Solving Systems of Linear Equations Using Elimination
Systems of Linear equations:
A system of linear equations is just a set of two or more linear equations.
In two variables , the graph of a system of two equations is a pair of lines in the plane.
There are three possibilities:
- The lines intersect at zero points. (The lines are parallel.)
- The lines intersect at exactly one point. (Most cases.)
- The lines intersect at infinitely many points. (The two equations represent the same line.)
How to Solve a System of Linear Equations Using The Elimination Method (aka The Addition Method, aka The Linear Combination Method)
- Step : Add (or subtract) a multiple of one equation to (or from) the other equation, in such a way that either the -terms or the -terms cancel out.
- Step : Then solve for (or , whichever's left) and substitute back to get the other coordinate.
Now, how do we know that a linear equation obtained by the addition of the first equation with a scalar multiplication of the second is equivalent to the first?
Let us take an example. Consider the system
Consider the equation obtained by multiplying the second equation by a constant and then adding the resultant equation with the first one.
That is, .
What we need to prove is that this equation is equivalent to the equation .
We have .
Since , subtract from the left side and from the right side of the equation which will retain the balance.
Cancelling common terms we get, which is equivalent to the first equation.
Therefore, the systems of equations and are equivalent.
In general, for any system of equations and , it ca be shown that is equivalent to .
Solve the system
Multiply the first equation by and add the result to the second equation.
Solve for .
Substitute for in either of the original equations and solve for .
The solution is .