Linear equations are equations of the first degree. This means that all the numbers in the equation must be written with the exponent 1. So, if any number had an exponent that is not equal to 1, the equation would no longer be classified as linear. The standard form of a linear equation is:
a is the coefficient, x is the unknown or the variable, and b is the constant.
This is known as the one-variable form of a linear equation. In order to solve it, you can use the following formula for x:
If the equation you have doesn’t fit this form, you need to convert it, and it is pretty simple. For example:
In order to solve it, you need to make sure the right side is equal to 0. To accomplish that, we need to transfer 25 to the left side. Keep in mind, once it switches sides, it also becomes negative.
Now, to solve it, you simply use the formula we mentioned before:
With two-variable linear equations, it is not that simple, however. When we have more variables, we also have more solutions. Logically, the more variables you have, the more combinations you can make with those variables in order to get the desired result. Two-variable linear equations take the form of:
Describing linear relationships
A linear relationship is a relationship between two numbers that, when put on a cartesian plane, form a line. Presenting them graphically would show this line, however they can also be demonstrated mathematically, using the form of:
In this form, m is the slope of the line, and b is the y-intercept.
In order to be a linear relationship, an equation must meet 3 criteria:
- The equation needs to have a maximum of two variables; if it has more than two, it is not a linear relationship
- Every variable has to have a power of 1; just like with a linear equation, every number must have 1 as its exponent, otherwise, the equation is not linear
- Once put on a plane, it has to graph with a straight line; this is the best indicator, as when you graph an equation, and make sure you did everything correctly, if you get a straight line it is a linear relationship
We actually encounter linear relationships in our everyday lives, even if we’re not consciously aware of it. For example, for calculating the speed a car goes on a highway. If a policeman knows for how long someone was travelling on a highway, and how much distance they covered in that time frame, they can calculate the average speed of that car. It is a very useful method for confirming if someone was speeding or not, even though it is not perfect.
Another commonly used type of linear relationship is correlation. Correlation demonstrates the power of a relationship between two variables, and it is shown through the correlation coefficient. The correlation coefficient has a maximum value of 1, and a minimum value of -1.
Having a value of 1 means the correlation is positive and perfect. Graphically, the two variables would move in the same direction, at the same pace.
Having a value of -1 would mean the correlation is negative and perfect. Graphically, the two variables would move in opposite directions. If the value is 0, however, it is considered that there is no linear relationship between the two variables at all.
How do you solve linear equations on a calculator?
With our linear equation calculator, solving linear equations has never been easier. In fact, it is as easy as entering the equation into the calculator, and getting the result basically instantaneously. Keep in mind that, because our calculator uses the formula for x we mentioned before, your equation does have to have 0 on the right side, otherwise the result will not be true.