In Mathematics, an equation is defined as the mathematical statement, which consists of an equal sign “=” between two algebraic expressions. The algebraic equation’s most important property is that the Left Hand Side (L.H.S) of an equation is equal to the Right Hand Side of the equation. In algebra, most of the concepts are based on solving the variables of an equation. For example 2x+5 = 12 is an equation. The process of finding the unknown variable “x” is called solving an equation.
Generally, an equation consists of coefficients, variables, constants, terms, exponents, and operators. From the equation mentioned above, 2 is the coefficient of x, 5, and 12 are constants, x is a variable, 2x is a term, and the operator used here is the “+” sign, which describes the addition operation. The basic algebraic equation consists of one variable or more than one variable. The system of equations is defined as the set or the collection of equations that are all working together. The simplest equation is the linear equations with two variables and two equations. The linear equations are always simpler than the nonlinear equations. The set of equations can be classified as linear equations, polynomial equations, nonlinear equations, differential equations, and so on.
We can easily solve the equation if it contains one variable. The most common procedure to solve the algebraic equation is to bring the variables on one side, and all the constant terms should be on the other side. For instance, 4x+2 = 10. This equation can be solved as follows:
4x = 10- 2
4x = 8
x = 2.
If we substitute x= 2 in the given equation, both sides of the equation should be equal.
Similarly, the system of equations can be solved using three different methods. Before discussing the three different methods, let us look at the different types of solutions to a system. The three different types of solutions that the system of linear equations may encounter are:
The system has only one solution when the graph intersects at a point. The answer should be in the ordered pair, i.e., (x, y), which should be the solutions to both equations.
If the two given lines are parallel to each other, the lines will never intersect. It shows that the system has no solution because they do not have any common point to intersect the lines.
Infinitely Many Solutions:
If two lines lie on top of each other, then the system has infinitely many solutions.
Now, let’s discuss the three methods used for solving the system of equations. They are a graphical method, substitution method, and elimination method.
If the equation contains more than two variables, it cannot be solved using a simple graph. But algebraically, we can solve the equations with more than two variables using substitution and elimination methods. Let’s have a look at these two important methods.
Solving the System of Equations Using Substitution Method
Step 1: Write down one of the equations in the form of “variable =.”
Step 2: Substitute that variable in the other equation.
Step 3: Solve the equation using the usual algebraic methods.
Solving the System of Equations Using Elimination Method
Step 1: Simplify the equations in the form of Ax+By = C, if required.
Step 2: Multiply one or both equations by a constant value, such that it creates the opposite coefficient for either of the variables x or y.
Step 3: Now, add the equation so that the equation is left with only one variable.
Step 4: Finally, solve the equation using simple algebraic methods.