Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture, we saw how systems of two linear equations in two variables
could be solved using the elimination method
to find one solution which simultaneously satisfies both equations.
This solution is represented graphically
by the point where the graphs of the two equations intersect.
As we saw, the elimination method involves adding the equations in the system
to produce a new equation that eliminates one of the variables.
The justification usually given for this process
is that since it is okay to add the same thing to both sides of an equation
then two equations can be added
since both sides of the second equation are equal
and are therefore the "same thing".
However, this does not tell the whole story
since adding two equations
leads to an equation whose graph
is different from either of the original equations.
In this lecture, we will more closely examine what happens when two equations are added
and why this procedure works for the elimination method.
The process of adding two equations is simple.
We just line up like terms and add them.
For instance, in this system we add the x terms
the y terms
and the constant terms
to produce a new equation.
But what is the significance of this new equation?
And what mathematical relationship does it have to the two equations of the system?
If we graph all three equations
we see that the graph of the new equation
has only one point in common with the two equations which were added.
All three equations intersect at this same point.
When adding two equations, the solutions of the equation which is produced
are generally different from the solutions of either equation in the system
and so its graph will be different as well.
However, the new equation will always share the one solution
which is common to both equations of the system.
We can prove this will always happen when any two equations are added
by showing that this is true in a general case.
When we simultaneously graph two equations
at any point of intersection, both equations are satisfied.
Let's call the two equations of this system "equation A"
and "equation B".
And let's call the expressions on either side of these equations
expressions A-one and A-two
and B-one and B-two.
Now let's add equations A and B to produce a new equation.
Since the intersection point of the graphs
represents the point at which equations A and B are both satisfied
at that point, both expressions in equation A have the same value
and both expressions in equation B have the same value.
So at that point, both sides of the new equation have the same value
and that equation will also be satisfied.
Since all three equations are satisfied at that intersection point
that point is included in all three equations' graphs
and all three graphs will pass through the same intersection point.
So when two equations are added, the resulting equation's graph
must pass through the intersection point of the two original equations.
However, when using the elimination method
it is sometime necessary to multiply one or both equations by constants
before they are added.
But after multiplying by constants
will the graph of the resulting equation still pass through the original intersection point?
To find out, let's start with equations A and B
and multiply equation A by a constant which we will call "small a"
and equation B by a constant "small b".
We know that multiplying both sides of an equation by a non-zero constant
does not change its solutions or graph.
So the graphs of both equations will remain unchanged
and they will still intersect at the same point as before.
Now, let's add the two equations.
Since at the intersection point, equations A and B are both satisfied
at that point, both expressions in equation A have the same value
and both expressions in equation B have the same value
so both sides of the new equation have the same value
and that equation will also be satisfied.
Once again, all three graphs must pass through the same intersection point.
As we have just shown, multiplying the equations by constants before they are added
maintains the original point of intersection
and the graph of the new equation will still share this common solution.
But other than passing through this point
what effect does this have on the rest of the resulting equation's graph?
At the beginning of this lecture
we added the equations "3x minus y equals 3"
and "x minus y equals negative 1"
producing the equation "4x minus 2y equals 2"
whose graph passes between the two original lines.
But let's say that before adding the equations in this system
we multiply both sides of the bottom equation by negative 2.
Now, when we add the equations
we get x
plus y
equals 5.
Although as we saw
this equation's graph must still pass through the intersection point
of the two lines of the system
it now has a different slope than before.
Now, let's see what happens
if in addition to multiplying the bottom equation by negative 2
we multiply the top equation by positive 2.
When we add the equations
we get 4x
plus zero
equals 8
or "4x equals 8"
which can be written more simply as "x equals 2".
Just as before, this equation's graph
passes through the intersection point of the original two lines
but now the resulting line is vertical.
This equation is especially informative
since it tells us that the x-coordinate of the intersection point is 2.
Likewise, if we take the original equations
and multiply the bottom equation by negative 3
when we add the equations
we get zero
plus 2y
equals 6
or "2y equals 6"
which we can write more simply as "y equals 3".
Once again, this equation's graph
passes through the intersection point of the original two lines
but now the resulting line is horizontal.
This equation tells us that the y-coordinate of the intersection point is 3.
By choosing various multipliers before adding the two equations
we can produce a new line with any orientation we like.
However, no matter which multipliers we choose
the resulting line will always cross the two original lines at their intersection point
and thus share the solution common to the two original equations.
To find the coordinates of the intersection point of two lines
we choose multipliers for the equations
that will cause the line produced by adding them
to be either vertical
or horizontal.
The equation of this line then tells us the intersection point's x-coordinate
or y-coordinate.
This is the basis for solving a system of equations using the elimination method.
So far, we have seen how the graphs of two linear equations
can intersect at a single point and thus share a common solution.
In the next lecture we will examine other relationships
which can occur between two linear equations
which do not share a single common solution.