Download 3.4 Solving Equations w/ Variables on Both Sides

3.5 Solving
Equations w/
Variables on Both
Sides
Goal:
Solving Equations with variables on both sides
What!?!
Variables on both sides!?!
• In some cases,
we may have
to solve
equations with
variables on
both sides of
the equal sign!
• Example 1:
2x - 4 = x - 2
• Example 2
4x - 4 + 2x = 7x - 4
• Example 3
3(x - 4) = 2(x - 4)+2x
So what can we do?
• We can undo or
eliminate
variables the
exact same way
we did
constants.
(numbers)
• What undoes a
+2?
• A -2 will undo it or
zero it out!
• What undoes a
+2x?
• A -2x will undo it
or zero it out!
Ex 1:Variables on Each Side
• Solve
7x +19 = -2x + 55
7x +19 + 2x = -2x + 55+ 2x
9x +19 = 55
9x +19 -19 = 55-19
9x = 36
9
36
x=
9
9
x=4
Pick one of the variables
to undo!
Lets undo the -2x by
adding 2x to each side!
Example 1: Again?
• Solve
7x +19 = -2x + 55
7x +19 - 7x = -2x + 55- 7x
What if we undo
the +7x instead?
19 = 55- 9x
19 - 55 = 55- 9x - 55
-36 = -9x
-36 -9x
=
-9
-9
4= x
No matter what variable we
undo first we end with the
same solution
Does it Matter what Variable
we start with?
• Both methods have a solution of 4.
• It does not matter what variable you
eliminate first, whatever you feel more
comfortable with.
• If you eliminate the the variable with the
smaller coefficient, your variable will stay
positive thus reducing chance for error.
Example 2
• Solve 80 − 9𝑦 = 6𝑦
Eliminate the – 9y
80 − 9𝑦 = 6𝑦
80 − 9𝑦 + 9𝑦 = 6𝑦 + 9𝑦because that side
has a constant
80 = 15𝑦
also, where the 6y
80
15𝑦
side only has a
=
does not
15
15
variable!
simplify to an
80
=𝑦
integer, so just
15
reduce the
16
=𝑦
fraction!!
3
Try these!
• 34 − 3𝑥 = 14𝑥
2=x
• 5𝑦 − 2 = 𝑦 + 10
y=3
• −6𝑥 + 4 = −8𝑥
-2 = x
• −10 − 3𝑥 = 2𝑥 + 15
-5 = x
Combine Like Terms First…
• Always look to simplify each expression before undoing any
operations!
4x + 4 - 5x = 9x -10 +14
4 - x = 9x -10 +34
4 - x = 9x + 24
4 =10x + 24
-20 =10x
-2 = x
You try…
• 5𝑥 − 3𝑥 + 4 = 3𝑥 + 8 −4 = 𝑥
• 6𝑥 + 3 = 8 + 7𝑥 + 2𝑥
−5
=𝑥
3
What happens if the variables
disappear?
• What happens if
the variables
eliminate or
undo each
other?
Number of Solutions…
• So far all the
equations we have
solved have had one
specific solution. We
end up with x = some
value.
• In some cases, the
variables can totally
eliminate each other
from the equation!
Infinite Solution or Identity
• 3 𝑥 + 2 = 3𝑥 + 6
3 𝑥 + 2 = 3𝑥 + 6
3𝑥 + 6 = 3𝑥 + 6
3𝑥 + 6 − 3𝑥 = 3𝑥 + 6 − 3x
6=6
• Notice the Variables eliminated each other…
• We are left with a balanced equation at the end.
• This means this is an identity and any value will
be a solution to this equation.
• We say this equation has infinite solutions.
No Solutions
• 3 𝑥 + 2 = 3𝑥 + 4
3 𝑥 + 2 = 3𝑥 + 4
3𝑥 + 6 = 3𝑥 + 4
3𝑥 + 6 − 3𝑥 = 3𝑥 + 4 − 3x
6≠4
• Notice the Variables eliminated each other…
• We are left with an unbalanced equation at the
end.
• This means this equation is impossible to solve.
• We say this equation has no solutions.
Remember…
• If the variables do
not eliminate, there
is one solution.
• If variables eliminate
and equation is
balanced there are
infinite solutions.
• If variables eliminate
and equation is
unbalanced there is
no solutions.
You try…
1) 2 𝑥 + 4 = 2𝑥 + 8
Infinite solutions/
identity
2) 2 𝑥 + 4 = 𝑥 − 8
X = -16
3) 2 𝑥 + 4 = 2𝑥 − 8
No Solutions
4) 2 𝑥 + 4 = 𝑥 + 8
X=0