Formation of a System of Linear Equations

A linear system can be built directly from a real situation. Suppose three shoppers, A, B and C, buy the same brands of rice and sugar in a supermarket: A buys 5 kg rice + 3 kg sugar for Rs.,440; B buys 6 kg rice + 2 kg sugar for Rs.,400; C buys 8 kg rice + 5 kg sugar for Rs.,720. Letting $x$ = price per kg of rice and $y$ = price per kg of sugar, each purchase gives one linear equation:

This is a system of linear equations in the unknowns $x,y$ -- here 3 equations in 2 unknowns. Values of $x$ and $y$ satisfying every equation simultaneously form a solution of the system. Since $x,y$ each appear only to the first degree, the equations are called simultaneous linear equations; geometrically, each equation is a straight line in the plane, and a common solution is a common intersection point. …