Solving Linear Equations

Date: 03/18/97 at 02:21:05
From: Anonymous
Subject: Solving Linear Equations

Dr. Math,

What are the steps to solve linear equations?

B2Q24GET

Date: 03/18/97 at 14:14:59
From: Doctor Rob
Subject: Re: Solving Linear Equations

To solve a single linear equation in one unknown, like

7*x - 4 = 4*x + 8,

The steps are:

1) Move all terms involving the unknown to one side of the equation,
   and all other terms to the other.

      7*x - 4*x = 8 + 4

2) Add like terms together to simplify.

            3*x = 12

3) Divide by the coefficient of the unknown.

              x = 12/3

4) Simplify.

              x = 4


Solving two linear equations in two unknowns can be done in several
different ways.  I will give one, not necessarily the best one.

Say we have 

     3*x + 7*y = 23
     2*x - 5*y = -4

1) Solve one of the equations for one of the variables using the
   method above.

             x = (-7*y + 23)/3

2) Substitute for that variable in the other equation.

      2*(-7*y + 23)/3 - 5*y = -4.

3) Solve the resulting equation for the second variable using the
   method above.

     46/3 + 4 = (14/3 + 5)*y
         58/3 = 29/3*y
            y = 58/29 = 2

4) Substitute in the result of step 1 and simplify.

            x = (-7*2 + 23)/3 = 9/3 = 3.

Similar methods work for three equations in three unknowns, four
equations in four unknowns, and so on.  The trick is to pick one
equation, solve it for one of the variables, and substitute it
into all the rest of the equations.  You are left with a set of
one fewer equations involving one fewer variable.  Repeat this
until you have one equation in one unknown, which you know how to
do.  This gives you the value of the last unknown.  Substitute it
into the previous results, and get the next-to-last unknown, then
the one before that, until you have all the variable values.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/