Solving equations
Consider the equation: 8 − 3 = 4 + 1.
What is the value of both the LHS and RHS ? Both have the value of
Let's try addition: Add 2 to both sides, which gives us
LHS: 8 − 3 + 2 =
RHS: 4 + 1 + 2 =
We see that the equality holds i.e. 7 = 7.
Thus, if we add the same number to both sides of an equality, the equality still holds.
Let's try subtraction: Subtract 2 from both sides, which gives us:
LHS : 8 − 3 − 2 =
RHS : 4 + 1 − 2 =
We see that the equality holds i.e. 3 = 3.
Thus, if we subtract the same number from both sides of an equality, the equality still holds.
Let's try multiplication: Multiply both the sides of the equality by 3, we get
LHS = 3 × (8 – 3) = 3 × 5 =
RHS = 3 × (4 + 1) = 3 × 5 =
We see that the equality holds i.e. 15 = 15.
Let's try division: Divide both sides of the equality by 2, we get
LHS = (8 – 3) ÷ 2 = 5 ÷ 2 =
RHS = (4+1) ÷ 2 = 5 ÷ 2 =
We see that the equality holds i.e.
Similarly, if we multiply or divide both sides of the equality by the same non-zero number, the equality still holds.
If we take any other equality, we shall find the same conclusions.
Suppose, we do not observe these rules. Specificially, suppose we add different numbers, to the two sides of an equality.
We shall find in this case that the equality
For example, let us take again equality 8 – 3 = 4 + 1
Adding 2 to the LHS and 3 to the RHS, we get:
LHS: 8 – 3 + 2 = 5 + 2 =
RHS: 4 + 1 + 3 = 5 + 3 =
The equality does not hold, because the new LHS and RHS are not equal.
Thus if we fail to do the same mathematical operation with same number on both sides of an equality, the equality may not hold.
The equality that involves variables is an equation.
Important Note: These conclusions are also valid for equations, as in each equation variable represents a number only.
As seen earlier at the beginning of the section, often an equation is said to be like a weighing balance.
Doing a mathematical operation on an equation is like adding weights to or removing weights from the pans of a weighing balance.
An equation is like a weighing balance with equal weights on both its pans, in which case the arm of the balance is exactly horizontal.
If we add the same weights to both the pans, the arm remains horizontal. Similarly, if we remove the same weights from both the pans, the arm remains horizontal. On the other hand if we add different weights to the pans or remove different weights from them, the balance is tilted; that is, the arm of the balance does not remain horizontal.
1. Consider the equation: x + 3 = 8
We shall subtract 3 from both sides of this equation.
LHS: x + 3 – 3 =
Thus, we earlier had: LHS =
Now, we have: LHS = x, RHS =
Since this does not disturb the balance, we have New LHS = New RHS (or) x = 5 which is exactly what we want, the solution of the equation.
Validation: To confirm whether we are right, we shall put x = 5 in the original equation. We get LHS = x + 3 = 5 + 3 =
2. Let us look at another equation: 5y = 35
We shall
New LHS =
Original LHS:
New LHS: y, New RHS:
Therefore, y = 7, which is the required solution.
By putting y = 7 in the original equation, we confirm that the solution is correct:
LHS of original equation =
This is equal to the RHS as required.
One can see that in the above examples, the operation we need to perform depends on the equation.
Our attempt should be to get the variable in the equation separated.
Sometimes, for doing so we may have to carry out more than one mathematical operation.
Example 5: Solve (a) 3n + 7 = 25
- Subtracting 7 from both sides, we get 3n =
- We divide 3 on both sides which gives us
= - Thus, we have found the value of n.
(b) 2p – 1 = 23
- Adding 1 to both sides, we get 2p =
- We divide 2 on both sides which gives us
= - Thus, we have found the value of p.
What about the Mind Game?
We are now in a position to go back to the mind-reading game and understand how it works.
We already know that the equation for the mind-game becomes:
Say, we have thought of the number:
Thus, the equation becomes:
Subtracting 8 from both sides:
Which gives us:
Divide both sides by 2: this will separate x.
We get