Section 4-1
(5) Solve by graphing: { 3x - y = 2    
x + 2y = 10    

[Graphics:Images/4-1_gr_2.gif]

By looking at the graph, we see that the solution is (x,y) = (2,4).

(8) Solve by graphing: { 3u + 5v = 15    
6u +10v = -30    
 

[Graphics:Images/4-1_gr_7.gif]

Since the lines are parallel, they never intersect. Therefore the system has NO SOLUTIONS.


(12) Solve by substitution: { 3x - y = 7
2x + 3y = 1
Solving the first equation for y gives us  y = 3x - 7.  Now, plugging that into the second equation yields
2x + 3(3x -7) = 1 2x + 9x -21 = 1
11x = 22 x = 2 Now that we've got a value for x, we plug it back into y = 3x - 7 to find y. (Plugging it into either
equation of the original system would work just as well.) y = 3(2) - 7 y = -1 Thus the solution is (x,y) = (2, -1).


(14) Solve by addition: { 2x - 3y = -8
5x + 3y = 1
Adding the equations: 2x - 3y = -8
5x + 3y = 1

7x = -7

Therefore we get that x = -1. Plugging this back into the second equation (the first would work just as well):
5(-1) + 3y = 1
-5 + 3y = 1
3y = 6
y = 2

Thus (x,y) = ( -1, 2) is the solution.


(24) Solve by addition: { 2x +4y= -8
x + 2y = 4
Add 1st to -2 times 2nd: 2x +4y = -8
-2x -4y = -8

0 = -16


Since we get a false statement, it's impossible for both equations to be satisfied by the same (x, y).
Thus the system has NO SOLUTIONS.