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).
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