Solve each system using elimination.
1. x + y = 7 2. 2x + y = –5
x – y = 3 3x – y = –10
3. x + 3y = 4 4. 2x + 3y = –12
–x + 2y = –4 –2x + y = 4
5. x – 3y = 27 6. 4x + 2y = 2
3x – 3y = 39 3x + y = 4
1. Multiply the first equation by -1 and add it to the second equation:
(-1)(x + y) = -1(7)
-2x - y = 5
2x + y = -5
2x + y + (-2x - y) = -5 + 5
0 = 0
Since 0 = 0, this system is dependent. The two equations are essentially the same line. The solution is infinite.
2. Multiply the first equation by -3 and add it to the second equation:
(-3)(2x + y) = (-3)(-5)
-6x - 3y = 15
3x - y = -10
3x - y + (-3x + y) = -10 + 15
0 = 5
Since 0 does not equal 5, this system is inconsistent. There is no solution.
3. Multiply the first equation by 2 and add it to the second equation:
(2)(x + 3y) = (2)(4)
2x + 6y = 8
2x + y = -4
2x + y + (-(2x + 6y)) = -4 + 8
0 = 4
Since 0 does not equal 4, this system is inconsistent. There is no solution.
4. Multiply the first equation by 2 and add it to the second equation:
(2)(-x + 2y) = (2)(-4)
-2x + 4y = -8
-2x + y = 4
-2x + y + (-(-2x + 4y)) = 4 + 8
0 = 12
Since 0 does not equal 12, this system is inconsistent. There is no solution.
5. Multiply the first equation by 3 and add it to the second equation:
(3)(x - 3y) = (3)(27)
3x - 9y = 81
4x + 2y = 2
4x + 2y + (-(3x - 9y)) = 2 + 81
x + 11y = 83
Solve the new equation for x:
x = 83 - 11y
Substitute x in the first equation:
83 - 11y - 3y = 27
-14y = -56
y = 4
Substitute y back into the new equation for x:
x = 83 - 11(4)
x = 83 - 44
x = 39
The solution to the system is x = 39, y = 4.
6. Multiply the first equation by 3 and subtract it from the second equation:
(3)(4x + 2y) = (3)(2)
12x + 6y = 6
12x + 4y = 8
12x + 4y - (12x + 6y) = 8 - 6
-2y = 2
y = -1
Substitute y back into the first equation:
x - 3(-1) = 27
x + 3 = 27
x = 24
The solution to the system is x = 24, y = -1.