Final answer:
The lines represented by the equations 6x + 2y = 5 and y + 3x = 2 will not intersect because they are parallel; they have the same slope but different y-intercepts.
Step-by-step explanation:
The question is asking how many times the two lines represented by the equations 6x + 2y = 5 and y + 3x = 2 will intersect. To find the intersection of two lines, we can solve the system of equations. If the system has one solution, it means that the two lines will intersect exactly once. If it has infinitely many solutions, the lines are coincident (the same line), and if the system has no solution, the lines are parallel and will not intersect. By manipulating the given equations into slope-intercept form, which is y = mx + b where 'm' is the slope and 'b' is the y-intercept, we can more easily determine the answer without graphing.
First, let's arrange the equations:
- 6x + 2y = 5 can be rearranged to: y = -3x + 5/2
- y + 3x = 2 can be rearranged to: y = -3x + 2
Both equations now have the form y = mx + b, where the slope 'm' is -3 for both lines. However, the y-intercepts are different ('b' is 5/2 for the first equation and 2 for the second equation). Because the slopes are equal and the y-intercepts are different, we can conclude that the lines are parallel and will not intersect. Therefore, the correct answer is option B: the lines will not intersect, because the system has one slope (meaning the lines have the same slope).