Linear Algebra
In this problem, we need to solve a system of equations. There are 4 equations and 4
variables.
x1
4x1
3x1
-
x2
x2
+
+
+
2x3
x3
2x3
+
x4
+
+
x4
x4
=
=
=
=
4.3
4.8
4.6
-1.1
(1)
(2)
(3)
(4)
Here are my steps to solving the problem (I’ll refer to the equations as (1) through (4),
then add more numbers as needed):
Add (1) and (4):
x1
-
x1
x2
x2
0
x1
+
+
2x3
2x3
+
+
+
x4
x4
2x4
=
=
=
4.3
-1.1
3.2
+
2x3
+
2x4
=
3.2
+
+
-
x4
-6x4
5x4
=
=
=
4.6
-9.6
-5
(5)
Multiply (5) by -3 and add it to (3):
3x1
-3x1
0
+
+
+
x3
-6x3
-5x3
Simplify to get x3 + x4 = 1, and plug this expression into (3):
3x1 + (1) = 4.6
x1 = 1.2
We know x1! Now, make new equations by plugging that value into (1) through (4):
-x2
x2
+
+
+
2x3
x3
2x3
+
x4
+
+
x4
x4
=
=
=
=
(1’)
(2’)
(3’)
(4’)
3.1
0
1
-1.1
Look at (7) ... now we also know x3 = 0
Plug that into (1’) through (4’):
-x2
+
x4
=
3.1
(1’’)
x2
+
+
x4
2x4
=
=
1
-1.1
We know by (3’’) that x4 = 1. This means that x2 = -2.1
The answer is:
(x1, x2, x3, x4) = (1.2, -2.1, 0.0, 1.0)
(3’’)
(4’’)