Transcript Slide 1

Solving Equations with Variables
on Both Sides of the Equal Sign
This lesson shows you step by step how to solve equations with
variables on both sides of the equal sign.
Just sit back and enjoy the equations being solved.
After the show, you will have the option of going more slowly
through the four examples.
Solving equations with variables on both
sides of the equal sign.
Find the smaller variable term
7x – 2 = 2x + 8
7x – 2 - 2x = 2x + 8 - 2x
7x - 2x – 2 = 2x - 2x + 8
5x – 2 = 8
2x is smaller than 7x so it is
the smaller variable term.
Subtract 2x from both sides
of the equation
Combine like terms
This is just a two-step
equation. Now solve for x
Follow the two-step procedure
to solve for x
5x – 2 = 8
5x – 2 + 2 = 8 + 2
1
1
5x = 10
5
5
x=2
2
1
Isolate the variable by adding 2 to
both sides of the equation.
Then divide both sides by 5.
Solve for x.
What does this solution mean?
What the solution means…
ALGEBRAICALLY
GEOMETRICALLY
When you substitute x=2 in the
equation, what happens?
x=2
7x – 2 = 2x + 8
7(2) - 2 = 2(2) + 8
14 - 2 = 4 + 8
12 = 12
You make the equation TRUE.
7x – 2 = 2x + 8
y = 7x - 2
y = 2x + 8
Where do they cross?
That wasn’t so bad
Mr. Anderson, let’s
try another!
A second equation with variables on both
sides of the equal sign.
Find the smaller variable term
5 - 3x = 2x + 10
5 - 3x + 3x = 2x + 10+ 3x
5 - 3x + 3x =2x + 3x + 10
5 = 5x + 10
-3x is smaller than 2x.
It’s Negative!
Subtract -3x from both sides
of the equation. That’s the
same as adding 3x.
Combine like terms
This is just a two-step
equation. Now solve for x
Follow the two-step procedure
to solve for x
5 = 5x + 10
5 - 10 = 5x + 10 - 10
-1
1
-5 = 5x
5 5
-1 = x
1
1
Isolate the variable by subtracting
10 from both sides of the equation.
Then divide both sides by 5.
Solve for X
What does this solution mean?
What the solution means…
ALGEBRAICALLY
GEOMETRICALLY
When you substitute x=-1 in the
equation, what happens?
x = -1
5 – 3x = 2x + 10
5 – 3(-1) = 2(-1) + 10
5 + 3 = -2 + 10
8=8
You make the equation TRUE.
5 – 3x = 2x + 10
y = 5 – 3x
y = 2x + 10
Where do they cross?
WOW, I think I can do that!
But, Mr. Anderson, is that
all there is to it?
Good question.
Let’s watch
and learn!
A special equation with variables on both
sides of the equal sign.
Find the smaller variable term
3x – 4 = 1 + 3x
3x – 4 - 3x = 1 + 3x - 3x
3x - 3x – 4 = 1 + 3x – 3x
-4 = 1
3x is the smaller variable term.
So, subtract 3x from both
sides of the equation
Combine like terms
WHOA! Where did the x go?
What is the answer?
-4 = 1
Think about what this equation says.
“Negative four is equal to one.”
BUT, negative four is NOT equal to one.
NO SOLUTION
Therefore, the answer is…
What does THIS solution mean?
What the solution means…
ALGEBRAICALLY
GEOMETRICALLY
For the given equation
3x – 4 = 1 + 3x
There exist NO value for x
which makes the left side of equation
equal the right side of the equation.
You can’t make the equation TRUE.
3x – 4 = 1 + 3x
y = 3x - 4
y = 1 + 3x
Where do they cross?
They will NEVER cross.
They are parallel.
So far, we have seen two types of solution.
1. One solution, like x = 2.
2. No solution, lines are parallel.
Let’s watch one more type of solution.
A third type of equation with variables on
both sides of the equal sign.
Find the smaller variable term
2x + 5 = 5 + 2x
2x + 5 - 2x = 5 + 2x - 2x
2x - 2x + 5 = 5 + 2x - 2x
5=5
2x is the smaller variable
term.
Subtract 2x from both sides
of the equation
Combine like terms
Again, the x – term has
disappeared.
5=5
Think about what this equation says.
“Five is equal to five.”
Duh, five is equal to five. Great.
IDENTITY
Therefore, the answer is…
What does THIS solution mean?
What the solution means…
GEOMETRICALLY
ALGEBRAICALLY
When you substitute ANY VALUE
in the equation, what happens?
2x + 5 = 5 + 2x
Say, x = 4
OR
Say, x = -3
2(4) + 5 = 5 + 2(4)
2(-3) + 5 = 5 + 2(-3)
8+5=5+8
-6 + 5 = 5 - 6
13 = 13
-1 = -1
2x + 5 = 5 + 2x
y = 2x + 5
y = 5 + 2x
You make the equation TRUE.
Where do they cross?
No matter what number you plug
in, you get a true equation!
They cross everywhere.
They are IDENTICAL lines!
What Have We Learned Today?
When Solving Equations with Variables on
Both Sides of the Equal Sign
•Find the smaller variable term and subtract.
•Combine like terms.
•Then, just solve the two-step equation, normally.
•There are THREE types of solutions:
-
One solution, like x = 2. The lines intersect once.
-
No solution. The lines are parallel.
-
Identity. The lines coincide. They are the same.