Solving Equations Slide 3 / 188 Table of...

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8th Grade Math

Solving Equations

2015-12-17

www.njctl.org

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Table of Contents

Review of Two Step Equations

Multi-Step Equations

Click on a topic to go to that section.

Glossary & Standards

Translating & Solving Consecutive Integer Problems

Solving Equations that Contain Fractions

Writing and Solving Algebraic Equations

Equations with the Same Variable on Both Sides

Comparing Expressions with the Same Variable

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Review of Two-Step Equations

Return to Table of Contents

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Two-Step Equations

A two-step equation is an equation that contains two operations. For example, it could contain multiplication and subtraction, like the equation below.

5x - 9 = 16

Or it could contain addition and division like this equation.

+ 11 = -6

Before we start solving two-step equations, let's review some tips for solving them.

x 4

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2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.

3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."

4. You can always switch the left and right sides of an equation.

Tips for Solving Equations

1. To "undo" a mathematical operation, you must perform the inverse operation.

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1. To "undo" a mathematical operation, you must do the opposite.

We learned earlier that for every mathematics operation, there is an inverse operation which undoes it: when you do both operations, you get back to where you started.

When the variable for which we are solving is connected to something else by a mathematical operation, we can eliminate that connection by using the inverse of that operation.

Tips Explained

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2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same to the other side.

If the two expressions on the opposite sides of the equal sign are equal to begin with, they will continue to be equal if you do the same mathematical operation to both of them.

This allows you to use an inverse operation on one side, to undo an operation, as long as you also do it on the other side.

You can just never divide by zero (or by something which turns out to be zero) since the result of that is always undefined.

Tips Explained

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3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."

The operations which are connected to a variable must be "undone" in the reverse order from the Order of Operations.

So, when solving for a variable, you:first have to undo addition/subtraction, then multiplication/division,then exponents/roots, finally parentheses.

The order of the steps you take to untie a knot are the reverse of the order used to tie it.

Tips Explained

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4. You can always switch the left and right sides of an equation.

Once an equation has been solved for a variable, it is typically easier to use if that variable is moved to the left side.

Mathematically, this has no effect since the both sides are equal.

Tips Explained

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Let's solve this equation for "y"

Solving for y

That means that when we're donewe'll have y alone

on the left side of the equation.

-4y - 11 = -27

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1 Is y already alone? If not, what is with it? Select all that apply.

A -4

B y

C -11

D -27

E it is already alone

-4y - 11 = -27

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-4y - 11 = -27

2 Which math operations connect the numbers to y? Select all that apply.

A Addition

B Subtraction

C Multiplication

D Division

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-4y - 11 = -27

3 Which math operation gets undone first?

A AdditionB SubtractionC MultiplicationD Division

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-4y - 11 = -27

4 What must we do if we add 11 to the left side?

A Subtract 11 from the left side

B Subtract 11 from the right side

C Add 11 to the left side

D Add 11 to the right side

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Solving for y

-4y - 11 = -27

Are we done?


2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.So we undo 9 being subtracted from -2y by adding 9 to both sides.

-4y = -16

-4y - 11 = -27 + 11 +11

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5 What math operation connects -4 and y?

A -4 is being added to y

B -4 is being subtracted by y

C -4 is being multiplied by y

D -4 is being divided by y

-4y = -16

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-4y = -16

6 What is the opposite of multiplying y by -4?

A Dividing y by 4

B Dividing y by -4

C Multiplying y by 4

D Multiplying y by -4

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-4y = -16 -4 -4


y = 4

Solving for y


-4y = -16

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7 Is y alone on the left? If not, what is with it? A -11B -4C yD it is alone

y = 4

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Let's solve this equation for "t"

Solving for t

15t

5 =

That means that when we're donewe'll have t alone

on the left side of the equation.

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8 Is t already alone? If not, what is with it?

A 5B 15C tD it is already alone

15t5 =

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9 What mathematical operation connects d to t?

A t is being divided by 15B 15 is being divided by tC 15 is being multiplied by tD t is being subtracted from 15

15t5 =

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10 What is the opposite of dividing 15 by t?

A dividing 15 by tB dividing 5 by tC multiplying 15 by tD multiplying t by 15

Rule 1. To "undo" a mathematical operation, you must do the opposite.

15t5 =

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11 What must we do if we multiply the right side by t?

A divide the left side by tB multiply the left side by tC divide the left side by 15D divide the left side by 5

Rule 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.

15t5 =

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12 Is there more than one mathematical operation acting on "t"?

YesNo

Rule 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations."

15t5 =

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Are we done?

Solving for t


2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other.So we undo d being divided by t, by multiplying both sides by t.

15t

5 =(t) (t)

5t = 15

15t

5 =

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13 What mathematical operation connects 5 to t?

A t is being divided by 15B t is being divided into 5C t is being multiplied by 5D t is being subtracted from 5

5t = 15

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14 What is the opposite of multiplying t by 5?

A dividing t by 5B dividing t by tC multiplying t by tD multiplying t by 5

5t = 15

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5


5

t = 3

Solving for t


5t = 15

5t 155

=

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15 Is t alone on the left? If not, what is with it? A 5B 15C tD it is alone

t = 3

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Review of Solving Two-Step Equations

The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:

https://www.njctl.org/courses/math/7th-grade/equations-inequalities-7th-grade/

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16 Solve the equation.

5x - 6 = -56

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14 = 3c + 2

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x 5

- 4 = 24

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5r - 2 = -12

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14 = -2n - 6

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+ 7 = 13 x 5

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+ 2 = -10 x 3

-

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23 Solve the equation.-2.5x - 4 = 3.5

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24 Solve the equation.3.3x - 4 = -13.9

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+ (-5.1) = -2.3-x 6

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26 Solve the equation.2.8x - 7 = -1.4

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Multi-Step Equations


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Steps for Solving Multiple Step EquationsAs equations become more complex, you should:

1. Simplify each side of the equation.

2. Use inverse operations to solve the equation.

Remember, whatever you do to one side of an equation, you MUST do to the other side!

(Combine like terms and use the distributive property.)

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Example:

12h - 10h + 7 = 25

Multiple Step Equations

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Example:

17 - 9f + 6 = 140

Multiple Step Equations

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Always check to see that both sides of the equation are simplified before you begin solving the equation.

When an equation is simplified there should be at most one term for each variable and one constant term.

Hint

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In order to solve for a variable, it can't be in parentheses with other variables or numbers.

If that is how the equation is given, you have to use the distributive property to get it on its own.

For example: You need to use the distributive property as shown below as a first step in solving for x.

5(x - 6) - 3x = 8 Distributive Property5x - 30 - 3x = 8 Now we can combine like terms and solve

Distributive Property

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Examples:

5(20 + 6) = 5(20) + 5(6) 9(30 - 2) = 9(30) - 9(2)

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

-2(4x - 7) = -2(4x) - (-2)(7)

Remember: The distributive property is a(b + c) = ab + ac


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Example:3(w - 2) = 9


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Example:6m + 2(2m + 7) = 54


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27 Solve.

3 + 2t + 4t = -63

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28 What is the value of n in the equation 0.6(n + 10) = 3.6?

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

A -0.4B 5C -4D 4

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29 Solve.

19 = 1 + 4 - x

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30 Solve.

8x - 4 - 2x - 11 = -27

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31 Solve.

-4 = -27y + 7 - (-15y) + 13

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32 Solve.

9 - 4y + 16 + 11y = 4

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33 Solve.

6(-8 + 3b) = 78

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34 Solve.

18 = -6(1 - 1k)

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35 Solve.

2w + 8(w + 3) = 34

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36 Solve.

4 = 4x - 2(x + 6)

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37 Solve.

3r - r + 2(r + 4) = 24

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38 What is the value of p in the equation 2(3p - 4) = 10?

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

A 1B 2 1/3C 3D 1/3

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Solving Equations that Contain Fractions


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You always need to have the variable for which you are solving end up in the numerator.

Also, it's often easier to just get rid of fractions as a first step.

Distributing Fractions

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(-3 + 3x) = 3 5

72 5

3(-3 + 3x) = 72 Distribute-9 + 9x = 72 Add 9 to both sides9x = 81 Divide both sides by 9x = 9


Multiply both sides of this equation by the least common denominator (LCD) of both sides to eliminate the fractions.

Then this just becomes like all the equations we've been solving.

In this case, the LCD is 5...so just multiply both sides by 5 to get:

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Example:

Multiply both sides by 21

Simplify fractions12(x + 9) = 14 Distribute the 1212x + 108 = 14 Subtract 108 from both sides12x = -94 Divide both sides by 12x = -7 10/12 Simplify fraction x = -7 5/6

4 7

(x + 9) = 2 3

(21)(4) 7

(x + 9) = (2)(21) 3


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Example: 3 5

(x + 2) = 9


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Example:

1 8

(x + ) = 7 16

1 2


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2x - 3 = 95

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2x + 3 = 4 3

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41 Which value of x is the solution of the equation ?

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

A

B

C

D

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42 Solve: 2 3

x + 5 6

= 7 12

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43 Solve

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45 Solve

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Equations with the Same Variable on Both Sides


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Now, we will be given an equation with the same variable on both sides. These equations will look similar to the following:

These require one additional step to get all the terms with that variable to one side or the other. It doesn't matter which side you choose to move the variables to, but it’s typically most helpful to choose the side in which the coefficient of the variable will remain positive.

Variables on Both SidesPreviously, you solved equations with variables on one side, similar to the following:

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When you have finished solving, discuss the meaning of your answer with your neighbor.

Meaning of Solutions

Before we encounter the new equations, let's practice how to solve an equation with the variable on only one side.

Solve for x:

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Meaning of Solutions

Remember that you always have the ability to check your answers by substituting the value you solved for back in to the original equation.

It isn't necessary to show on each problem, but is encouraged if you feel unsure about your answer.

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Which side do you think would be easiest to move the variables to?

Variables on Both Sides

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Now, solve the equation.


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Which side do you think would be easiest to move the variables to?

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Now, solve the equation.

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Example:What do you think about this equation? What is the value of x?


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Example:What do you think about this equation? What is the value of x?


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47 Solve for f:

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48 Solve for h:

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49 Solve for x:

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Sometimes, you get an interesting answer.What do you think about this?

What is the value of x?

3x - 1 = 3x + 1 -3x -3x -1 = +1

Since the equation is false, there is no solution!

No value will make this equation true.

No Solution

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How about this one?What do you think about this?

What is the value of x?

3(x - 1) = 3x - 3 3x - 3 = 3x - 3 -3x -3x -3 = -3

Since the equation is true, there are infinitely many solutions! The equation is called an identity.

Any value will make this equation true.

Identity

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50 Solve for r:

A r = 0

B r = 2

C infinitely many solutions (identity)

D no solution

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51 Solve for w:

A w = -8

B w = -1


D no solution

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52 Solve for x:

A x = 0

B x = 24


D no solution

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54 In the equation n is equal to?


AB

C

D

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55 In the accompanying diagram, the perimeter of ∆MNO is equal to the perimeter of square ABCD. If the sides of the triangle are represented by 4x + 4, 5x - 3, and 17, and one side of the square is represented by 3x, find the length of a side of the square.

5x – 3

4x + 4

17

O

M

N 3x

A B

C D


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56 Solve this equation for x.

0.5(5 - 7x) = 8 - (4x + 6)

From PARCC sample test - PBA non calc #2

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57 Solve for x.

9(3 - 2x) = 2(10 - 8x)

From PARCC sample test - EOY non-calc #1

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http://practice.parcc.testnav.com/#



· When solving an equation with variables on both sides, choose a side to move all of them to, then continue working to isolate the variable.

· When solving an equation where all variables are eliminated and the remaining equation is false, there is No Solution.

· When solving an equation where all variables are eliminated and the remaining equation is true, there are Infinite Solutions.

RECAP

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Comparing Expressions with the Same Variable


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Recall that an expression is a mathematical phrase that contains ordinary numbers, variables and operations. Also, an equation is a statement indicating that two expressions are equal.

In the last lesson, you were solving an equation that contained variables on both sides of the equation. In some cases, you were able to find a solution. In others, there was either no solution, or infinitely many solutions.

What if you were given two expressions and trying to determine the relationship between them, instead. Could it be done?

Expressions with the Same Variable

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You can easily compare expressions to one another.

Take this example:

What is the relationship between the expressions (x + 4) and (x + 8)?

Since we don't know the relationship between the expressions yet, we can write an equation, but replace the "=" sign with a question mark.

x + 4 ? x + 8


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What is the relationship between the expressions (x + 4) and (x + 8)?

x + 4 ? x + 8

Treat this like an equation and try to solve it by first subtracting x from both sides. Now, we have

4 ? 8

We can conclude that 4 < 8. Therefore, no matter what the value of x is, (x + 4) will always be less than (x + 8).


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Let's try another example:

What is the relationship between the expressions 4x and -6x?

Since we don't know the relationship between the expressions yet, we can write an equation, but replace the "=" sign with a question mark.

4x ? -6x


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4x ? -6x

Treat this like an equation and try to solve it by first adding 6x to both sides. Then, we have

10x ? 0

We can finish solving by dividing both sides by 10. The result will be

x ? 0


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x ? 0

In the expression above, there is no set value for x. Can you determine the different answers and provide evidence to support them?

Discuss.


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x ? 0

Here are the cases that your groups should have come up with.

Case 1: If x > 0, then 4x > -6x e.g. if x = 2, 4(2) > -6(2), or 8 > -12 Case 2: If x < 0, then 4x < -6x e.g. if x = -1, then 4(-1) < -6(-1), or -4 < 6

Case 3: If x = 0, 4x = -6x e.g. if x = 0, then 4(0) = -6(0), or 0 = 0


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59 What is the relationship between the expressions (x - 10) and (x - 15)?

A x - 10 < x - 15

B x - 10 > x - 15

C x - 10 = x - 15

D all of the above, depending on the value of x

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60 What is the relationship between the expressions

and ?

A <

B >

C =


x 5

x -3 x -3 x -3

x -3

x 5 x 5 x 5

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61 What is the relationship between the expressions (4x - 3) and (4x + 3)?

A 4x - 3 < 4x + 3

B 4x - 3 > 4x + 3

C 4x - 3 = 4x + 3


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62 What is the relationship between the expressions

3x - 7 and (9x - 21)?

A 3x - 7 < (9x - 21)

B 3x - 7 > (9x - 21)

C 3x - 7 = (9x - 21)


1 3

1 3 1 3 1 3

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Summer is considering the expressions

2(3x + 20) and (6x - 12) + 4(x + 11)

Part AShe claims that the first expression is greater than the second expression for all values of x. Is Summer correct? Explain your answer.

Part BWrite another expression that is always less than both of these expressions.


1 3

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2(3x + 20) and (6x - 12) + 4(x + 11)


If we set up our expressions with a question mark in between them, and simplify them, we would have

2(3x + 20) ? (6x - 12) + 4(x + 11)

6x + 40 ? 2x - 4 + 4x + 446x + 40 ? 6x + 40


1 3

1 3

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2(3x + 20) and (6x - 12) + 4(x + 11)


6x + 40 ? 6x + 40

Clearly, we can replace the "?" with an "=". Therefore, Summer is incorrect.


1 3

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2(3x + 20) and (6x - 12) + 4(x + 11)

Part BWrite another expression that is always less than both of these expressions.

Since both of them simplified to 6x + 40, we can write any algebraic expression that simplifies to 6x and any number less than 40. Here are 2 sample answers:

6x + 20

(12x - 10) = 6x - 5


1 3

1 2

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Recall from 6th grade and 7th grade, we translated phrases/sentences into expressions and equations. Now, we can extend this understanding to writing and solving algebraic equations, and using them to solve word problems.

Let's start off by reviewing key words for our operations. What are some of them?


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Review of Writing and Solving Algebraic Equations

The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at:


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70 Which equation can be made from the sentence below:The difference between 15 and a number is 18.

A n - 18 = 15

B 18 - n = 15

C n - 15 = 18

D 15 - n = 18

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71 What is the solution to the equation that represents the sentence below?The difference between 15 and a number is 18.

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72 Which equation can be made from the sentence below:Twelve more than the quotient of a number and four is -3.

A - 12 = -3

B + 12 = -3

C - 12 = 4

D + 12 = 4

n 4 n 4 n -3 n -3

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73 What is the solution to the equation that represents the sentence below?Twelve more than the quotient of a number and four is -3.

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74 Which equation can be made from the sentence below:Two fifths of a number increased by 7 is -1.

A n + (-1) = 7

B n - (-1) = 7

C n + 7 = -1

D n - 7 = -1

2 5 2 5 2 5 2 5

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75 What is the solution to the equation that represents the sentence below?Two fifths of a number increased by 7 is -1.

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76 Translate and solve.

10 equals the sum of four and the quotient of a number n and 12.

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77 Translate, then solve the equation.

Four more than three times a number is 13.

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How old am I if 400 reduced by 2 times my age is 342?

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Twice the quantity of x plus eight is negative 2.

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Negative seven times the sum of eighteen and a number s is equal to 49.

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132 is equal to negative twelve times the quantity of the sum of a number x and eighteen.

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Negative one is the same as the quotient of the sum of four and x, and six.

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We can use our algebraic translating skills to solve other problems.

We can use a variable to show an unknown.A constant will be any fixed amount.

If there are two separate unknowns, relate one to the other.


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The school cafeteria sold 225 chicken meals today. They sold twice the number of grilled chicken sandwiches than chicken tenders. How many of each were sold?

2c + c = 225

chicken sandwiches

chicken tenders

total meals

c + 2c = 225 3c = 225 3 3 c = 75The cafeteria sold 150 grilled chicken sandwiches and 75 tenders.


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Many times with equations there will be one number that will be the same no matter what (constant) and one that can be changed based on the problem (variable and coefficient).

Example: George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?


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George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $127.00. How many games did he buy in all?

Notice that the video games are "per game" so that means there could be many different amounts of games and therefore many different prices. This is shown by writing the amount for one game next to a variable to indicate any number of games.

30g

cost ofone videogame

number of games


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Notice also that there is a specific amount that is charged no matter what, the flat fee. This will not change so it is the constant and it will be added (or subtracted) from the other part of the problem.

30g + 7


number of games

the cost of the shipping


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"Total" means equal so here is how to write the rest of the equation.

30g + 7 = 127


numberof games

the total amount

the cost of shipping


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Now you can solve it.


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Walter is a waiter at the Towne Diner. He earns a daily wage of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?



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83 Lorena has a garden and wants to put a gate to her fence directly in the middle of one side. The whole length of the fence is 24 feet. If the gate is 4 feet, how many feet should be on either side of the fence?

12

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84 Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. Which equation shows this problem?

A 12p + 27 = 147 B 12p + 27p = 147 C 27p + 12 = 147 D 39p = 147

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85 Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. How many people went to the amusement park WITH Lewis?

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86 Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? Which equation represents the situation?

A 9 + 68 = 239B 9d + 68 = 239C 68d + 9 = 239D 77d = 239

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87 Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle?

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88 You are selling t-shirts for $15 each as a fundraiser. You sold 17 less today then you did yesterday. Altogether you have raised $675.

Write and solve an equation to determine the number of t-shirts you sold today.

Be prepared to show your equation!

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89 Rachel bought $12.53 worth of school supplies. She still needs to buy pens which are $2.49 per pack. She has a total of $20.00 to spend on school supplies. How many packs of pens can she buy?

Write and solve an equation to determine the number of packs of pens Rachel can buy.


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90 The product of -4 and the sum of 7 more than a number is -96.

Write and solve an equation to determine the number.


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91 A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200.

Write and solve an equation to determine the number of subscribers they had each year.


How many subscribers last year?

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Teachers:

Use this Mathematical Practice Pull Tab for the next 2 SMART Response slides.

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92 The length of a rectangle is 9 cm greater than its width and its perimeter is 82 cm.

Write and solve an equation to determine the width of the rectangle.


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93 The perimeter of a regular hexagon is 13.2 cm.

Write and solve an equation to determine the length of a side of the hexagon.


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94 Two angles are complementary. One angle has a measure that is five times the measure of the other angle. What is the measure, in degrees, of the larger angle?


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95 Two angles are supplementary. One angle has a measure that is 40 degrees larger than the measure of the other angle. What is the measure, in degrees, of the larger angle?


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Together two laptops cost $756. Laptop #1 is one third the cost of laptop #2. What is the cost of the cheaper laptop?

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97 Filipo is building a rectangular sandbox for his younger brother. The length of the sandbox is 1 foot longer than twice the width of the sandbox. The perimeter of the sandbox is 29 feet.

Part A Which equation could be used to determine w, the width, in feet, of the sandbox?

A w + w + 2 = 29

B w + 2w + l = 29

C 2w + 2(w + 2) = 29

D 2w + 2(2w + l) = 29

From PARCC sample test - EOY calculator #2

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98 Part B (Continued from previous question)

What is the width, in feet, of the sandbox?

From PARCC sample test - EOY calculator #2

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Translating & Solving Consecutive Integer

Problems


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Another type of word problem that you might encounter involve consecutive integers.

Consecutive integers are integers that are in a row and have a difference of 1.

For example

1, 2, 3 -6, -5, -4 99, 100, 101

If we need to solve a consecutive integer word problem, it is necessary to assign a variable to each unknown and then set them equal to the given value.

Let's see an example...

Consecutive Integers

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The sum of two consecutive integers is 57. Find the two integers.

Begin by assigning the 1st integer as x and the 2nd integer as x + 1.

Then add both integer representations and set the expression equal to the number given.

x + x + 1 = 57 solve the equation2x + 1 = 572x = 56x = 28 once you solve for x, then substitute the value in to find the value of the next integer.x + 1 = 29The integers are 28 & 29

Consecutive Integer Problems

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Try the examples below.

The sum of two consecutive integers is -33. Find the integers.

1st integer = x2nd integer = x+1x + x + 1 = -332x + 1 = -332x = -34x = -17x + 1 = -16

The sum of two consecutive integers is 205. Find the integers.

1st integer = x2nd integer = x+1x + x + 1 = 2052x + 1 = 2052x = 204x = 102x + 1 = 103

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1st integer = x2nd integer = x+2x + x + 2 = 1142x + 2 = 1142x = 112x = 56x + 2 = 58

1st integer = x2nd integer = x+13rd integer = x+2x+x+1+x+2+933x + 3 = 933x = 90x = 30x + 1 = 31x + 2 = 32

The sum of three consecutive integers is 93. Find the integers.

The sum of two consecutive even integers is 114. Find the integers.

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99 The sum of two consecutive integers is 35. Find the larger of the two.

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100 The sum of two consecutive integers is 201. Find the smaller of the two integers.

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101 The sum of three consecutive integers is 615. Find the middle one.

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102 The sum of two consecutive odd integers is 624. Find the smaller integer.

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Glossary & Standards

Return toTable ofContents

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Back to

Instruction

Consecutive IntegersIntegers that are in a row and have a

difference of 1

1, 2, 3 -9, -8, -7 x, x + 1, x + 2

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Back to

Instruction

Distributive PropertyFor all real numbers a, b, c,

a(b + c) = ab + ac and a(b - c) = ab - ac.

3(2 + 4) = (3)(2) + (3)(4) =

6 + 12 = 183(2 - 4) =

(3)(2) - (3)(4) =6 - 12 = -6

3(x + 4) = 48 (3)(x) + (3)(4) = 48

3x + 12 = 48 3x = 36 x = 12

a(b + c) = ab + ac

a(b - c) = ab - ac

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Back to

Instruction

EquationA mathematical statement, in symbols, where two things are exactly the same

(or equivalent).

4 + 9 = 13

1 + 2 = 322 = 20 + 2 3y + 2 = 14

14 - 1 = 3z + 1

7x = 28(where x = 4)

3x + 6 = 11

7 + 4 = 905 = -3 + 10

(where x = 4)(where z = 4)

(where y = 4)

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Instruction

ExpressionNumbers, symbols and operators

(such as + and ×) grouped together that show the value of something.

7x 3y + 22 - 9b

-0.5a7 x 6

7x = 2111 = 3y + 2

11 - 1 = 3z + 1

Remember!

7x "7 times x"

"7 divided by x"

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Back to

Instruction

Identity An equation that has infinitely many solutions.

3(x - 1) = 3x - 3 3x - 3 = 3x - 3

-3x -3x -3 = -3

7(2x + 1) = 14x + 7 14x + 7 = 14x + 7

-14x -14x 7 = 7

3x - 1 = 3x + 1 -3x -3x

-1 = +1

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Back to

Instruction

Inverse OperationThe operation that reverses the effect

of another operation.

Addition

Subtraction

Multiplication

Division

+ _

x ÷

11 = 3y + 2- 2- 2

9 = 3y÷ 3÷ 3

3 = y

- 5 + x = 5

x = 10+ 5 + 5

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Instruction

Like Terms Terms whose variables (and their

exponents) are the same.

3x

5x15.7x

x 1/2x

-2.3x

27x3

-2x3

x3

1/4x3

-5x3

2.7x3

5x3

5x

5x25

5x4

NOT LIKE

TERMS!

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Back to

Instruction

No SolutionAn equation that is false.

3x - 1 = 3x + 1 -3x -3x

-1 = +1

8x - 4 = 8x + 6 -8x -8x

-4 = 6

3(x - 1) = 3x - 3 3x - 3 = 3x - 3

-3x -3x -3 = -3

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Back to

Instruction

Two-Step EquationAn equation that contains two operations

3x - 1 = 11 + 8 = 6 3(x - 1) + x = 25 x 3

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Instruction

Variable

A symbol for a number that is unknown.

7x = 2111 = 3y + 2

11 - 1 = 3z + 1x y z

u v

any letter towards end of alphabet!

*Sometimes Greek!

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Standards for Mathematical Practices

Click on each standard to bring you to an example of how to

meet this standard within the unit.

MP8 Look for and express regularity in repeated reasoning.

MP1 Make sense of problems and persevere in solving them.

MP2 Reason abstractly and quantitatively.

MP3 Construct viable arguments and critique the reasoning of others.

MP4 Model with mathematics.

MP5 Use appropriate tools strategically.

MP6 Attend to precision.

MP7 Look for and make use of structure.

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http://www.corestandards.org/Math/Practice/#CCSS.Math.Practice.MP1

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