simultaneous equations

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Two or more algebraic equations that contain two or more unknown quantities that may have a unique solution. For example, in the case of two linear equations with two unknown variables, such as:

(i) x + 3y = 6

and

(ii) 3y - 2x = 4

the solution will be those unique values of x and y that are valid for both equations. Linear simultaneous equations can be solved by using algebraic manipulation to eliminate one of the variables, coordinate geometry, or matrices (see matrix).

For example, by using algebra, both sides of equation (i) could be multiplied by 2, which gives 2x + 6y = 12. This can be added to equation (ii) to get 9y = 16, which is easily solved: y = 16/9. The variable x can now be found by inserting the known y value into either original equation and solving for x. Another method is by plotting the equations on a graph, because the two equations represent straight lines in coordinate geometry and the coordinates of their point of intersection are the values of x and y that are true for both of them. A third method of solving linear simultaneous equations involves manipulating matrices. If the equations represent either two parallel lines or the same line, then there will be no solutions or an infinity of solutions, respectively.

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