Describe when it would be easier to use the substitution method versus the elimination method.

Question

Not sure how to answer this.

Answer 1

Phyllis,

 

Just to add a bit more information, "Elimination" Can have a variety of other interpretations.  Elimination techniques typically refer to 'row reduction' to achieve 'row echelon form.'  Do not worry if you have not heard of these terms.  They are used in Linear Algebra when referring to "Elimination techniques"

 

1. Gaussian Elimination
2. Gauss-Jordan Elimination
3. LU-Decomposition
4. QR-Decomposition

 

These techniques for elimination are preferred for 3rd order systems and higher.  They use "Row-Reduction" techniques/pivoting and many subtle math tricks to reduce a matrix to either a solvable form or perhaps provide an inverse of a matrix (A^-1)of linear equation AX=b.  Solving systems of linear equations (n>2) by elimination is a topic unto itself and is the preferred method.  As the system of equations increases, the "condition" of a matrix becomes extremely important.  Some of this may sound completely alien to you.  Don't worry about these topics until Linear Algebra when systems of linear equations (Rank 'n')  become larger than 2.

 

Substitution is the preferred method for 2 equations in 2 unknowns.  The constants are unimportant other than having a non-zero determinant.  It is always easy to find multiplicative factors using LCMs of one variable or the other to allow substitution into the other equation:

 

Example:

 

4X + 5Y = 9

5X -  4Y = 1

 

Notice that 20 is a LCM of either the X or Y variable.  So multiply the first by 4 and the second by 5 and then adding the two (Y's will drop out allowing for substitution)

 

4(4X + 5Y = 9)

5(5X -  4Y = 1)  

 

Multiplying to produce the LCM factors:

 

16X + 20Y = 36

25X -  20Y = 5

 

Adding the equations

 

41X = 41

X = 1

 

Substitution into either equation yields

Y = 1

 

Elimination techniques are preferred for Rank-n>3

 

Answer 2

Hi Phyllis;

ELIMINATION...simple equations, one or neither variable has coefficient(s).

The elimination method is used to cancel out one variable in sets of simple equations which have more than one variable.

For example...

2x+y=10

x-y=-1

The coefficient of y in both equations is 1.  The coefficient of x in the first equation is 2.  In the second equation it is 1.

Let's add the two equations together to eliminate y...

3x+0=9

x=3

Let's subtract the second equation from the first to effectively plug-in x...

x+2y=

3+2y=11

2y=8

y=4

This would not be possible if both variables initially had different coefficients in each equation.

 

SUBSTITUTION...complex equations, variables have coefficients.

The substitution method should be applied to complex equations which use two variables and each has a different coefficient..  For example...

x+2=5y and

2x+4=y

Plug-in 2x+4 into the first equation...

x+2=5(2x+4)

x+2=10x+20

Now apply the addition method by subtracting 10x from both sides to combine like terms...

-10x+x+2=10x+20-10x

-9x+2=20

Again apply the addition method by subtracting 2 from both sides to again combine like terms...

-9x+2-2=20-2

-9x=18

x=-2

With x solved, y can also be resolved.

 

Answer 3

The substitution mthod is better to be applied if one of variables in one of equations has a coefficient equal 1, such as

 

                 x + by = c    or  mx +y = d

 

In this case you can write

 

                       x = c - by  or  y = d - mx

 

and substitute it into second equation of the system. In many instances both methods are qually easy

to apply,