Solving Equations

Please view this tutorial and answer the follow-up questions on loose leaf to

turn in to your teacher.

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 you’re correct.

One Step Equations

Let’s take a look at an example.

In one-step equations, you only need to do one step to get x by itself.

4 = x + 3

You want to get x by itself. What do you think your first step

should be?

One Step Equations

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

4 = x + 3 - 3- 3

1 = x + 01 = 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

Since 4 is equal to 4 you know you’re correct!

4 = (1) + 34 = 4 ✓

One Step Equations

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

For Example: 32 = -4xWhat is happening to the

x term?

It is being multiplied by -4.

Since the operation is multiplication what

operation must you do to get rid of the -4?

You should DIVIDE both sides by -4.

-4-4

-8 = 1 x-8 = x

Remember to check your work!

32 = -4x32 = -4(-8)32 = 32 ✓

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 = -4x32 = -4(-8)32 = 32 ✓

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?

Two Step Equations

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.

6x – 14 = -26 +14 +14 6x + 0 = -12 6x = -12

Two Step Equations

What will be the next step?

6x – 14 = -26 +14 +14 6x + 0 = -12 6x = -12

You must divide by six to get “x” by itself on one side of the equation.

6 6 1x = -2 x = -2

Two Step Equations

6x – 14 = -26

x = -2

Remember that it is important to check your work by substituting your answer in for x!

6(-2) – 14 = -26-12 – 14 = -26

-26 = -26 ✓

Two Step Equations

4 – 2x = -6

Let’s 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.

Two Step Equations4 – 2x = - 6

- 4 - 40 – 2x = -10

You need to subtract 4 fromboth sides first since youneed to get the “x term” byitself on one side of theequation.

– 2x = -10

Do not forget to bring down the negative sign with the 2x.

– 2 -2 1x = 5 x = 5

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.

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.

…Why are you not finished yet?

Two Step EquationsYou 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 ✓

Two Step EquationsLet’s say someone got the answer

x = 5for the equation

-8x – 5 = 35.

Check to make sure this person got the correct answer.

-8(5) – 5 = 35- 40 – 5 = 35 - 45 = 35

The person got the wrong answer since -45 is not equal to 35. Let’s take a look at their work to see what they did wrong.

Two Step EquationsTake a look at the person’s work to find their

mistake. Find the mistake before 15 seconds is up.

-8x – 5 = 35 + 5 + 5

8x = 40 8 8 x = 5

They forgot to bring down the negative sign next to the 8.

Two Step EquationsTake a look at the person’s work to find their

mistake.

-8x – 5 = 35 + 5 + 5 8x = 40 8 8 x = 5

-Let’s see what the answer would be if they would have brought down the negative sign.

- --

Equations with Distributive Property

Sometimes you’ll 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 +248x + 0 = 8

8x = 8 8 8 x = 1 Be sure to check your answer.

Equations with x’s 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.

Equations with x’s on both sides of the equation

x + 8 = 11 – 2xWe need to get rid of the x-term on the right side by doing the opposite operation.

NOTE: the 2x is negative.

+ 2x

Make sure that when you add 2x to the left side that you adding to the other ‘x-term’.

+2x 3x + 8 = 11 + 0

3x + 8 = 11

From here you just solve the two step equation.

- 8 - 8 3x = 3 3 3

x = 1

So in this case we need to add 2x to both sides.

Equations with x’s on both sides of the equation

Let’s check to make sure that we’re correct.

x + 8 = 11 – 2x x = 1

(1) + 8 = 11 – 2(1) 9 = 11 – 2

9 = 9 ✓

Equations with fractionsSome 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).

Equations with Fractions

Here is an example:

½ x – 5 = 3

What would your first step be?

Add five to both sides just like you would in any other two step equation.

+ 5 + 5½ x + 0 = 8 ½ x = 8

Now to get rid of the ½ you must multiply by the reciprocal (flip the fraction).

What would the reciprocal be in this case?

The reciprocal would be 2/1 or 2. Now multiply both sides of the equation by 2 and solve for x.

(2) ½ x = 8 (2) 1x = 16 x = 16

Equations with Fractions

Make sure to check your answer!

½ x – 5 = 3

x = 16

½ (16) – 5 = 3

8 – 5 = 3 3 = 3 ✓

Other equationsYou may also have equations where you must

combine like terms before you solve for x.

For example:

5x – 8 + 3x + 2 = 15What would the equation be after you combine like terms?

8x – 6 = 15

You will then solve the equation as a two step equation.

Follow-Up Questions

Answer the following questions on loose leafand hand them in to your teacher.

Follow-Up QuestionsSolve the following equations algebraically for x. Show all work.

1) 6)

2) 7)

3) 8)

4) 9)

5) 10)

x−4 =−32

3x =6

5x + 8 =−42

−3x + 6 = −6

8−2x=14

5 =18 −x

2x + 4 =3x−1

−3x − 8 = −10x − 78

4(2x−3)−3x=33

3

4x +10 =4