To determine the equation of the line passing through the point (3, 17) with a slope of 6, follow these steps:
1. Recall the Slope-Intercept Form: The equation of a line in slope-intercept form is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Plug in the Slope (m): Here, the slope [tex]\( m \)[/tex] is given as 6. So, the equation becomes:
[tex]\[
y = 6x + b
\][/tex]
3. Use the Given Point (3, 17): The line passes through the point (3, 17), meaning when [tex]\( x = 3 \)[/tex], [tex]\( y = 17 \)[/tex]. Substitute these values into the equation to find the y-intercept [tex]\( b \)[/tex]:
[tex]\[
17 = 6 \cdot 3 + b
\][/tex]
4. Solve for [tex]\( b \)[/tex]:
[tex]\[
17 = 18 + b
\][/tex]
To isolate [tex]\( b \)[/tex], subtract 18 from both sides:
[tex]\[
17 - 18 = b
\][/tex]
[tex]\[
-1 = b
\][/tex]
5. Construct the Equation: Now that we have the slope [tex]\( m = 6 \)[/tex] and the y-intercept [tex]\( b = -1 \)[/tex], the equation of the line is:
[tex]\[
y = 6x - 1
\][/tex]
So, the equation of the line passing through the point (3, 17) with a slope of 6 is:
[tex]\[
y = 6x - 1
\][/tex]
