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It is discovering one variable in one equation and substituting that value into the other equation to solve for other variable.

Substituting is a technique utilized to solve a system of equations.
Overall, one variable-value of one equatoin is substituted into the variable-value of other. 
In substituting, there can be one solution, no solution or infinite solutions.
STEPS TO SOLVING SYSTEM OF EQUATIONS: SUBTITUTING
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 = b, then no solution,
if 0 = 0, then infinite solutions

Steps to solving an equation by elimination
Does the equations have similiar aligned variables? YES, proceed

1) Are both equations in slope intercept form? NO, then, goto #2, otherwise proceed:    
  • Equate expressions and solve for the x-variable             
  • Substitue the x-value into one of the equations to solve for the y-value             
  • SOLUTION (X, Y) VALUES (one solution, no solution, or infinite solutions)
 
2) Are both equations in standard form? NO, then, goto #3, otherwise proceed:    
  • Convert one equation to slope-intercept form?
  • Substitute the slope-intercept expression into y-variable of the standard form.
  • SOLUTION (X, Y) VALUES (one solution, no solution, or infinite solutions)

3) Is one equation in standard form and the other slope-intercept form?,YES, then:
  •  Substitute the slope-intercept expression into y-variable of the standard form.
  •  SOLUTION (X, Y) VALUES (one solution, no solution, or infinite solutions)
ELIMINATION: EXAMPLE 1
EQ 1:  Standard form:  y = 2x + 4
EQ 2: Standard form:  y =  -3x   +   1 
2           2
Does the equations have similiar aligned variables? YES

1) Are both equations in slope intercept form? YES, then, make the expressions of each equation equal to each other.

---Equate the expressions and solve for x
 EQ. 1 expression:  2x + 4
EQ. 2 expression:   -3x/2 + 1/2
         2x + 4  =  -3x +  1  
                               2       2
2x(2) + 4(2) = -3x (2) +  1 (2)  --multiply by  2 to begin to solve for x
                                  
             4x + 8 = -3x + 1
      4x + 8 - 8 = -3x + 1 - 8  -- subtract 8 from both sides
              4x + 0 = -3x -7
           4x + 3x = -3x + 3x - 7  --add 3x to both sides
                 7x =  0 - 7
                       7x =  -7   --updated expression
                       7x =  -7
                         
                         x =   -1
 7         7
--divide both sides by 7
---Substitue the x-value into one of the equations to solve for the y-value
Substitute the x value for EQ. 1 or EQ. 2
This problem chooses EQ. 1
y = 2x + 4
y = 2(-1) + 4  --substitute -1 for x
y = -2 + 4 
y = 2
Rule a = b (x=1, y=2) , therefore ONE SOLUTION to the equation.
The equations are independent, however one y-value depend on one x-value to solve the equation.
These equations are consistent and considered intersecting lines.
 2             2
ELIMINATION: EXAMPLE 2
EQ 1:  Standard form:  2x + y = 5
EQ 2: Standard form:  2x + y = 1
Does the equations have similiar aligned variables? YES

1)  Are both equations in slope intercept form? NO, then, goto #2, otherwise proceed:
2) Are both equations in standard form? YES

---Convert one equation to slope-intercept form?
This problem choses EQ. 1 to convert to standard form
         2x + y = 5
2x - 2x + y = -2x + 5  -subtract -2x from both sides
            0 + y = -2x + 5 
                   y = -2x + 5
New equation for EQ. 1
y = -2x + 5
---Substitute the slope-intercept expression into y-variable of the standard form.
Substitute the y-variable into the expression of EQ 2.
EQ. 2:   2x + y = 1
                     2x + y =  1
     2x + (-2x + 5) =  1  --substitute (-2x -5) for y
            2x - 2x + 5 =  1  
                        0 + 5 =  1  
                  0 + 5 -5 =  1 - 5  -- subtract 5 from both sides
                                0 = -4
Rule 0 = c  (0 = -4), therefore there is NO SOLUTION to the equation.
The equations are independent, however,  y-variables do not depend on any x-variables.
These equations are inconsistent and considered parallel lines.
ELIMINATION: EXAMPLE 3
Discovering solutions in slope-intercept form and standard form
EQ 1:  Slope-intercept:  y = 2x + 4
EQ 2: Standard form: 4x - 2y = -8
Does the equations have similiar aligned variables? YES

1) Are both equations in slope intercept form? NO, then, goto #2, otherwise proceed:
2) Are both equations in standard form? NO, then, goto #3, otherwise proceed: 
3) Is one equation in standard form and the other slope-intercept form? YES

---Substitute the slope-intercept expression into y-variable of the standard form.
Substitute the y-variable in EQ. 2 with the expression of EQ 1.
EQ. 1: expression 2x + 4
             4x - 2y = -8  
4x - 2(2x + 4) = -8  --distributive property
      4x - 4x - 8 = -8  
                  0 - 8 = -8
                -8 + 8 = -8 + 8  --add 8 to both sides
                  0 + 0 = 0  
                         0 = 0
Rule 0 = 0 (0, 0) , therefore  INFINITE SOLUTIONS to the equation.
The equations are dependent on each other. This means every y-value depends on every corresponding x-value for a solution.
These equations are consistent and considered same linear equation lines.
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