Welcome to a lesson on the types of
linear equations. In this video we will
define and identify an identity
conditional equation and contradiction
equations. We will also illustrate the three
types of linear equations graphically.
Let's first consider the type of linear
equation that we're used to solving. A
conditional equation is an equation that
is only true for a specific value of the
variable, which is the solution. If the
conditional equation is not linear it is
possible to have more than one solution.
As an example let's solve 2x minus 1
equals 5. This is the basic two-step
equation so for the first step we'll
undo the subtraction by adding 1 to both
sides of the equation. Notice how minus 1
plus 1 simplifies to 0 so now we have 2x
equals 6. So the second step will be to
divide both sides by 2 giving us a
solution of x equals 3, which means this
equation is only true when x equals 3.
Therefore this is a conditional linear
equation. To verify this graphically we
can graph the left side and right side
of this equation separately on the
coordinate plane. For example we can
graph y1 equals 2x minus 1 and y2 equals
positive 5. Let's do this and see what we
find.
So we'll press y equals. we'll first type
in 2x minus 1 press Enter, and then type
in five. To make sure we have the
standard window let's go ahead and press
zoom 6. There's a graph of y1. There's a
graph of y 2. Notice how these two linear
equations intersect at this point here
where the x coordinate is equal to
positive 3, which is the solution to our
conditional equation. To verify this
point of intersection
let's press 2nd trace option 5 and then
enter 3 times.
Notice at that this point is on both
lines has an x coordinate of 3 which is
our solution.
And the y-coordinate is 5 which means
both sides of this equation or equal to
5 when x equals 3.
To verify this numerically we can press
2nd graph to access the table. Notice
when x equals positive 3
both y1 and y2 are equal to positive 5.
Now let's talk about an identity. An
identity is an equation that is always
true and has infinite solutions. For
example, 1 equals 1 is an identity
because it is always true. The solution
of a linear equation is an identity can
be expressed as having infinite
solutions or we can say X can be any
real number or X is in the open interval
from negative infinity to positive
infinity, indicating X can be any real
value. For example, if we're asked to
solve this given linear equation, let's
first clear the parenthesis by
distributing. So we would have 2x minus 6
equals on the right side we also have 2x
minus 6. So we should recognize at this
point that for any value of x the left
side would equal the right side since
the expressions are the same. And
therefore we have an identity, but if we
didn't recognize this we can continue
solving by subtracting 2x on both sides
of the equation. However notice when we
do this both on the left side and on the
right side the X terms simplify out
leaving us with negative 6 equals
negative 6, which is always true
indicating that we have an identity.
Therefore the linear equation has an
infinite number of solutions or we could
also indicate that X can be any real
number or X can be any real number in
the open interval from negative infinity
to positive infinity. Let's verify this
graphically as well, so we'll graph y1
equals 2x minus 6 and y2 equals the
right side or 2 times the quantity X
minus 3. We'll press y equals clear out
the old equations
and type in the new equations. So we have 2x minus 6 and we have 2 times the
quantity X minus 3. Now to make sure we
can tell these two linear equations
apart I'm going to go to the far left
for y to press ENTER once so y2 will be
a thick line. y1 will be a thin line. So
I'll press graph. Notice how it's the exact
same line which means for any value of x
the point would be on both lines at the
same time, indicating we have an infinite
number of solutions or the equation is
an identity. To verify this numerically,
if we press 2nd graph. Notice how for any
value of x y1 and y2 are equal. Again
indicating we have an identity on an
infinite number of solutions. The last
type of equation is a contradiction. A
contradiction is an equation that is
always false and has no solution. We can
also say that the equation is
inconsistent. For example 0 equals 1 is a
contradiction because it is always false.
You can write the solution to a
contradiction using the empty set symbol
using this notation here or this
notation here or by just stating we have
no solution. So if we want to solve the
given linear equation again let's first
clear the parentheses so we'll
distribute. This would give us 1/2 X
minus 1/2 equals 1/2 X plus 2. We may
recognize at this point that since the
variable term is the same and the
constants are different, this equation
will have no solution and therefore is a
contradiction. Again if we don't, we can
continue solving by subtracting 1/2 x on
both sides. Once again notice that on
both sides the X terms simplify out
leaving us with negative 1/2 equals
positive 2. But in this case, we know this
is not true and therefore this is always
false. Indicating
we have no solution or indicating our
linear equation is a contradiction
equation. We can also say that X belongs
to the empty set. Let's finish by
verifying this graphically. So we'd have
y 1 equals 1/2 times the quantity X
minus 1 and y 2 equals 1/2 X plus 2.
Let's go back to the calculator again
clear out the old equations first
equation is 1/2 times the quantity X
minus 1 second equation is 1/2 X plus 2.
Let's press graph. There's y1.
There's y2. Notice how these two lines
are parallel and therefore they never
intersect, indicating there are no values
of X for which the Y values would be the
same or that the left side of the
equation would not equal the right side
therefore we have no solution and an
equation that is a contradiction. If we
press 2nd graph you can see numerically
that for any value of x y1 will never
equal y2. That's going to do it for this
lesson. I'll leave you with a summary of
the three types of linear equations.  I
hope you found this explanation helpful.
 
