A system of a linear equation comprises two or more equations
and one seeks a common solution to the equations. In a system of
linear equations, each equation corresponds with a straight line
corresponds and one seeks out the point where the two lines
Solve the following system of linear equations:
Since we are seeking out the point of intersection, we may graph
We see here that the lines intersect each other at the point x =
2, y = 8. This is our solution and we may refer to it as a graphic
solution to the task.
But how does one reach a solution if the lines never intersect?
One cannot, the system of equations have no solution.
One may also arrive at the correct answer with the help of the
elimination method (also called the addition method or the linear
combination method) or the substitution method.
When using the substitution method we use the fact that if two
expressions y and x are of equal value x=y, then x may replace y or
vice versa in another expression without changing the value of the
Solve the systems of equations using the substitution method
We substitute the y in the top equation with the expression for
the second equation:
To determine the y-value, we may proceed by inserting
our x-value in any of the equations. We select the first
We plug in x=2 and get
We have thus arrived at precisely the same answer as in the
The elimination method requires us to add or subtract the
equations in order to eliminate either x or y,
often one may not proceed with the addition directly without first
multiplying either the first or second equation by some value.
We now wish to add the two equations but it will not result in
either x or y being eliminated. Therefore we must
multiply the second equation by 2 on both sides and get:
Now we attempt to add our system of equations. We commence with
the x-terms on the left, and the y-terms
thereafter and finally with the numbers on the right side:
The y-terms have now been eliminated and we now have an
equation with only one variable:
Thereafter, in order to determine the y-value we insert
x=2.5 in one of the equations. We select the first:
Videolesson: solve the system of equations: