Simultaneous Linear Equations
      System of two linear equations in two unknowns (variables)
         The system has, a single solution, no solution or has infinitely many solutions
         Solving systems of equations
      Substitution and comparison method
      Addition or elimination method
System of two linear equations in two unknowns (variables)
Two linear equations in two unknowns x, y form a system if they can be written in the standard form:
that is, in the form where the variables in both equations are in order on the left side and the constant term c is on the right. 
The numbers, a1 and a2, are the coefficients of the x variable and, b1 and b2, are the coefficients of the y variable.
Underlying indicates that the two equations should be solved simultaneously.
The solution of a linear equation in two unknowns is every ordered pair of numbers (x, y), when substituted into equation, satisfy the equation.
All these ordered pairs of numbers are the points of the strait line in a coordinate plane. Therefore, simultaneous equations can be solved graphically.
The solution of the system of equations, are the values of the variables that make both equations true at the same time.
Graphically it is the intersection point of the two lines, because the intersection point is the only point that lies on both lines at the same time.
The system has, a single solution, no solution or has infinitely many solutions
The following three cases are possible for any given system of linear equations:
a) The system has a single solution, lines intersect, and the coordinates (x, y) of the intersection point are
the one and only solution to the system of equations.
In this case the ratios of the coefficients of the unknowns x, y are not equal,  
In this case the system of equations is called independent system.
b) The system has no solution.
Graphically, lines are parallel, i.e., have the same slope, but are not identically the same line and they will
never intersect, that is:  
Equations are said to be inconsistent.
c) The system has infinitely many solutions. Equations describe the same line.
The three ratios are equal:  
This is called dependent system.
Solving systems of equations
When solving a system of simultaneous equations in two unknowns, we use algebraic methods to get such equivalent system at which one of two equations will contain only one variable.
Then, solve that equation, and substitute its solution in another equation. That way, this equation also turns into a single equation with a single unknown.
Finish by solving that single equation.
Substitution and comparison method
With the substitution method, we solve one of the equations for one variable in terms of the other, and then substitute that into the other equation.
Example:  Solve the system of equations.
Similar is the comparison method.
Solve both equations for the same variable in terms of the other, and then compare them, as is shown in the example.
Example:  Solve the system of equations using comparison method.
Addition or elimination method
To eliminate one of the variables from either of equations to obtain an equation in just one unknown, make one pair of coefficients of the same variable in both equations negatives of one another by multiplying both sides of an equation by the same number.
Upon adding the equations, that unknown will be eliminated.
Example:  Solve the system of equations using elimination method.
Intermediate algebra contents
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