Solving linear equations in one variable (Part 1)

Before we proceed to understand how to solve a linear equation in one variable, we need to know what does "solving" an equation mean.

Solving an equation means getting a definite value for each variable in the equation, which when substituted into the equation makes the equation balanced (that is, RHS = LHS). Thus, the "solution" of an equation is a value which when put in place of the variable/s, makes the equation "true". Here "true" means that on simplifying the equation, its left and right hand sides are equal.

For example, the solution of the linear equation `3x + 1 = 7` is `x = 2` because when we put `x = 2` in the equation, we get

`3(2) + 1 = 7`

`6 + 1 = 7`

`7 = 7`

In the above statement, `7 = 7` means that the left and right hand sides of the equation are same. Thus, the equation holds true for `x = 2`, and hence `x = 2` is a solution of the equation.

Now that we are clear as to what "solving" an equation means, we can proceed to learn how to solve a linear equation.

The golden rule of solving a linear equation

When all numbers in the equation are on one side of the = sign, and the variable of the equation is isolated on the other side, the equation is said to be solved and the value on the one side is called the solution of the equation.

Rules to be remembered when solving a linear equation

1. Any number added, subtracted, multiplied or divided on one side of a linear equation should be similarly added, subtracted, multiplied or divided on the other side.
2. Any term containing a variable added, subtracted, multiplied or divided on one side of the linear equation should be similarly added, subtracted, multiplied or divided on the other side.

Now we will solve a simple linear equation step by step to understand how it is solved:

Consider the linear equation

`x + 4 = 6`

In order to solve this linear equation, we need to move all the numbers in the LHS (Left hand side) to the RHS (right hand side) and hence isolate the variable 'x' in the LHS. There is only one number 4 in the LHS which has to be moved to the RHS. We know that `4 - 4 = 0`, thus we can subtract 4 from both sides of the equation to get rid of the 4 in the LHS.

`x + 4 - 4 = 6 - 4`

Since `4 - 4` is 0,

`x + 0 = 2`

We can remove the 0,

`x = 2`

Notice that in the above line, 'x' is isolated in the LHS. Thus it is the solution of the equation given above.

The above linear equation was solved in a single step. Thus, we call it a one-step linear equation.

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Solved Examples: One-step linear equations