Solve Systems of Linear Equations in Two Variables

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solving linear equations worksheets, solving linear equations calculator, solving linear equations with fractions, solving linear equations online, solving linear inequalities

What is the system of linear equations?
A "system" of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.

Solve Systems of Linear Equations in Two Variables

There are three ways to solve systems of linear equations with two variables:


  1. solving linear equations by substitution method
  2. solving linear equations by elimination method
  3. solving linear equations by graphing 


The substitution method for solving linear systems

A way to solve a linear system algebraically is to use the substitution method. The substitution method functions by substituting the one y-value with the other. We're going to explain this by using an example.
y=2x+4
3x+y=9

We can substitute y in the second equation with the first equation since y = y.
3x+y=9
3x+(2x+4)=9
5x+4=9
5x=5
x=1

This value of x can then be used to find y by substituting 1 with x e.g. in the first equation
y=2x+4
y=2·1+4
y=6

The solution of the linear system is (1, 6).
You can use the substitution method even if both equations of the linear system are in standard form. Just begin by solving one of the equations for one of its variables.

Back to : Solving Linear Equations


The elimination method for solving linear systems

Another way of solving a linear system is to use the elimination method. In the elimination method you either add or subtract the equations to get an equation in one variable.
When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

Example :
3y+2x=6
5y-2x=10

We can eliminate the x-variable by addition of the two equations.
3y+2x=6
5y-2x=10_
<=> 8y=16
<=> y=2

The value of y can now be substituted into either of the original equations to find the value of x
3y+2x=6
3·2+2x=6
6+2x=6
x=0
The solution of the linear system is (0, 2).

To avoid errors make sure that all like terms and equal signs are in the same columns before beginning the elimination.
If you don't have equations where you can eliminate a variable by addition or subtraction you directly you can begin by multiplying one or both of the equations with a constant to obtain an equivalent linear system where you can eliminate one of the variables by addition or subtraction.

Example :
3x+y=9
5x+4y=22

Begin by multiplying the first equation by -4 so that the coefficients of y are opposites
3x+y=9 ( multiplying by (-4))
5x+4y=22

-12x-4y=-36
5x+4y=22_
<=> -7x = -14
<=> x=2

Substitute x in either of the original equations to get the value of y
3x+y=9
3·2+y=9
6+y=9
y=3
The solution of the linear system is (2, 3)

Next: Solving Quadratic Equations