LINEAR EQUATIONS WITH TWO UNKNOWN VARIABLES.

System of Simultaneous Equations : Two or more simultaneous equations having the same variables is called a system of simultaneous equations.

Solution : A solution of a couple of simultaneous equations is the values of the two variables which satisfy the simultaneous equations.

Methods to solve Simultaneous Equations : 
Graphical method : Using one equation, find the coordinates (x,y). Represent the given system of simultaneous equations on a graph using the variables x and y corresponding to the two axes X and Y by plotting the points. Draw lines passing through them and the coordinate points of intersection of these lines are the solutions of the simultaneous equations. 
Determinant Method : Take the coefficients of all terms from the two equations and represent them in a 2X2 Matrix with the coefficients of the first equation in the top and coefficients of second equation in bottom row. Write 3 such matrices namely D, Dx and Dy. Matrix of Dxwill contain the two RHS and two coefficients of variable y in the left and right column respectively; similarly for Dy. Now cross multiply and subtract 'ad-bc' and find value of D, Dx and Dy. Then find x and y by using x = Dx / D and y = Dy / D. 

Condition of Consistency of Equations : We can decide the nature and number of the solutions by using the following chart, where a1, a2 are coefficients of x, b1, b2 coefficients of y and c1, c2 the RHS.
 Simultaneous Equations a1 / a2  b1 / b2  c1 /  c2  Graphical InterpretationAlgebraic Interpretation Consistency 
 x + y = 3
x - y = 1
1 / 1  1 / -13 / 1  Intersecting LinesUnique Solution Consistent 
 2x - y = -1
2x - y = 4
2 / 2  -1 / -1 -1 / 4 Parallel Lines No SolutionInconsistent 
x - y = -2
2x - 2y = -4 
 1/2-1 / -2 -2 / -4 Coincident Lines  Infinite SolutionsConsistent 
Equations Reducible to Simultaneous Equations : Some equations can be made into simultaneous equations by making suitable substitutions. For example,
4/x + 3/y = 1, 8/x + 9/y = 7 
Substituting 1/x = m and 1/y = n we get,
4m + 3n = 1, 8m + 9n = 7