Linear Equations in Two Variables Vikas Patel X-B Roll.no -35 Maths Project Work Kendriya Vidyalaya, Ujjain Lesson 3
A system of equations is a collection of two or more equations, each containing one or more variables. A solution of a system of equations consists of values for the variables that reduce each equation of the system to a true statement. When a system of equations has at least one solution, it is said to be consistent ; otherwise it is called inconsistent . To solve a system of equations means to find all solutions of the system.
If each equation in a system of equations is linear, then we have a system of linear equations .
If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. Solution y x
If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. x y
If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent. x y
Rules for Obtaining an Equivalent System of Equations (Elimination Method) 1. Interchange any two equations of the system. 2. Multiply (or divide) each side of an equation by the same nonzero constant. 3. Replace any equation in the system by the sum (or difference) of that equation and any other equation in the system.
Multiply (2) by 2 Replace (2) by the sum of (1) and (2) Equation (2) has no solution. System is inconsistent. Use Method of Elimination to solve: (1) (2)
Multiply (2) by 2 Replace (2) by the sum of (1) and (2) The original system is equivalent to a system containing one equation. The equations are dependent. Use Method of Elimination to solve: (1) (2)
This means any values x and y , for which 2x -y = 4 represent a solution of the system. Thus there is infinitely many solutions and they can be written as or equivalently
A linear equation in two unknowns is an equation that can be written in the form
ax + by = c with a, b and c being real numbers, and with a and b not both zero.
A solution to a linear equation in the two unknowns x and y consists of a pair of numbers: a value for x and a value for y that satisfy the equation. By the same token, a solution to a system of two or more linear equations in x and y is a solution that satisfies all of the equations in the system.
We can solve such a system of equations either graphically, by drawing the graphs and finding where they intersect, or algebraically, by combining the equations in order to eliminate all but one variable and then solving for that variable.
A system of two linear equations in two unknowns has either:
(1) A single (or unique) solution. This happens when the lines corresponding to the two equations are not parallel, so that they intersect at a single point.
(2) No solution. This happens when the two lines are parallel and different.
(3) An infinite number of solutions. This occurs when the two equations represent the same straight line. In this case, we represent the solutions by choosing one variable arbitrarily, and solving for the other.