To find the equation of the line with a given slope that passes through a specific point, we will use the point-slope form of the equation of a line. The point-slope form is given by:
[tex]\[ y - y_1 = m(x - x1) \][/tex]
where:
- [tex]\( m \)[/tex] is the slope,
- [tex]\( (x1, y1) \)[/tex] are the coordinates of the given point.
Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{-7}{2} \)[/tex] and the given point is [tex]\( (0, 2) \)[/tex].
1. Substitute the slope [tex]\( m = \frac{-7}{2} \)[/tex] and the point [tex]\( (0, 2) \)[/tex] into the point-slope form equation.
[tex]\[ y - 2 = \frac{-7}{2}(x - 0) \][/tex]
2. Simplify the equation. Since [tex]\( x - 0 = x \)[/tex], it becomes:
[tex]\[ y - 2 = \frac{-7}{2}x \][/tex]
3. To get this equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-7}{2}x + 2 \][/tex]
So, the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( b \)[/tex] is the y-intercept is:
[tex]\[ y = \frac{-7}{2}x + 2 \][/tex]
Therefore, the slope of the line is [tex]\( -3.5 \)[/tex] (since [tex]\(\frac{-7}{2} = -3.5\)[/tex]), and the y-intercept [tex]\( b \)[/tex] is [tex]\( 2 \)[/tex].
Hence, the equation of the line with the given slope passing through the point [tex]\( (0, 2) \)[/tex] is:
[tex]\[ y = -3.5x + 2 \][/tex]
