Substitution and elimination are the common algebraic methods used to solve a system of linear equations in two variables. Against the graphical method of solving a system which often can gives only an approximate solution, the algebraic methods yield exact solution of the system. In elimination method, we add the both equations together so that one of
the variables eliminated. Sometimes it is necessary to multiply one or
both equations by a constant in order for the terms to have opposite
signs. So when we add the equations, one of the variables will be
Elimination method involve solving for one of the variable by eliminating the
other variable. This method also named as addition method. Multiply the
original equations by a constant before it eliminate one of the
variables by adding or subtracting the equations.
Solving Linear Equations by Elimination
Elimination method uses the addition and subtraction property of equality for solving the systems of equations. Substitution method makes use of solving an equation literal for a variable. Elimination method is useful whenever one of the variables cannot solved as a simple expression of the other. Elimination method also makes the solving process easier if fractions are involved in the equations. If the coefficients of any variable are seen same in the given
equations, the variable can be removed by addition or subtraction
without multiplying to get the equivalent equations. Of course, you have to use the elimination method when the problem direction says so.
Solve the system of equations
$- 3x + 4y = 11$
$5x - 2y = 5$ Solution:
$- 3x + 4y = 11$ ..................(1)
$5x - 2y = 5$ ...................(2)
can eliminate any one of the variables x or y. But for the above system
it is easier to eliminate y. By multiplying the second
equation by 2 we can make coefficient of y as -4.
To eliminate y, multiply each side of equation (2) by 2
$2(5x - 2y) = 2(5)$
$10x - 4y = 10$ ........................(3)
Add equation (1) and equation (3)
- 3x + 4y + 10x - 4y = 11 + 10
=> -3x + 10x = 21
=> 7x = 21
=> $x = 3$
Put x = 3 in equation (2)
=> 5 * 3 - 2y = 5
=> 15 - 2y = 5
=> -2y = 5 - 15 = -10
=> y = 5
Hence the solution to the system is x = 3 and y = 5.
Elimination method is a algebraic methods used to solve a system of linear equations in two variables. Eliminate that variable whose coefficients can be made equal by easy multiplication or multiplication done on one equation only. If one equation is a multiple of the other equations, the equations are coincident. Hence the system has infinitely many solutions. On multiplying one of the equation suitably by a constant if the resulting equation varies from the second equation only by the constant term, then the two lines are parallel and the system has no solution.
Solve the system of equations
$x + 2y = 12$
$2x + 6y = 24$ Solution:
x + 2y = 12 ....................(1)
2x + 6y = 24 .....................(2)
Eliminate any one of the variables x or y.
To eliminate y, multiply each side of equation (1) by 3
=> 3(x + 2y) = 12 * 3
=> 3x + 6y = 36 ................(3)
Subtract equation (2) from equation (3) to eliminate y.
=> 3x + 6y - (2x + 6y) = 36 - 24
=> 3x + 6y - 2x - 6y = 12
=> x = 12
Put x = 12 in equation (1)
=> 12 + 2y = 12
=> 2y = 12 - 12 = 0
=> y = 0
Hence the solution to the system is x = 12 and y = 0.