Solving Equations

1
Solving Equations

• Please view this tutorial and answer the
  follow-up questions on loose leaf to turn in to
  your teacher.

2
Solving Equations Basics

• The goal of solving an equation is to find a
  value for x that when substituted makes both
  sides of the equation equal.
• Essentially, you are trying to get x by itself on
  one side of the equation so that x is equal to a
  number.
• Whatever operation you do on one side of an
  equation, you have to do on the other to keep the
  equation balanced.
• Be sure to substitute your answer back into the
  equation for x to check to see if youre correct.

3
One Step Equations
In one-step equations, you only need to do one
step to get x by itself.
Lets take a look at an example.
4 x 3
You want to get x by itself. What do you think
your first step should be?
4
One Step Equations
4 x 3 - 3

• Since the 3 is positive you must subtract 3 to
  make it 0.

- 3
1 x 0 1 x
Since you subtracted 3 on the right side of the
equation you must subtract 3 on the left side of
the equation to keep it balanced.
Substitute 1 for x into the original equation.
4 x 3
4 (1) 3
4 4 ?
Since 4 is equal to 4 you know youre correct!
5
One Step Equations

• You may also have one step equations where the
  only step you have to do is division.

For Example
32 -4x
-4
-4
What is happening to the x term?
-8 1 x
It is being multiplied by -4.
-8 x
Since the operation is multiplication what
operation must you do to get rid of the -4?
Remember to check your work!
32 -4x
You should DIVIDE both sides by -4.
32 -4(-8)
32 32 ?
6
One Step Equations

• You may also have one step equations where the
  only step you have to do is division.

Note The 4 is NOT being subtracted from the x.
So Do NOT add 4 to undo the operation!
32 -4x
-4
-4
-8 1 x
-8 x
Remember to check your work!
32 -4x
32 -4(-8)
32 32 ?
7
Two Step Equations

• Two step equations involve adding/subtracting on
  both sides first and then dividing/multiplying on
  both sides second
• Example
• 6x 14 -26

What do you think the first step is?
8
Two Step Equations
6x 14 -26
14 14
6x 0 -12
6x -12
If you said add 14 to both sides you are
correct. You must add first so that only the
6x remains on left side of the equation.
9
Two Step Equations
6 6
1x -2
x -2
What will be the next step?
You must divide by six to get x by itself on
one side of the equation.
10
Two Step Equations
Remember that it is important to check your work
by substituting your answer in for x!
6x 14 -26
x -2
6(-2) 14 -26
-12 14 -26
-26 -26 ?
11
Two Step Equations
Lets try another example. Write down the
following problem first and begin solving for x.
Before clicking to the next step make sure you
have done what you think the next step is. This
way you should be checking to see if you did each
step correctly.
4 2x -6
12
Two Step Equations
4 2x - 6
You need to subtract 4 from both sides first
since you need to get the x term by itself on
one side of the equation.
- 4 - 4
Do not forget to bring down the negative sign
with the 2x.
0 2x -10
2x -10
In the next step, you would need to divide by -2
on both sides so that only x is left on one
side of the equation.
2 -2
NOTE You should NOT add 2 to both sides since
the x is being multiplied by -2. So, in order to
get rid of the -2 you have to divide.
1x 5
x 5
Why are you not finished yet?
13
Two Step Equations
You need to check your work to make sure you are
correct!!
4 2x - 6
x 5

• 4 2(5) - 6

• 4 10 - 6

- 6 - 6 ?
14
Two Step Equations
Lets say someone got the answer x 5 for the
equation -8x 5 35. Check to make sure this
person got the correct answer.
The person got the wrong answer since -45 is not
equal to 35. Lets take a look at their work to
see what they did wrong.
-8(5) 5 35
- 40 5 35
- 45 35
15
Two Step Equations
Take a look at the persons work to find their
mistake. Find the mistake before 15 seconds is
up.
-8x 5 35
They forgot to bring down the negative sign next
to the 8.
5 5
8x 40
8 8
x 5
16
Two Step Equations
Take a look at the persons work to find their
mistake.
-8x 5 35
5 5
Lets see what the answer would be if they would
have brought down the negative sign.
-
8x 40
-
-
8 8
-
x 5
17
Equations with Distributive Property
Sometimes youll need to use the distributive
property before you can solve the equation.
-4(-2x 6) -16
First, distribute the -4.
8x - 24 -16
After distributing, you can solve the equation
just like a two step equation.
24 24 8x 0 8
8x 8
8 8
x 1
Be sure to check your answer.
18
Equations with xs on both sides of the equation
Sometimes equations will have an x-term on both
the left and right side of the equation.
When this happens, you must get rid of the
x-term on the right by doing the opposite
operation.
To keep the equation balanced, make sure that you
do the same operation on the left side of the
equation as you did on the right.
19
Equations with xs on both sides of the equation
We need to get rid of the x-term on the right
side by doing the opposite operation. NOTE the
2x is negative.
Make sure that when you add 2x to the left side
that you adding to the other x-term.
x 8 11 2x
2x
2x
3x 8 11 0
3x 8 11
- 8 - 8
From here you just solve the two step equation.
3x 3
So in this case we need to add 2x to both sides.
3 3
x 1
20
Equations with xs on both sides of the equation
Lets check to make sure that were correct.
x 8 11 2x
x 1
(1) 8 11 2(1)
9 11 2
9 9 ?
21
Equations with fractions
Some two step equations will involve fractions.
If x is being multiplied by a fraction, you need
to get rid of it by multiplying by the reciprocal
(the fraction flipped).
22
Equations with Fractions
Here is an example
What would your first step be?
Add five to both sides just like you would in any
other two step equation.
½ x 5 3
5 5
Now to get rid of the ½ you must multiply by the
reciprocal (flip the fraction). What would the
reciprocal be in this case?
½ x 0 8
½ x 8
(2) ½ x 8 (2)
1x 16
The reciprocal would be 2/1 or 2. Now multiply
both sides of the equation by 2 and solve for x.
x 16
23
Equations with Fractions
Make sure to check your answer!
½ x 5 3
x 16
½ (16) 5 3
8 5 3
3 3 ?
24
Other equations
You may also have equations where you must
combine like terms before you solve for x. For
example 5x 8 3x 2 15
What would the equation be after you combine like
terms?
8x 6 15
You will then solve the equation as a two step
equation.
25
Follow-Up Questions
Answer the following questions on loose leaf and
hand them in to your teacher.
26
Follow-Up Questions

• Solve the following equations algebraically for
  x. Show all work.

1) 6) 2) 7) 3) 8) 4)
9) 5) 10)