The equation which in a general view is written down ah +bu + with =0 is called the linear equation with two variables. Such equation in itself contains an infinite set of decisions therefore in tasks it is always complemented with something – one more equation or the limiting conditions. Depending on the conditions provided by a task it is necessary to solve the linear equation with two variables in different ways.
It is required to you
- - linear equation with two variables;
- - the second equation or additional conditions.
Instruction
1. If the system from two linear equations is given, solve it as follows. Choose one of the equations in which coefficients before variables it is less and express one of variables, for example, x. Then substitute this value containing at in the second equation. In the received equation there will be only one variable at, transfer all parts with at to the left part, and free members – in right. Find at and substitute in any of the initial equations, find x.
2. It is possible to solve a system from two equations and in a different way. Increase one of the equations by such number that the coefficient before one of variables, for example, before x, was identical in both equations. Then subtract one of the equations of another (if the right part is not equal 0, do not forget to subtract similarly and the right parts). You will see that the variable x disappeared, and there was only one variable at. If solve the received equation, then and substitute the found value at in any of initial equalities. Find x.
3. The third way of the solution of a system of two linear equations – graphic. Draw the system of coordinates and represent schedules of two straight lines which equations are specified in your system. For this purpose substitute any two values x in the equation and you find corresponding at – it will be coordinates of the points belonging to a straight line. It is the most convenient to find crossing with axes of coordinates – to substitute values x =0 and at =0 enough. Coordinates of a point of intersection of these two lines will also be the solution of a task.
4. If in statements of the problem only one linear equation, so you were given additional conditions thanks to which it is possible to find a solution. Attentively read a task to find these conditions. If variables x and at designated distance, speed, age, weight – safely put restriction h0 and u0. Quite perhaps, under x or at the number of children, apples, trees, etc. disappears – then integers can only be values. If x – age of the son, it is clear that he cannot be more senior than the father therefore specify it in statements of the problem.
5. Construct the schedule of a straight line corresponding to the linear equation. Look at the schedule, perhaps, on it will be only several decisions meeting all conditions – for example, the whole and positive numbers. They will also be solutions of your equation.