An equation of the form mx + b = 0 is a linear equation or an equation of a straight line. Any other equations, which can be rewritten to that form, are also linear equations, that is, they have one variable x, which is raised to the first power.
Usually the multiplication sign between the constant and the variable of an equation is omitted, like for instance 2x. In the following example, however, we will write instead of to make it more clear, that 2 must be multiplied by x.
Examples of linear equations
Solving a linear equation
In order to solve a linear equation, you must rewrite it, until the variable x is isolated on one side of the equation.
Every time you rewrite an equation you put a “if and only if”-symbol ⇔, indicating that the equation in its initial form is true if and only if the equation in its rewritten form is also true.
As you can see all three expressions above are equivalent, which is shown with ⇔ after each expression. It is the same equation in different forms.
How to rewrite an equation
You can add or subtract the same number on both side of the equation.
You can multiply or divide by the same number (except 0) on both side of the equation.
Guide to solve an equation
When you want to solve an equation, the first thing you do is to get rid of all fractions appearing in it. You can get rid of a fraction by multiplying it by its own denominator, because then you get its numerator. But if you do so, you must multiply by the same number on the other side of the equation as well, according to the rules above.
If an equation contains parentheses, they must be removed as well. (see more about the properties of parentheses on our site concerning parentheses)
Isolation of x
Finally you can isolate x on one side of the equation in accordance with the rules above concerning addition, subtraction and multiplication.
The linear equation and the Zero Product Property
Sometimes you will come across equations like:
In this case you can solve the equation on the basis of the Zero Product Property, which states that when two quantities multiply to get zero, either one or both of the quantities must be zero.
So instead you have to solve these two linear equations.
There will be two solutions to the initial equation, because there is one solution for each of the two equations.
Examples of solving a linear equation
Simple linear equation
We add 4 on both sides:
We divide by 4 on both sides:
Linear equation with parentheses
The first thing we do is to get rid of the parentheses. According to the distributive property the term outside the parentheses distributes across all terms inside the parentheses. Be aware of the signs + and – in front of each term, which also distribute equally across the parentheses.
We subtract 12 from both sides.
We add 6x on both sides.
We divide by 10 on both sides.