What is a linear equation?
The general form of a linear equation:
– One variable is ax + b = 0, where a ≠ 0 and x is the variable.
– Two variables is ax + by + c = 0, where, a ≠ 0, b ≠ 0 , x and y are the variables.
– Three variables is ax + by + cz + d = 0 where a ≠ 0, b ≠ 0, c ≠ 0, x, y, z are the variables.
How to solve Linear Equations?
For the equations involving 2 variables, we need to have two equations to solve for the two variables. There are various methods to solve these equations. Some of them are as follows:
- Cross multiplication method
- Method of substitution
- Method of elimination
- Matrix method
- Determinant methods
Linear equations in one variable
Linear Equations are those algebraic expression is equated to a particular constant or an algebraic expression. In a linear equation there will be definite values for the variables to satisfy the condition of the equation.
For example, Solution for the system of linear equations
x + y = 12,
2x + 3y = 32
x = (4, 8)
Linear equations in two variables
Linear equations of two variables are of the form, ax + by + c = 0. To solve for the variables x and y, we need to have two equations. Otherwise for every values of x there will be a corresponding value of y. Hence a single equation has infinite number of solutions and each solution is the point on the line.
Question: Solve x + y = 5
We have x + y = 5
=> y = 5 – x
when x = -3, y = 5 – (-3) = 5 + 3 = 8 , the solution is ( -3, 8)
when x = 0, y = 5 – 0 = 5 , the solution is ( 0, 5)
When x = 5, y = 5 + 5 = 10, the solution is ( 5, 0)
Graphing Linear Equations
- Graph of linear equations of one variable is a point on the real number line.
For Example: Draw graph for the linear equation, 2x + 5 = 9
Given 2x + 5 = 9
=> 2x + 5 – 5 = 9 – 5
=> 2x = 4
=> x = 2
This can be represented on the real number line as follows:
In the above number line we can see that the solution, x = 2 is marked.
- Graph of linear Equations of 2 variables will be a straight line, which can be shown on co-ordinate graph.
Let us draw the graph for the equation, x + y = 5
We have the points, (-3, 8), (0, 5), (5, 0)
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System of Linear Equations
System of linear equations are of the form:
a1x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0
These system of equations have following types of solutions according to the ratio of the corresponding coefficients. The above system of equations may have unique solution or no solution or infinite number of solutions.