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Solve an equation, inequality or a system.

Example: 2x-1=y,2y+3=x

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Solving a system of two linear equations

Solving a system of linear equations by substitution

Example 1

The substitution method involves substituting an expression for one variable in terms of the other in another equation of a system. For example, to solve the system

solving a system of 2 equations

by substitution, replace y in the first equation

solving a system of 2 equations

noticing that parentheses are required. Then solve this equation for x.

finding out the first variable in the system

Replace x with 1 in y = x + 3 to find that y = 1 + 3 = 4. The solution set for this system is {(1, 4)}.

Example 2:

Solve the system

another system of 2 equations in x and y

Begin by solving one equation for one of the variables in terms of the other. For example, solving the first equation for x gives :

solving the linear equation for x

Now substitute this result for x into equation (2).

using substitution to solve for the second variable

To eliminate the fraction on the left, multiply both sides of the equation by 2 and then solve for y.

using distributive property, combining terms and dividing

Substitute y = 5 back into equation (3) to find x

solving for the second equation in the system

The solution set for the system is ((-2, 5)}. Check by substituting -2 for x and 5 for y in each of the equations of the system.


Solving a system of linear equations by addition

Another method of solving systems of two equations is the addition  method. With this  method, we first multiply the equations on  both sides by suitable numbers, so that when they are added, one variable is  eliminated.  The  result is an equation in one variable that can be solved by methods used for linear equations. The solution is then substituted into one of the original equations,  making it possible to solve for the other variable. In this process the given system is replaced by new systems that have the same solution set as the original system.  Systems that have the same solution set are called equivalent  systems. The addition method is illustrated by the following examples.

Solve the system

system in variables x and y

eliminate x,  multiply both sides of equation (4) by -2 and both sides of equation (S) by 3 to get equations (6) and (7).

first step in solving the system by addition

Although this new system is not the same as the given system,   it will have the same solution set.

Now add the two equations to eliminate x, and then solve the result for y.

eliminating one variable

Substitute 2 for yin equation (4) or (5). Choosing equation (4) gives

finding value of x

The solution set of the given system is { (3,  2)},   which can be checked by substituting 3 for x and 2 for y  in equation (5).

Since the addition method of solution results in the elimination of one variable from the system, it is also called the elimination  method.