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Mathematicians enjoy being meticulous and seeking out all possible connections. So simply deriving from an algebraic rule to creating a graph isn’t sufficient for them. They want to be able to travel in both directions. It’s not difficult to move from the graph to the rule because you can select the information you require. However, you will be given only small parts of knowledge, and it will be up to you to fill in the blanks.
What is a Linear Function?
A linear function is an algebraic equation in which each term is either a fixed or the product of a steady and a single variable. For example, the equation y=mx+b is a linear function since it fits both criteria using the variables x and y and the constants mm and bb. It is linear: the xx term’s exponent is one (first power), and it follows the definition of a function: there is exactly one output for each input (x) (y). Its graph is also a straight line.
Steps for solving linear equations
To solve a linear equation in algebra, it’s common to have to undo many operations. While complex equations may necessitate significant thought, and as previously said, numerous strategies for solving the same problem may exist, a typical starting point is to follow these steps:
Expand the brackets first if your equation has them (if you need a refresher on how to do this, refer to the Key terms and techniques page of this module).
If your equation contains like terms, group them first (if necessary, adding or removing them from one side of the equation), and then simplify the equation (again, refer to the Key terms and techniques page of this module if required).
If your equation needs the removal of more than one constant from the variable, undo each action in the following order:
Remove anything you’ve added or subtracted from the variable.
Remove anything that has been multiplied or divided by the variable.
Verify that the solution you got is correct by inserting the value back into the original equation and evaluating the result. If it’s not correct, go back and try again, seeing if you can figure out where you went wrong.
The standard form of a linear equation
The standard form of linear equation, often known as the “generic form,” is as follows:
Standard Form (Linear Equation):
ax+by=c
Coefficients are represented by the letters aa, bb, and cc. The letters aa, bb, and cc are all substituted with real numbers in standard form. The independent variable is represented by the letter xx, and the dependent variable is represented by the letter yy.
Observations on Standard Form:
The aa word must be a whole number.
a, b, a, b, and cc cannot be fractions or decimals.
Why use the standard form?
When solving a system of equations, the standard form of a linear equation can be quite useful. For example, when solving a system of equations with the elimination approach, we can easily align the variables using a standard form.
Equation system in the conventional form
Let’s look at an example. If we were given the following equations as a system of equations:
y=−4x+9
The term “linear equation in one variable” refers to an equation with only one variable. It signifies that there is just one solution to this linear equation. The standard or generic form of one-variable linear equations is expressed as,
Ax + B is equal to 0
A and B are both integers, and
The only variable is x.
For example, 3x + 6 = 12 is the typical form of a single-variable linear equation, and there is only one solution for the value of x, which is 4.
Conclusion
That’s the answer to what is a linear equation and its standard form! A linear equation in algebra can be signified in three different ways. The general form of linear equations is Ax + By = C, which is also known as the standard form. A linear equation is defined as an equation with a maximum degree of 1. This means that in a linear equation, no variable has an exponent greater than 1. A linear equation’s graph is always a straight line.
