Systems of Linear Equations

Graphing a system of two linear equations in the variables and
     
     ,
we get two straight lines.
Solving the system means finding the point where the two lines intersect.
A point on the graph is given by two coordinate numbers
( first coordinate, second coordinate), so we need to determine these two numbers.

Several things can happen:
If the lines are parallel, they do not intersect. The system has no solution.
For example,
     
     .

If the lines are coincident - one line lies on top of the other line -
then all the points on one line are also all the points on the other line.
The system has an infinite number of solutions.
For example,
       
    .

If the lines are not parallel and not coincident, then the lines intersect
in a unique point.
For example,
     
    .

There are two methods of solution - by elimination or by substitution.
In this section we will show

The Method of Substitution.
Here is the idea of the method, applied to the system above.

Step 1)
Pick one of the equations in the system, which one is not important.
To be specific, pick the first equation.
     

Note: Although, mathematically, which equation we pick is irrelevant -
we will get the same correct answer no matter which equation we pick
to start the process - it is best for us humans to pick the equation that
leads to the easiest algebra or arithmetic.
If we solve the equation for , we get
      .
If we solve the equation for , we get
      .
This expression for looks simpler than the expression for
which involve a fraction and a minus sign.

Step 2)
Substitute this equation for into the second equation of the system
               ,
so
      .

Solve this equation for :
         by the distributive property.

     

        to get alone on the left side of the equation

     .
Now, divide by to cancel that factor on the left side of the equation
and simplify the fraction that results on the right side of the equation.

       

We have found the second coordinate of the point where the two lines intersect,
           .
What remains is to find the first coordinate, the value of the   .


Step 3)
Substitute      into   .


      .

Step 4)
Write down the point of intersection in coordinate notation.

                    

See Examples 1 - 2, pages 90 - 91.


                         top
                         next     Linear Inequalities