To graph the equation of a line, all you need are the coordinates of two points on the line. It is often convenient to use (x, y) values such that x or y is zero.
3) For the first equation, when x=0, you have -3y = 2, so y = -2/3. That means (0, -2/3) is one point on the line. When y=0, you have x = 2, so (2, 0) is another point on the line. Draw the graph by plotting these points and draing a straight line through them.
For the second equation, when x=0, you have 9y = -6, so y = -6/9 = -2/3. That means (0, -2/3) is also a point on the second line. When y=0, you have -3x = -6, so x=2 and (2, 0) is also a point on the second line.
The second line is identical to the first line, so it has an infinite number of points in common with it. The system of equations has an infinite number of solutions.*
4) Repeat the exercise as for problem 3 to find that the first line goes through points (5/2, 0) and (0, -5). The second line goes through points (-1/2, 0) and (0, 1). These lines are parallel, so never intersect. The system of equations has zero solutions.**
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* We say these equations are "dependent." If you multiply the first equation by -3, you get the second equation.
** We say these equations are "inconsistent."
