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A system of linear equations in two variables has a single solution, no solutions, or an unlimited number of solutions. There may be ‘n’ variables in a linear system of equations. The equation with the highest degree of one is known as a linear equation. This means that in a linear equation, no variable has an exponent greater than 1. The graph of a linear equation is always a straight line.
Linear Equations
A linear equation is an algebraic equation in which each component has an exponent of 1 and produces a straight line when graphed. This is what a ‘linear equation’ means. There are two types of linear equations: one-variable and two-variable. We will learn how to recognize linear and non-linear equations using the examples below.
Linear Equation In One Variable: A linear equation with only one variable is known as a one-variable equation. The formula is Ax + B = 0, where A and B are any two real numbers and x is an unknown variable with just one solution. It’s the most basic way of expressing a mathematical claim. In one variable, the degree of a linear equation is always one (1). Solving a one-variable linear equation is quite straightforward. The variables are separated and moved to one side of the equation, while the constants are combined and moved to the other, to find the value of the unknown variable.
Example – 21 x + 15 =11
y- 5 = 10
Linear Equations In Two Variables
Ax + By + C = 0 is a two-variable linear equation with the variables x and y each having a degree of one. Simultaneous linear equations are two linear equations that have the same solution. For example, 6x + 2y + 9 = 0 is a two-variable linear equation. Methods for solving linear equations in two variables include the graphical technique, substitution method, cross multiplication method, elimination method, and determinant method.
Example – 4x-3y=6
X-2y=22.
Solution of Linear Equation in two variables
Every one-variable linear equation has a single solution. However, a pair of linear equations has two solutions, one for x and one for y, which satisfy both equations.
- Unique Solution
When a set of linear equations intersects at a location, the solution for both equations is unique. The slopes of the lines should differ from one another in order to generate a unique solution for a set of equations.
- No Solution
The lines will be parallel if the slopes of two two-variable equations are equivalent,
- Infinite Solutions
If a pair of linear equations are consistent, the lines will have both unique and infinite solutions, i.e. they will intersect or coincide.
Method of Solving Linear Equation in two variables
There are five ways to solve a two-variable system of linear equations. The following are the methods,
- Graphical Method
- Substitution Method
- Cross Multiplication Method
- Elimination Method
- Determinant Method
Linear Equation with Two Variables – Applications
We employ linear equations on a regular basis even if they don’t use a line graph since the scenarios they confront may involve an unknown number that may be expressed as a linear equation, such as calculating mileage rates, income over time, and so on. The four major arithmetic operations – addition, subtraction, multiplication, and division – are crucial.
The basic goal of linear equations or linear systems applications is to answer diverse issues with two variables, one of which is known and the other of which is unknown and also dependent on the first. The following are some examples of linear equation applications:
- Some Geometry problems can be solved using two variables.
- Some Money problems can be solved by using two variables
- A mixture of problems can be solved by using two variables.
- Distance-Rate-Time problems can be solved by using two variables.
Conclusion
- When using either the substitution or elimination methods to solve the equations:
- When we have a valid equation, we know that the system has an endless number of solutions.
- If we receive a false equation, it suggests the system doesn’t have a solution.
