2D (Binary) Systems and Gaussian Elimination
Here are three elimination methods for solving systems (sets) of
linear equations
- Substitution
- Comparison
- Equation (or Row) Addition and Subtraction, as is or after multiplication.
Learn all three, and watch for situations in which one requires less work
than the others. The last method only works for linear systems (sets) of
simultaneous equations.
Remember to check your answers by making sure all the
equations in the systems being solved, the original one, are satisfied.
If the check fails, you have made a mistake in obtaining the answer or doing
the check. If you answer differ from someone else's, at least one of you has
made a mistake, possibly both. Here there is a difference between the method
for checking answers and the methods for obtaining answers.
Substitution (Replacement) Method
Example 1: The set or system of equations
x = 25 + 2y
x + 3y = 10
becomes a system in one unknown when we eliminate the x from the second
equation by replacing x in the second by the expression 6 +2y.
The second equation
x + 3 y = 10
then gives
(25+2y) + 3 y = 10
or
25 + 5y = 10.
So 5y = -15 and hence y = -3 (from -15 divided by 5)
Example 1 Revisited: Now consider the set or system of equations
x - 2y = 25
x + 3y = 10
In this system, neither x nor y is expressed in terms of the other. But
the first equation implies x = 25 + 2y as before by adding 2y to both
sides of it. So both equations together give the previous set of equations
x = 25 + 2y
x + 3y = 10
What we did here was express x in terms of the other unknown to allow us
to convert the original system into a system with essentially one unknown.
See the first example again.
Example 2: Animated

Example 3: Now lets us consider the third binary or two unknown
system
-2x + y = 8
3x + 4y =21
Here the first equation implies y = 8 -2x. So the third system gives
y = 8+ 2x
3x + 4y =21
So we replace y in the second equation 3x + 4y =21 by the expression 8 + 2x
that should have the same value as y. This implies
3x + 4 (8+2x) =21 or
3x + 32 + 8x =21 or
11x+32 = 21 or 11x = -11
So x = -1. Now y = 8 + 2x gives y = 8 +2(-1) = 6.
Note 4 (8+2x) = 32 + 8x due to the distributive law, namely
a(b+c) = ab + ac whenever a, b and c are (real) numbers.
I will leave to you to check that
-2x + y = 8
3x + 4y =21
when x = -1 and y=6.
Example 4: Animated

|