Final answer:
To find the slope-intercept form of the line with a slope of 2 that intersects the line 2x - 3y = 6 at x = 3, we substitute x into the equation, find y, and then use the point of intersection and the slope to write the equation in the form y = mx + b. The correct slope-intercept form is y = 2x - 6.
Step-by-step explanation:
The student is asking for the slope-intercept form of a line that has a slope of 2 and intersects another line at a certain point. First, to find the y-value where the line intersects at x = 3, we substitute x into the given equation of the line 2x - 3y = 6 and solve for y:
- 2(3) - 3y = 6
- 6 - 3y = 6
- -3y = 6 - 6
- -3y = 0
- y = 0
Now that we have the point of intersection as (3, 0), we can use the slope of 2 and the point to write the slope-intercept form y = mx + b where m is the slope and b is the y-intercept. Since the y-value of the intersection is 0 (also the y-intercept), the equation of the line is:
y = 2x + b
Therefore, with b being the y-intercept, which is 0 in this case, the final equation is:
y = 2x
The correct choice that represents the slope-intercept form with these conditions is answer (a) y = 2x - 6, because it has a slope of 2 and a y-intercept of -6, which provides the necessary conditions when the x-value is 3.