To find the equation of the line in the form [tex]\(y = -3x + b\)[/tex], we need to determine the y-intercept, [tex]\(b\)[/tex].
The given line equation is [tex]\( y = -3x + b \)[/tex], where the slope [tex]\( m \)[/tex] is -3.
The slope-intercept form of a line equation is [tex]\( y = mx + b \)[/tex], where:
- [tex]\( m \)[/tex] is the slope,
- [tex]\( b \)[/tex] is the y-intercept.
In this case, the slope [tex]\(m\)[/tex] is given as -3. To find the y-intercept [tex]\(b\)[/tex], we need a point [tex]\((x, y)\)[/tex] that the line passes through.
Assume the line passes through the origin, which is the point [tex]\( (0, 0) \)[/tex]. By substituting [tex]\((x, y)\)[/tex] = (0, 0) into the equation:
[tex]\[ y = -3x + b \][/tex]
we get:
[tex]\[ 0 = -3(0) + b \][/tex]
Simplifying the above equation:
[tex]\[ 0 = 0 + b \][/tex]
Thus:
[tex]\[ b = 0 \][/tex]
So, the y-intercept [tex]\( b \)[/tex] is 0. Therefore, the equation of the line simplifies to:
[tex]\[ y = -3x + 0 \][/tex]
or simply:
[tex]\[ y = -3x \][/tex]
This indicates that the line passes through the origin and has a slope of -3.
