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It is the method of eliminating one variable, so that you can solve for the other variable.

Eliminating is a technique utilized to solve a system of equations.
It is best to solve in the standard format. 
It doesn't matter which x- or y-value is eliminated, first.
The variable-value eliminated in one equation can now solve for the missing variable-value in the other.
There can be no solution, one solution or infinite solutions to a system of equations.
ELIMINATION: STEPS TO SOLVING SYSTEM OF EQUATIONS
Discovering solutions to system of equations

Rule:
if a & b are real numbers, then
if x = a, y = b,  & a = b, then one solution
if 0 = c, then  no solution
if 0 = 0, then infinite solutions

Steps to solving an equation by elimination
  • Does the equations have similar aligned variables?
  • Are the equations in standard form?
  • What similar coefficient values can be eliminated whose sum can equal to zero?
  • What is the LCM of the two coefficient values?
  • Choose which equation to begin the elimination?
  • What factor should be multiplied to eliminate the coefficient values?
  • Multiply that value by every number in the equation:
  • Choose either equation & substitute that value for the other value (x or y).
  • SOLUTION (X, Y) VALUES (one solution, no solution, infinite solutions)
 -3x + y =  5
 -2x - y =  -10
 -5x + 0 =  -5
         -5x = -5  -- divide by 5
         
               x =  1
-5       -5
ELIMINATION: EXAMPLE 1
EQ 1:  Standard form:  -3x + y = 5
EQ 2: Standard form:     2x + y = 10
  • Does the equations have similar aligned variables? YES
  • Are the equations in standard form? YES
  • What similar coefficient values can be eliminated whose sum can equal to zero? 1y
  • What is the LCM of the two coefficient values? 1
  • Choose which equation to begin the elimination? (note: either one)? EQ. 2
  • What factor should be multiplied to eliminate the coefficient values? -1
  • Multiply that value by every number in the equation:
             2x  +  y  =  10
2x(-1) + y(-1) = 10(-1)  -- multiply -1 by every term
              -2x - y =  -10  -- updated equation
  • Add both equations to get the first part of the solution (x or y): Solve for x
  • Choose either equation, then substitute that value for the other value (x or y): Solve for y
 Substitute the value x = 1 into original EQ.1 OR EQ. 2

EQ.1:
     -3x + y = 5
  -3(1) + y = 5  -- substitute 1 for x
-3 + 3 + y = 5 + 3  --add 3 to both sides
          0 + y = 8
              y = 8
Rule a = b (x = 1, y = 8), therefore ONE solution to the equation.
The equations are independent, one y-value depends on one x-value to solve the equation.
These equations are consistent and considered intersecting lines. 
ELIMINATION: EXAMPLE 2
EQ 1:  Slope-intercept:  y = -2x + 5
EQ 2: Slope-intercept:  y = -2x + 1
  • Does the equations have similar aligned variables? YES
  • Are the equations in standard form? NO
  • What similar coefficient values can be eliminated whose sum can equal to zero? 2x & 2x
  • What is the LCM of the two coefficient values?  2
  • Choose which equation to begin the elimination? note: either one)?  EQ. 1
  • What factor should be multiplied to eliminate the coefficient values?  -1
  • Multiply that value by every number in the equation:
                2x + y = 5
     2x(-1) + y(-1) = 5(-1)  --multiply -1 by every term
               -2x - y = -5  --updated new EQ. 1
  • Add both equations to get the first part of the solution (x or y): Begin by solving for x
Standard Forms
2x + y = 5
2x + y = 1
EQ. 1:
             y = -2x + 5  --updated EQ 1
   2x + y = -2x + 2x + 5  -- add 2x to both sides
  2x + y = 0 + 5
   2x + y = 5
EQ. 2:
             y = -2x + 1  --updated EQ 2
   2x + y = -2x + 2x + 1  -- add 2x to both sides
  2x + y = 0 + 1
   2x + y = 1
  -2x - y =  -5
    2x + y =   1   
      0 + 0 =  -4
              0 = -4
EQ. 1:
EQ. 2:   
Rule 0 = c (0 = -4) , therefore NO solution to the equation.
The equations are independent, no y-value depend on a x-value to solve the equation.
These equations are inconsistent and considered parallel lines.
EQ. 1:
               y = -2x + 2
                          
        (3)y = (3) -2x + 2(3)  -- multiply 3 by every term    
            3y = -2x + 6
  2x + 3y = -2x + 2x + 6  --add 2x to both sides
  2x + 3y = 0 + 6  
   2x + 3y = 6
3
3
ELIMINATION: EXAMPLE 3
EQ 1:  Slope-intercept:  y  =  -2x/3  + 2
EQ 2: Standard form:    4x + 6y = 12
  • Does the equations have similar aligned variables? YES
  • Are the equations in standard form? NO, not both
Only one variable is not in standard form
Standard Form
   2x + 3y = 6
 -4x - 6y = 12
  • What similar coefficient values can be eliminated whose sum can equal to zero? 2x & -4x
  • What is the LCM of the two coefficient values?  4 
  • Choose which equation to begin the elimination? EQ. 1
  • What factor should be multiplied to eliminate the coefficient values? -2 
  • Multiply that value by every number in the equation:
                     2x + 3y = 6
    2x(-2) + 3y(-2) = 6(-2)  -- multiply -2 by every term
                   -4x - 6y = -12  -- updated EQ. 1
  • Add both equations to get the first part of the solution (x or y): Begin by solving for x
  -4x - 6y = -12
    4x + 6y =  12
            0 + 0 =  0
                    0 = 0
EQ. 1:
EQ. 2:   
Rule (0 = 0) , therefore infinite solutions to the equation.
The equations are dependent, every y-value depends on every x-value to solve the equation.
These equations are consistent and are on the same linear lines.
ELIMINATION: EXAMPLE 4 (Challenging problem)
EQ 1:  Standard Form:    4x + 3y = 10
EQ 2: Standard Form:    9x - 4y = 1
  • Does the equations have similar aligned variables? YES
  • Are the equations in standard form? YES
  • What similar coefficient values can be eliminated whose sum can equal to zero? 4x & 9x OR 3y & 4y 
​
NOTE:
You can observe that the values 4x & 9x  NOR 3y & 4y can be quickly eliminated like the former steps.  Therefore, you must consider the LCM of both of the coefficient values.
​
  • What is the LCM of the two coefficient values?  if we choose 4x & 9x, then 36 is LCM
  • Choose which equation to begin the elimination? In this instance you must use both equations
  • What factor should be multiplied to eliminate the coefficient values? Because the product of 4 * ? cannot be eliminated to equal 9 OR 9 * ?  cannot be eliminated to equal 4, then we need two different factors. Both have common multiple of 36, therefore, 4 * 9 = 36 and 9 * 4 = 36. 
  • Multiply that value by every number in the equation: The factor for 4 is 9, and the factor or 9 is -4. 
Common Factors of 4 & 9
4: 4, 8, 12, 16, 20, 24, 28, 32, 36
9: 9, 18, 27, 36

The LCM is 36.

Next consider what factors will give you the product value of 36.

4 * 9 = 36
9 * -4 = -36

Multiply the both equations by there respective factors. This will cause the  x coefficents to be eliminated.                                      
Add both equations to get the first part of the solution (x or y): Solve for y
                     4x + 3y = 10
                     9x - 4y =  1

        [4x * 9] + [3y *9] =  [10 *9]  --multiply 5 by every term
 [9x * -4] - [4y * -4]  = [1 * -4] --multiply 2 by every term

                   36x   +   27y      =  90
                 -36x   -   (-16y) =   -4
                       0     +       43y =  86
                              43      43

                                  y = 2
  • Choose either equation, then substitute that value for the other value (x or y): Solve for x
 Substitute the value y = 2 into original EQ.1 OR EQ. 2

EQ.1:
     9x - 4(2)  =  1
             9x - 8 =  1
             9x - 8 + 8 =  1 + 8
                      -9x = -9
                       -9     -9

                       x = 1
Rule a = b (x = 1, y = 2), therefore ONE solution to the equation. The equations are independent, one y-value depends on one x-value to solve the equation. 
These equations are consistent and considered intersecting lines. 
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