**Solve using the linear combination method**
1.
4x + y = 4
2x + y = 6
2.
x - y = 10
5x + 2y = 12
3.
5/2x + 2y = -2
-x + y = 2
The linear combination method is also known as the "addition" method
1. We can pick either variable to eliminate....I'll pick y
4x + y = 4 (1)
2x + y = 6 multiply this through by -1 ⇒ -2x - y = -6 (2)
Add (1) and (2)
2x = - 2 divide both sides by 2
x = -1
And using (1) to solve for y we have
4(-1) + y = 4
-4 + y = 4 add 4 to both sides
y = 8
2.
x - y = 10 multiply through by 2 ⇒ 2x - 2y = 20 (1)
5x + 2y = 12 (2)
Add (1) and (2)
7x = 32 divide both sides by 7
x = 32 / 7
Using (1) to solve for y
(32/7) - y = 10 multiply through by 7
32 - 7y = 70 subtract 32 from both sides
-7y = 38 divide both sides by -7
y = -38/7
3.
5/2x + 2y = -2 multiply through by 2 to clear the fraction ⇒ 5x + 4y = -4 (1)
-x + y = 2 multiply through by 5 ⇒ -5x + 5y = 10 (2)
Add (1) and (2)
9y = 6 divide through by 9
y = 6/9 = 2/3
Put this into -x + y = 2
-x + (2/3) = 2 subtract 2/3 from both sides
-x = 2 - 2/3 = 4/3
-x = 4/3
x = -4/3