Case 3

Same Line – Infinitely Many solutions

x + 3y = 9
2x + 6y = 18

Graphically	Algebraically
Rewrite both equations in slope-intercept form (y = mx + b). x + 3y = 9 –x –x 3y = –x + 9 = + y = –x + 3 2x + 6y = 18 –2x –2x 6y = –2x + 18 = + y = –x + 3 Notice that when rewritten in slope-intercept form, the equations look exactly alike. They have the same slope and same y-intercept. When these equations are graphed, they will graph the same line. This means that every point on the line will be a solution for both equations. Therefore, there are infinitely many solutions. Check out the graph of this system of equations.	Since the variables are already lined up, let’s solve using elimination. To cancel the x’s, you need to multiply the first equation by –2 and leave the second equation alone. –2(x + 3y) = –2(9) –2x – 6y = –18 2x + 6y = 18 2x + 6y = 18 Now combine the equations. 0 = 0 TRUE! Notice, the variables cancelled and created a true statement. Therefore, there are infinitely many solutions.

Graphically

Algebraically

Rewrite both equations in slope-intercept form
(y = mx + b).

Since the variables are already lined up, let’s solve using elimination. To cancel the x’s, you need to multiply the first equation by –2 and leave the second equation alone.