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A linear equation is one in which the maximum power of the variable is always 1. Its another name is a one-degree equation. A linear equation in one variable is written in the usual form Ax + B = 0. In this equation, x is a variable, A is a coefficient, and B is a constant. A linear equation with two variables can be written in the form of Ax + By = C, where A and B are coefficients, and C is a constant in this equation.
Linear Equations
A linear equation can be defined as, the equation whose maximum degree of one. This signifies that no variable in a linear equation has an exponent bigger than 1. A linear equation’s graph is always a straight line.
Definition of Linear Equation: A linear equation is an algebraic equation in which each component has an exponent of 1 and when graphed, the result is always a straight line. This is why it is called a ‘linear equation.’ There are one-variable linear equations and two-variable linear equations. With the help of the examples below, we will learn how to recognise linear and non-linear equations.
Standard Form of Linear Equation: Ax + B = 0 is the standard or general form of linear equations in one variable, where A and B are real values and x is the single variable. Ax + By = C is the conventional form of a two-variable linear equation, where A, B, and C are any real numbers, and x and y are the variables.
Linear Equation In One Variable: A one-variable linear equation is one in which only one variable is present. It has the formula Ax + B = 0, with A and B being any two real numbers and x being an unknown variable with just one solution. It’s the most straightforward approach to express a mathematical assertion. The degree of linear equation in one variable is always equal to one (1). A one-variable linear equation is fairly simple to solve. To determine the value of the unknown variable, the variables are separated and moved to one side of the equation, while the constants are combined and moved to the other.
Example- 2x+5=11
5y-2=3
Linear Equations In Two Variables: Ax + By + C = 0 is a two-variable linear equation in which A, B, and C are real integers, and x and y are the two variables, each having the degree of one. Two linear equations with the same solution are referred to as simultaneous linear equations. 6x + 2y + 9 = 0 is a two-variable linear equation, for example. The graphical approach, substitution method, cross multiplication method, elimination method, and determinant method are all methods for solving linear equations in two variables.
Example- 4x-3y=6
X-2y=22.
Application Of Linear Equation: Linear equations have many applications in mathematics and in everyday life. An algebraic expression is called an equation when it consists of variables that are equated to each other using an equal “=” sign. A linear equation is an equation having a degree of one. Mathematical knowledge is typically applied through word problems, and the use of linear equations to answer such word problems is seen on a large scale. Here is a lengthy overview of linear equation applications and how they fit in the actual world. Linear equations have a wide range of applications and can be used in a variety of real-world scenarios. To deal with real-life events using algebra, we convert them into mathematical statements. So that the relationship between the unknown variables and the known information is clearly illustrated. The stages involved in reiterating a circumstance into a mathematical statement are as follows:
- Convert the real problem into a mathematical statement and put it in the form of an algebraic expression that expresses the problem scenario clearly.
- Determine the situation’s unknowns and assign variables to these unknown numbers. Read the situation several times over and cite the data, phrases, and keywords.
- Organise the collected data in a sequential manner.
- Create an equation with the algebraic expression and the facts from the statement and solve it using systematic equation-solving approaches.
- Reframe the solution to the problem statement and assess its suitability.
By this step, the word problems can easily be solved.
Representation of Linear Equations
Solving linear equations and expressing the solutions in a coordinate plane is what graphing a linear equation means. The two pairs (x,y) are required for plotting the equation on a graph. However, because the two points can always be linked and shown as a line, we can’t tell if there were any errors in collecting these values. As a result, it’s a good idea to plot one more point to make sure the solutions to the given linear equation are correct. For charting a linear equation with one variable, use the following steps:
- Make sure the linear equation is y = mx + b in y-intercept form.
- Find the value of (x, y) up to three pairings that fulfil the linear equation using the trial and error method.
- Find the equation’s x-intercept and y-intercept. Substitute the value of x = 0 in the equation for the y-intercept. As a result, x = a; in the equation, insert the value of y = 0 for the x-intercept. As a result, y = c.
- The points are thus (a, 0) and (b, 0). (0, c). Make a tabular form and fill in the values for x and y.
- On the graph paper, plot all of the points.
- Join all of the points on the graph to form a straight line that graphically represents the specified linear equation.
Conclusion
- The solution or root of a linear equation is the value of the variable that makes the equation true.
- When the same number is added, subtracted, multiplied, or divided into both sides of a linear equation, the solution remains unchanged.
- A linear equation with one or two variables always has a straight line as its graph.
