Simultaneous Equations - Part I

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We have studied how to solve equations that have a single variable like 2x + 1 = 3, 3m +  9 = 0, 
(x/3) - 1 = 2, etc. The trick to solve these equations was to first get the constants on one side and then try and make the coefficient of the variable as 1. All the operations have to be done without violating the equality side. This means that whatever operations are done on the left hand side of the equation, the same operation has to also be done on the right hand side of the equation.

In this chapter we will see how to solve equations that have two variables or unknowns.

What we will study in this chapter :

1. What are simultaneous equations?
2. How to find solutions for simultaneous equations
3. Graphical Method of solving simultaneous equations

1. What are simultaneous equations?
Consider the equation 2x + 5y = 10. Will you be able to find unique values of x or y[1]? No. You need some more information to find uniquely the values of x or y. 
Suppose the additional information is x - y = 0

Then you have two equations : 2x + 5y = 10
                                and         x  - y   = 0

With the additional information you can say that

x ñ y = 0  means x = y.

Substituting this in the first equation, we get :

2x  + 5 (x) = 10
7x = 10
x = 10/7
and y = 10/7
The values of x and y make the equation 2x + 5y = 10 valid.

LHS :  2 * 10/7 + 5 * (10/7) =  70/7 = 10
LHS = RHS

The values of x and y are thus correct.

Equations that have to be solved simultaneously (at the same time) are called simultaneous equations. In the above example, the values of x and y have to simultaneously satisfy the two equations 2x + 5y = 10 and x ñ y = 0.

In general, there is a rule that if there are two unknown variables in an equation, you need another equation with the same variables, with which we will be able to find the values of the two variables; if there are three variables, you need three simultaneous equations. Going further, if there are ëní number of variables in an equation, then you need ëní number of simultaneous equations to find the individual ëní variable values.

In this chapter we will consider simultaneous equations with only two variables, and hence only two equations. Since the index of variables used is 1, theses equations are sometimes called simultaneous linear equations also. (In case the index of one of the variables is 2, then the equations are called quadratic equations[2].)

[1] Sometimes we do encounter such a situation. Then one can solve the equation analytically. This means that given a series of values of x for which we can find corresponding series of values of y. Solving equations analytically is beyond this syllabus.

[2] Equations containing many variables with various indices can be solved but the methods are more intricate. Matrix method is one of the ways of solving these equations. But the discussion of this is not included in the present syllabus.

 

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