Solving Linear Equations

Performing Mathematical Operations on Equations

When we are doing mathematical operations on a linear equation, we should do it on both sides of the equality otherwise the equality won’t hold true.
Suppose,  4x + 3 = 3x +7 is a linear equation. If we want to subtract 3 from the given equation, then we do it on both sides of the equality, so that the equality holds true.
$4x+3-3=3x+7-3$
$\Rightarrow 4x=3x+4$
Similarly, if we want to multiply or divide the equation, we multiply or divide all the terms on the left side of the equality and to the right side of the equality by the given number.
Note: we can not multiply or divide the equation by 0.

Solving Equations with Linear Expression on one side and numbers on the other Side

Suppose we have to find the solution of $2x-3=7$, where the linear expression is on the left-hand side, and numbers on the right-hand side.
Step 1: Transpose all the constant terms from the left-hand side to the right-hand side.
$2x=7+3=10\Rightarrow 2x=10$

Step 2: Divide both sides of the equation by the coefficient of the variable.
In the above equation 2x is on the left-hand side. The coefficient of 2x is 2.
On dividing the equation by two, We get:

$1/2 \times 2x= 1/2\times 10$
$\Rightarrow x=10/2=5$, Which is the required solution.

Algebraic Equation

The statement of equality of two algebraic expressions is an algebraic equation. It is of the form $P=Q$, where $P$ and $Q$ are algebraic expressions.
6x + 5 and 5x + 3  are algebraic expressions. On equating the algebraic expressions we get an algebraic equation.
6x + 5 = 5x + 3 is an algebraic equation.

Linear Equations in One Variable

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable, where the highest power of the variable is one.
If the linear equation has only a single variable then it is called a linear equation in one variable.
For example, 7x + 4 = 5x + 8 is a linear equation in one variable.