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Linear Equation


What is a Linear Equation?

A Linear Equation is any equation that can be written in the form
$$ax + b = 0$$ where
  • • $x$ represents a variable or an unknown

  • • a and b are constants

  • • a ≠ 0



Note that the power (or exponent) of variable $x$ is 1. If the exponent of any variable is greater than 1, the equation is non-linear.

A linear equation in one variable can always be written as $$ax = b$$
If $a ≠ 0$, the above equation can be written as $$x = \frac{b}{a}$$
Here are some examples of linear equations
• $2x + 8 = 0$
• $\frac{m + 2}{4} + 5 = 9m$


Please note
• $9x^2 + 5 = 0$ is not a linear equation as $x^2$ term has a power greater than 1. It is a quadratic equation.
• $4y^{\frac{1}{3}} = 5$ is also not a linear equation as power of $y$ is not equal to 1



Linear Equations with more than one variable

Linear equations can have one or more variables.
Consider the equation $ax + by + cz + d = 0$ where a, b, c and d are constants and a, b and c are ≠ 0.
This is also a linear equation but number of variables/unknowns are more than one.

A common form of linear equation in two variables $x$ and $y$ is written as $$y = mx + b$$ where $m$ and $b$ are constants.

The linear equation is called so as the solution of such equations form a straight line in a plane. Refer next section to check out.
In the above equation $m$ represents slope of the straight line and $b$ represents the y intercept (point at which the line crosses y axis)

Here are some examples of linear equations with two variables:
• $2x - 5y = 0$
• $25c = 50d$
• $\frac{m + 2}{4 + n} + 5 = 9$




Solving a Linear Equation

Solving a linear equation involves finding the values of variable(s) which satisfy the equation.


• First and the most important point to remember while solving an equation (linear or otherwise) is that whatever we do on one side of the $ “ = ” $ sign, the exact same things should be done on the other side of the sign.
While solving linear equations, we will make heavy use of this fact.
If a constant is added to Left Hand Side of the equation (LHS), we need to add the same constant on the Right Hand Side (RHS) of the equation.
This is true for all operations - addition, subtraction, multiplication, division etc

• If there are fractions on one or both sides of the equation, use Least Common Denominator (LCD) to clear the fractions.

• Simplify both sides by solving brackets (parenthesis) and combining like terms.

• Try to isolate the variable.

Consider an equation
$4x - 8 = 0$

• Add 8 on each side
⇒ $4x - 8 + 8 = 8$ ⇒ $4x = 8$
• Divide both sides by 4
⇒ $\frac{4x}{4} = \frac{8}{4}$ ⇒ $x = 4$

We can check if our answer is right by placing this value of $x$ in the equation above.
$4x - 8 = 0$
$4 × 4 - 8 = 0$
$8 - 8 = 0$
$ 0 = 0 $
LHS and RHS tally. Hence our answer is right.


Geometric Interpretation

The graph of a linear equation is a straight line.
The values of m and b determine the slope and the y intercept of this line.




As we change the value of m and b, see how the slope and y intercept changes.
$y = x$ ⇒ For this line, slope is 1 (m = 1) and y intercept is 0 (b = 0). The line crosses y axis at (0,0).
$y = -x$ ⇒ For this line, slope is -1 (m = -1) and y intercept is 0 (b = 0). The line crosses y axis at (0,0).
$y = 2x + 2$ ⇒ For this line, slope is 2 (m = 2) and y intercept is 2 (b = 2). The line crosses y axis at (0,2).