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A linear equations in two variables x and y is of the form ax + by + c = 0
(
) where a, b, c are real numbers. To find a solution for this equation, we can assign any value for one of the variables and find the value of the other variable such that the two sides of the equation are equal. Hence we say a linear equations in two variables has indefinitely many solutions.
But we often study more than one equations in the same two variables at the same time. The equations under consideration are said to form simultaneous equations. We shall study in this chapter, a pair of linear equations in two variables, and methods of finding solutions common to both the equations.
Simultaneous Equation
Solving two equations simultaneously means to find the common solution of both the equations, i.e., a solution which satisfies both the equations. (Such a common solution, if it exists, can be shown to be unique.)
Method of Elimination (by Addition)
Solve the Systems of linear equations by Method of Elimination (by Addition): 3x - 4y = 20 …(i) 5x + 6y = 8 …(ii).
Substitution Method
Solve the Systems of linear equations by Method of Substitution: 2x - 9y = 0 …(i) x - 18y = 27 … (ii).
Problems on Simultaneous Equations
Solve the following Systems of linear equations:
1.If one number is thrice the other and their sum is 60, find the numbers.
2.Find the fraction which becomes 1/2 when the denominator is increased by 5 and is equal to 1/3 when the numerator is diminished by 4.
Summary
Finding the solution by the method of substitution.
(i) Coefficients of one of the variables (say x) in the two equations are made equal, by multiplying them with suitable factors.
