Solving Equations

Solving Equations

One Step Equations

• Isolate the variable by using inverse operations

• Addition & subtraction are inverses; so are multiplication and division

• In one step, your variable should be isolated
  

• Check your answer!

Multiple Choice

What is happening to the variable? 5x = 12

• 5 is being added to it

• 5 is being subtracted from it

• It's being divided by 5

• It's being multiplied by 5

5x = 12

• x is being multiplied by 5

• The inverse of multiplying by 5 is dividing by 5
  

• Remember to divide BOTH SIDES by the same thing

• x = 12/5 or x = 2 2/5

5x = 12 Check

• We said that x = 12/5. This means that the value 12/5 makes the statement true. Substitute in 12/5 wherever you see x, follow PEMDAS, and make sure both sides are the same.

• 5(12/5) = 12

• 60/5 = 12

• 12 = 12, so x = 12/5 is correct
  

x + 12 = 14

• 12 is being added to the variable

• The inverse of adding 12 is subtracting 12

• When you subtract 12 from BOTH SIDES 
  
  

Open Ended

What are the steps to check your answer for x + 12 = 14?

Multiple Choice

What is happening to the variable? x/4 = 5

• Having 4 added to it

• Having 4 subtracted from it

• Being multiplied by 4

• Being divided by 4

x/4 = 5

• x is being divided by 4

• The inverse of dividing by 4 is multiplying by 4

• Multiply both sides by 4 to get x = 20
  
  

• Check: 20/4 = 5?

• 5 = 5 so x = 20 is correct

Two Step Equations

• Instead of taking just one step to isolate the variable, you will now use 2 steps.

• Usually, the first step is to add/subtract your constant from each side.

• After that, it should look like the one step equation we already looked at, so you can solve based on those instructions
  
  

Two Step Equations

• Usually, a two step equation will be the variable term +- another term = a number

• The first goal is to isolate the variable term, which will usually require adding/subtracting the constant

• Then, you will have a one step equation that you know how to solve

Multiple Choice

5x + 7 = 27 - What is the first step to solve?

• Add 27 to each side

• Divide by 5

• Add 7 to both sides

• Subtract 7 from both sides

5x + 7 = 27

• First step is to subtract 7 from both sides and get

• 5x = 20

• Then, to isolate the variable, divide both sides by 5 and get

• x = 4

• Check:

• 5(4) + 7 = 27 --> 20+7=27 --> 27 = 27

MultiStep Equations

• Multistep equations will take more than 2 steps to solve

• You want to take steps to make it look like a two step equation

• These steps will be outlined

Distributing

• You may need to first distribute a number to other terms inside parentheses

• Eventually, you will also need to be able to "undistribute" or factor

• Remember to distribute the number outside of the parentheses by ALL numbers inside of them

Multiple Choice

What do you get when you distribute 2(6x + 12)

• 6x + 12

• x + 2

• 12x + 12

• 12x + 24

Combining Like Terms

• Once you have distributed, look to see if you have any like terms

• You are only looking for like terms on the SAME side of the equal side

• Remember, just combine the terms together. You do not need to add/subtract anything from either side.

Multiple Choice

Simplify: 2x + 3 + 17x - 9 = 12x

• 13x = 12x

• 19x - 6 = 12x

• 19x - 9 = 12x

• 5x + 8 = 12x

Move Constant Terms to 1 Side, Variable Terms to the Other

• If you have like terms on OPPOSITE sides of the equation, you will actually be moving the term, not just combining like terms

• The goal is to have all constant terms on one side of the equation and all variable terms on the other side of it so that it looks like a one step equation

Move Constant Terms to 1 Side, Variable Terms to the Other

• 5x + 12 = 8x - 9

• Subtract 5x from each side to get

• 12 = 4x - 9

• Add 9 to get 21 = 4x
  

• Divide 21 by 4 to get your variable isolated so x = 5.25 or 5 1/4

Multi --> 1/2

• If you use (in order) The Distributive Property, combining like terms, and moving constants to one side/variables to the other then you will be able to turn a multistep equation into a one or two step equation

Multi --> 1 or 2   5(x + 3) + 2x = 1 + 5x(

• Step 1: Distribute to get 5x + 15 + 2x = 1 + 5x

• Step 2: Combine like terms to get 7x + 15 = 1 + 5x

• Step 3: Move all variable terms to one side (left) by subtracting 5x from both sides to get 2x + 15 = 1

• Step 4: Move all constant terms to the other side (right) by subtracting 15 to get 2x = -14 
  

• Step 5: Isolate the variable. Look at what is happening to the variable and do the inverse. In this example, x is multiplied by 2, so divide on both sides by 2 to get x = -7

Special Cases

• Sometimes, there will be more than 2 terms in the numerator. If that is the case, multiply by the denominator first (unless distributon in the numberator needs to happen first)

• If that's the case, multiply by the denominator FIRST to eliminate it on one side

(x + 4)/5 = 20

• x + 4 gives us 2 terms in the numerator

• Multiply by 5 on both sides before continuing to get x + 4 = 100

• We now have a one step equation. 4 is being added to the variable, so subtract 4 from both sides to isolate and get x = 96

• Check: (96 + 4)/5 = 20 --> 100/5 =  20 --> 20 = 20 so x = 96 is correct                     

Multiple Choice

 $7\ =\ \frac{\left(12\ +\ x\right)}{-4}$  

• x = -16

• x = -40

• x = 16

• x = 40

Multiple Choice

 $-4\ =\ \frac{x}{5}+\ 6$  

• x = 50

• x = 10

• x = -10

• x = -50

Multiple Choice

 $\frac{1}{3}x\ +\ 2\ =\ 5$  

• x = -9

• x = 13

• x = 9

• x = 1