To derive the equation of the line in slope-intercept form (which is [tex]\( y = mx + b \)[/tex]), we need two pieces of information: the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex]. We are given:
- A point the line passes through: [tex]\((-3, 5)\)[/tex]
- The slope of the line [tex]\( m = -3 \)[/tex]
Step 1: Substitute the given point and slope into the slope-intercept form equation.
The general equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\( x = -3 \)[/tex], [tex]\( y = 5 \)[/tex], and [tex]\( m = -3 \)[/tex] into the equation:
[tex]\[ 5 = (-3)(-3) + b \][/tex]
Step 2: Solve for the y-intercept [tex]\( b \)[/tex].
[tex]\[ 5 = 9 + b \][/tex]
Subtract 9 from both sides to isolate [tex]\( b \)[/tex]:
[tex]\[ 5 - 9 = b \][/tex]
[tex]\[ b = -4 \][/tex]
Step 3: Substitute the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] back into the slope-intercept form equation.
So, the slope [tex]\( m = -3 \)[/tex] and the y-intercept [tex]\( b = -4 \)[/tex]. Plugging these values in, we get:
[tex]\[ y = -3x - 4 \][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = -3x - 4 \][/tex]
