1) \)[/tex]
- This simplifies to [tex]\( -1 - 1 = -4(-3 - (-3)) \)[/tex] which is incorrect.
- Option 4: [tex]\( 3 - y_1 = -4(1 - x_1) \)[/tex]
- This simplifies to [tex]\( 3 - 1 = -4(1 - (-3)) \)[/tex] which is incorrect.
Therefore, the correct option is:
[tex]\[ y - 1 = -4[x - (-3)] \][/tex]
Hence, the point-slope form of the line is accurately represented by Option 2. The correct answer is:
[tex]\[ \boxed{2} \][/tex]
