Before we look at simultaneous equations let us brush up some of the fundamentals. First, we define what is meant by an equation. It is a statement which indicates that two algebraic expressions are equal. For instance, let 3x - 4 be an expression and 5x - 10 be another expression. If these two expressions are related to each other by an equality sign in the fashion shown below we call it as an equation.
3x - 4 = 5x - 10 .......... (1)
The side on which we have the expression 3x - 4 is referred to as Left Hand Side (LHS) and the one which has 5x - 10 as the Right Hand Side (RHS). If we substitute x = 3 in the above equation we find that both sides of the equation gives us 5. Now we substitute some other value say x = 2. We find that the LHS gives us 2 whereas the RHS gives us 0. Looking at these two cases we conclude that only when x = 3, the equation holds and for other values of x it does not. But consider an equation which is shown below.
3x + 2 + 2x - 5 = 5x - 3 .......... (2)
The LHS and the RHS of this equation gives us the same values for any value of x. In other words, this equation holds for any value of x. Equations like these are called identities and the one we have seen before are referred to as equation of condition or more simply as an equation.
Above we have seen that only when we have substituted x = 3 in equ.(1), it holds true. That is, the value of x = 3 is said to be satisfying the equation. Since we are expected to find the value of x for which the equation holds true, the quantity x is known as the unknown quantity. The value of x found after solving the equation is called the solution or the root of the equation.
While solving equations, we have to remember these points.
1. If we are to add or subtract any quantity from/to one side of the equation, we should do so for the other side also. We look at this by taking an example.
For instance, we are required to solve the equation
x + 3 = 15
That is, on the LHS we ought to have only x. Can we subtract 3 from the LHS so that +3 and -3 cancel each other leaving behind only x? We can. But as stated above this operation should be done on both sides of the equation. That is, we will have
x + 3 - 3 = 15 - 3
x + 0 = 12
x = 12
If we do not perform this operation on both the sides, the balance which exists between the sides gets disturbed as a result the equality sign loses its relevance and it no longer has any meaning.
We take another example and check the same for addition. We have an equation x - 3 = 12, for which we have to obtain a solution.
x - 3 + 3 = 12 + 3
x = 15
As we are aware of this, while solving equations we directly transpose the quantity to the other side of the equation with its sign changed. Here we introduced a new word "Transpose". What is meant by Transposing? Bringing any term from one side of the equation to the other side is called transposing.
2. If we are to multiply or divide a particular element or the whole expression on one side of the equation, then we should do the same on the other side of the equation also. Let us take an example and understand this. We have to find the solution for the equation 3x + 5 = 20. We begin by subtracting 5 from both the sides. That will be
3x + 5 - 5 =20 - 5
3x + 0 =15
Since only x ought to be there on the LHS (i.e. solving for x), we divide the LHS by 3 and do a similar operation on the RHS also. We have
3x/3 = 15/3
Therefore, x = 5 is the solution of the given equation.
Suppose we are given an equation like
and asked to solve, how should we proceed?
We begin by multiplying both the sides of the equation by 3. We have
(x-4)/3 x 3 = 6 x 3
x - 4 =18
x - 4 + 4 =18 + 4