Linear Systems of Equations

Linear systems of equations are equations whose variables have exponents equal to one, and the system contains one equation for each variable named.

For example:

                 2x + 3y + 4z = 10

                 x + 12y - 8z = 16

                 6x - 3y + 16z = -12

The variables are x, y and z; three variables, so we need three equations to solve for all three x, y and z.

Invariant operations on a linear equation:

Consider the following equation:

3x + 6 = 12

Solving for x we get x = 2

Here's how:

- 6 + 3x + 6 = 12 + - 6 add a -6 to both sides

3x = 6 simplify

1/3(3x) = 1/3(6) divide both sides by 3

x = 2 simplify

MULTIPLICATION:

Now what happens if we first multiply the original equation by 10.

          30x + 60 =  120

Solving for x we get:

          -60 + 30x + 60 = 120 – 60

                           30x = 60

                 x = 60 / 30 = 2

Notice that the answer remains the same. We did not change the original equation multiplying by 10. We scaled all numbers by the same factor, 10.

DIVISION:

Now let's divide that equation by 3.

                        1/3 ( 3x + 6 ) = 1/3 (12)

                1/3 ( 3x) + 1/3 ( 6 ) = 4

                                    x + 2 = 4

                   x = 2 same answer!

The point of these two examples is that we can multiply or divide an equation and not change that equation as far as its answer is concerned.

SOLVING SYSTEMS OF LINEAR EQUATIONS:

Let's take a two variable example to solve for x and y.

                       3x + 5y = 10

                       6x - 12y = 9

The strategy to solve a set of linear equations is to eliminate one unknown after the other by multiplying one equation by a suitable constant, combining equations (that is add them together), and solve for one unknown. Once you find that value, substitute it back into any equation to solve for the other. Now you've solved the system of linear equations.

So, looking at these equations I see that if I multiply the first equation by -2, then when adding the equations together x will cancel, like so:

                  -2(3x + 5y) = -2 (10)

                  -6x -10y = -20 now, write the second equation under this one

                  6x - 12y = 9

adding we have

                  0x - 22y = -11 solving for y we get

                 y = -11 / -22 = 1/2

Now that we have y, we substitute this value into one of the equations and solve for x,

like so:

               6x - 12 (1/2) = 9

                         6x - 6 = 9

                             6x = 9 + 6

                             6x = 15          

                      1/6 (6x) = 1/6 (15)

                              x = 15 / 6 = 5/2

So, the solution to this set of linear equations is ( 5/2 , 1/2 ).

            You'll notice that these equations are equations of two separate lines. This solution is the coordinate of their intersection.

This is another FREE Algebra PRINTABLE presented to you from the Algebra section of K12math.com