Certainly! To write the equation of a line in slope-intercept form given a point [tex]\((x_1, y_1)\)[/tex] that the line passes through and the slope [tex]\(m\)[/tex], follow these steps:
### Given:
- Point [tex]\((x_1, y_1) = (1, 6)\)[/tex]
- Slope [tex]\(m = 9\)[/tex]
### Step 1: Recall the slope-intercept form
The slope-intercept form of the equation of a line is:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
### Step 2: Utilize the point-slope relationship
We know the line passes through the point [tex]\((1, 6)\)[/tex] and has a slope [tex]\(9\)[/tex]. To find the y-intercept [tex]\(b\)[/tex], we use the point-slope form of the line equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substitute the given point and slope into this equation:
[tex]\[ y - 6 = 9(x - 1) \][/tex]
### Step 3: Solve for [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex]
Simplify the equation above to find [tex]\(b\)[/tex].
[tex]\[ y - 6 = 9x - 9 \][/tex]
To find [tex]\(b\)[/tex], we rearrange this to the slope-intercept form:
[tex]\[ y = 9x - 9 + 6 \][/tex]
Therefore:
[tex]\[ y = 9x - 3 \][/tex]
### Conclusion
The equation of the line in slope-intercept form that passes through the point [tex]\((1, 6)\)[/tex] and has a slope of 9 is:
[tex]\[ y = 9x - 3 \][/tex]
So, the requested equation in slope-intercept form is:
[tex]\[ y = 9x - 3 \][/tex]
