To find the equation of a line that passes through the point [tex]\((1,9)\)[/tex] and has a slope of [tex]\(-9\)[/tex], we can use the point-slope form of a linear equation. The point-slope form 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 of the line. In this case, the point given is [tex]\((1, 9)\)[/tex] and the slope ([tex]\(m\)[/tex]) is [tex]\(-9\)[/tex].
Substitute the given point and slope into the point-slope form:
[tex]\[
y - 9 = -9(x - 1)
\][/tex]
Next, simplify the equation. First, distribute the slope [tex]\(-9\)[/tex] on the right side:
[tex]\[
y - 9 = -9x + 9
\][/tex]
To convert this into the slope-intercept form [tex]\( y = mx + b \)[/tex], add 9 to both sides of the equation to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -9x + 18
\][/tex]
Thus, the equation of the line in slope-intercept form is:
[tex]\[
y = -9x + 18
\][/tex]
Therefore, the equation for the line that passes through the point [tex]\((1, 9)\)[/tex] with a slope of [tex]\(-9\)[/tex] is:
[tex]\[
y = -9x + 18
\][/tex]
