*the line is parallel to the 3x-2y=6
*the line has the same y-intercept as the line y-4=2(x+1)
*line will never pass through the fourth quadrant
To determine the line that satisfies the given conditions, let's break down each requirement separately:
1. The line is parallel to the equation 3x-2y=6.
To find a line parallel to a given line, we need to determine the slope of the given line. The equation 3x - 2y = 6 can be rearranged to slope-intercept form (y = mx + b), where m is the slope.
3x - 2y = 6
-2y = -3x + 6
y = (3/2)x - 3
Comparing this equation to y = mx + b, we can see that the slope (m) is 3/2. Therefore, the new line should also have a slope of 3/2.
2. The line has the same y-intercept as the line y-4=2(x+1).
To find the y-intercept of a line, we need to rearrange the equation into slope-intercept form.
y - 4 = 2(x + 1)
y - 4 = 2x + 2
y = 2x + 6
Comparing this equation to y = mx + b, we can see that the y-intercept (b) is 6. Therefore, the new line should also have a y-intercept of 6.
3. The line will never pass through the fourth quadrant.
The fourth quadrant is the region where both x and y values are negative. If a line does not pass through the fourth quadrant, it means either the x-values or the y-values are always positive (or zero).
Since we already know the y-intercept of the line is 6, which is positive, the line will have a positive or zero y-values. Therefore, the line will not pass through the fourth quadrant.
In conclusion, the equation of the line that satisfies all the given conditions can be represented as y = (3/2)x + 6.