Combining Like Terms, Simplifying Expressions, Solving Equations

Combining Like Terms, Simplifying Expressions, Solving Equations

By Shalynne Orth

Vocabulary

• Variable - the letter that represents an unknown value ex. y, x, n

• Coefficient - The number infront of the variable ex. 6x, 7y, 1n

• Constant - Any number being added or subtracted in the expression ex. 6 + 1 - 2

• Terms - All the variables with their coefficient and constants that create an expression ex. 3 + 2x + 4y + 1 + 6x This expression has 5 terms.

• Like Terms - same variable to the same power

Multiple Choice

In the following expression, the number 5 is an example of a _____________?

- 7x + 5y + 11

• term

• constant

• coefficient

• variable

Combining Like Terms

AKA simplifying expressions

Multiple Choice

Simplify by combining like terms:

d + d + d + 3 + 2

• 3d + 5

• d + 5

• 2d + 7

• 3d + 4

Multiple Choice

Let's kick it up a notch!

Start by identifying all the like terms, then combine.

Pay attention to the signs before each term!

6f + 3 - 4f + 5 + 10f

• 20f

• 12f + 8

• 24f + 8

• 28 + f

Multiple Choice

This time, you need to distribute THEN combine like terms!

3(y + 5) - 2y

• 5y + 5

• 6y

• 5y + 15

• y + 15

Multiple Choice

Using the Distributive Property, which expression is equivalent to

$$5\ \left(3x\ +\ y\right)$$  

• $$15x\ +\ 5y$$  

• $$15x\ +\ y$$  

• $$20xy$$  

• $$15x5y$$  

Multiple Choice

$$-12\left(m\ +4n\ -5p\right)$$ Using Distributive Property, simplify this expression 

• $$-12m\ +48n\ +60p$$  

• $$12m\ +48n\ -60p$$  

• $$-12m\ -8n\ -17p$$  

• $$-12m\ -48n\ +60p$$  

Expressions vs Equations

Before solving an equation, it is helpful to simplify each expression.

We simplify by combining like terms... but what does that mean?

One-Step Equations

Solving equations

• use inverse operations to isolate the variable

• keep the equation balanced by performing the same operation on both sides of the equals sign

3 = z + 6

• subtract 6 from both sides

• z = -3

Multiple Choice

How do we solve f - 12 = -4

• add -12 to both sides

• subtract 12 from both sides

• add 12 to both sides

• add 4 to both sides

-16 = -4p

• divide both sides by -4

• p = 4

k/3 = -30

• multiply both sides by 3

• k = -90

Multiple Choice

How do we solve -2k = -100?

• multiply by 2 on both sides

• divide by 2 on both sides

• add 100 to both sides

• divide by -2 on both sides

Multiple Choice

z/4 = -8

• z = -32

• z = 32

• z = 2

• z = -2

Two Step Equations

How to Solve

• Get rid of the 3rd Wheel: Add or Subtract to get rid of the number

• Break up the happy couple: Multiply or Divide both sides of the equation by the coefficient

Multiple Choice

What is the first step in this equation?

3x + 5 = 20

• Add 5 and 20

• Subtract 5 from both sides of the equation

• Add 3 and 5

• Multiply 3 by 5

Multiple Choice

After you subtract 5 from both sides of the equation (3x + 5 = 20) what will the equation look like?

• 3x = 25

• 3x = 20

• 3x = 3

• 3x = 15

Multiple Choice

What is the next step in the equation?

3x = 15

• Divide both sides by 3

• Add 3 and 15

• Subtract 3 and 15

• Multiply 3 and 15

Multiple Choice

What is the final answer?

• 45

• 18

• 5

• 12

Multipstep Equations

1) Distribute

2) Combine like terms

3) complete two step equation steps

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Multiple Choice

$$x+2+6=8$$  

• 0

• 14

• 8

• 4

Multiple Choice

$$96=6\left(k+8\right)$$  

• $$12$$  

• $$8$$  

• $$-4$$  

• $$9$$  

Multiple Choice

10(h+1)-4=76

• h=10

• h=4

• h=7

• h=76

Equations with variables on both sides

Multiple Choice

How many x's (rectangles) can we remove from both sides?

• 1

• 2

• 3

• 4

Now we only have the variables on one side

We can move on to solving equations like you are used to

By removing the constants (on both sides) so that we have the variable (x) by itself

Multiple Choice

How many ones (yellow squares) can we remove from both sides?

• 2

• 4

• 6

• -2

We conclude that each x (rectangle) is the same as 4

Multiple Choice

How many n's can we remove from both sides?

3n + 5 = 2n + 7

• 3n

• 2n

• 5

• 7

Multiple Choice

Solve

What's the next step?

n + 5 = 7

• Remove 5 from both sides

• Remove 7 from both sides

• Multiply by 5

• Multiply by 7

Remember

• The goal is to have the variables on one side

• We remove as many as we can from both sides

• Once there are variables only on one side, continue solving for x

Multiple Choice

Solve for x

5x + 2 = 4x + 4

• 1

• 2

• 3

• 4

Multiple Choice

What is the solution to the equation  $$3+12(3p+2)=9(p-3)$$ ?  

• -20

• -13

• -2

• -18

Multiple Choice

• A
• B
• C
• D

How Many Solutions?

A review of the 3 types of solutions to equations

Infinitely Many Solutions

• True mathematical statement

• Example 3=3

• The coefficients on the variables will be the SAME on both sides of the equation, and the remaining numbers to be the SAME.

• Example: 4x+15=4x+15

One Solution

• Solve for the variable, ends with the variable equal to ONE number.

• Ex. x=3

• The coefficients on the variables will be DIFFERENT

• Ex. 5x+30=7x+30

No Solution

• Ends with a FALSE mathematical statement

• Ex. 4=8

• The coefficient on the variable will be the SAME, much like with a true statement. But, the remaining numbers will be DIFFERENT.

• Ex. -7x+10= -7x+15

Multiple Choice

3x+2=2+3x

• One Solution

• No Solution

• Infinitely Many

Multiple Choice

-3x-5+5x=2(x+7)

• One Solution

• No Solution

• Infinitely Many