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

Skill in arithmetic with fractions is a must for algebra. 

Folder Chapters -
lesson groups

Proper Use of Equal Sign
A. Letters and Lengths
B.. Solving Linear Eq'ns - WS
C. Solving Linear Eq'ns - No Sticks
D. Almost One
E: 2D Systems - Sub Method.
E:  Continued
E: Still More
F. Larger Systems

Area Intro
(i) x + 20 = 29 WS
(ii) 2x + 5 = 20 WS
(iii) 3x + 10 = 32 WS
(iv) 5a + 16 = 3a+ 24 WS
(v)  (½)x + 8 = 24½ WS
(vi)  (¾)a + 16 = (¼)a+ 24 WS
(vii) (¾)q + 17 = 32 WS
(viii) 13 =[2/3]x +7 twice WS
(ix) Animated Examples WS
(a) Integral Coefficients (A)
(b) Integral Coefficients (B)
(c) Fractional Coefficients
(d) With parameters

Up
Proper Use of Equal Sign
A. Letters and Lengths
B.. Solving Linear Eq'ns - WS
C. Solving Linear Eq'ns - No Sticks
D. Almost One
E: 2D Systems - Sub Method.
E:  Continued
E: Still More
F. Larger Systems
 

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