Linear Equations
Linear polynomial is a polynomial having degree 1. Equation means two expressions separated by sign of equality. Thus, linear equation involves only linear polynomials - An equation in which the highest power of the variable is 1.
General form of a linear equation in one variable is ax + b = 0, a ≠ 0, and a, b are real numbers.
Solution of linear equation in one variable: The value of the variable for which LHS of the given equation becomes equal to RHS.
Formation of a linear equation in one variable: Represent the unknown by an alphabet say x, y, z, m, n, p etc. and translate the given statement into an equation.
Linear equation in two variables: ax + by + c = 0 is a linear equation in two variables x and y. Linear equation in two variables have infinitely many solutions.
In ax + by + c = 0 for each value of y, we get a unique value of x. Graph of a linear equation in two variables is always a straight line.
System of linear equations: A pair of linear equations in two variables is said to form a system of linear equations written as:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
System of linear equations can be solved by graphical or any algebraic method.
Graphical method: Draw the graph of both equations on same graph paper. If the graph is intersecting lines then the point of intersection gives unique solution of system. If two lines coincide, system has infinitely many solutions. If graph is parallel lines, the system has no solution.
Algebraic methods
Substitution Method: Find the value of one variable in terms of other variable from one equation and substitute it in second equation, second equation will be reduced to linear equation in one variable.
Elimination Method: Multiply both equations by suitable non-zero constants to make the coefficients of one of the variables numerically equal. Now add or subtract one equation from another to eliminate one variable, we get an equation in one variable.
Word Problems: Translate the given information (data) into linear equations and solve them.