__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, D

_{x}and D

_{y}. Matrix of D

_{x}will contain the two RHS and two coefficients of variable y in the left and right column respectively; similarly for D

_{y}. Now cross multiply and subtract 'ad-bc' and find value of D, D

_{x}and D

_{y}. Then find x and y by using x = D

_{x}/ D and y = D

_{y}/ D.

__Condition of Consistency of Equations :__We can decide the nature and number of the solutions by using the following chart, where a

_{1}, a

_{2}are coefficients of x, b

_{1}, b

_{2}coefficients of y and c

_{1}, c2 the RHS.

Simultaneous Equations | a_{1} / a_{2} | b_{1} / b_{2} | c_{1} / c_{2} | Graphical Interpretation | Algebraic Interpretation | Consistency |

x + y = 3
x - y = 1
| 1 / 1 | 1 / -1 | 3 / 1 | Intersecting Lines | Unique Solution | Consistent |

2x - y = -1
2x - y = 4
| 2 / 2 | -1 / -1 | -1 / 4 | Parallel Lines | No Solution | Inconsistent |

x - y = -2
2x - 2y = -4
| 1/2 | -1 / -2 | -2 / -4 | Coincident Lines | Infinite Solutions | Consistent |

__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