Solving Linear Systems of Equations by Elimination
Use elimination to find the solution of this system.
Solution
To make the equations easier to work with:
â€¢ Clear the fractions in the
first equation by multiplying
both sides by 6, the LCD of
the fractions. 

â€¢ Clear the decimals from
the second equation by
multiplying both sides
by 10. 
10(0.3x + 0.2y = 1) → 3x
+ 2y = 10 
To make the xcoefficients
opposites, multiply the
transformed second equation
by 1.
Add the equations. 

Both variables are eliminated. The result is the false statement 0 = 31.
When the result is a false statement, the graphs of the equations never
intersect. The graph confirms that the lines are parallel and have no points
in common.
This system has no solution because the lines never intersect.
The system is inconsistent. (It has no solution.)
The equations of the system are independent. (Their graphs are not
identical.)
Note â€”
Solving a Linear System: Special Cases
When using either substitution or elimination, if both variables are
eliminated there are two possible outcomes:
â€¢ If the resulting equation is an identity, such as 5 = 5, then the
lines coincide.
The system has infinitely many solutions.
The solutions may be stated as the set of all points on the line.
â€¢ If the resulting equation is a false statement, such as 0 = 4, then
the lines are parallel and never intersect.
The system has no solution.
The system is inconsistent and the equations are independent.
