In the earlier classes, you have come across several algebraic expressions and equations. Some examples of expressions we have so far worked with are: We however, restrict to expressions with only one variable when we form equations. Moreover, the expressions we use to form equations are linear. This means that the highest power of the variable appearing in the expression is 1.Here we will deal with equations with linear expressions in one variable only. Such equations are known as linear equations in one variable. The simple equations which you studied in the earlier classes were all of this type.
Let us recall the technique of solving equations with some examples. Observe the solutions; they can be any rational number.
We begin with a simple example.
Sum of two numbers is 74. One of the numbers is 10 more than the other. What are the
We have a puzzle here. We do not know either of the two numbers, and we have to
find them. We are given two conditions.
(1) One of the numbers is 10 more than the other.
(2) Their sum is 74.
An equation is the equality of the values of two expressions. In the equation 2x – 3 = 7, the two expressions are 2x – 3 and 7. In most examples that we have come across so far, the RHS is just a number. But this need not always be so; both sides could have expressions with variables. For example, the equation 2x – 3 = x + 2 has expressions with a variable on both sides; the expression on the LHS is (2x – 3) and the expression on the RHS is (x + 2).
The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. What can be the original number? Solution: Take, for example, a two-digit number, say, 56. It can be written as 56 = (10 × 5) + 6. If the digits in 56 are interchanged, we get 65, which can be written as (10 × 6 ) + 5.
Multiplying both sides of the equation by 6,
Observe that the equation is not a linear equation, since the expression on its LHS is not linear. But we can put it into the form of a linear equation. We multiply both sides of the equation by (2x + 3).