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Table of Contents. Inverse Operations. Click on a topic to go to that section. One Step Equations. Two Step Equations. Multi-Step Equations. Variables on Both Sides. More Equations. Transforming Formulas. Inverse Operations. Return to Table of Contents. What is an equation?

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Table of Contents

Inverse Operations

Click on a topic to go to that section.

One Step Equations

Two Step Equations

Multi-Step Equations

Variables on Both Sides

More Equations

Transforming Formulas


Inverse Operations

Return to Table

of Contents


What is an equation?

An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in

2 + 3 = 5

9 – 2 = 7


Equations can also be used to state the equality of two

expressions containing one or more variables.

In real numbers we can say, for example, that for any given value of x it is true that

4x + 1 = 14 - 1

If x = 3, then

4(3) + 1 = 14 - 1

12 + 1 = 13

13 = 13


When defining your variables, remember...

Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand.

While letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes.

Try It!

Write an equation with a variable and have a classmate identify the variable and its value.


An equation can be compared to a balanced scale.

Both sides need to contain the same quantity in order for it to be "balanced".


For example, 20 + 30 = 50 represents an equation because both sides simplify to 50.

20 + 30 = 50

50 = 50

Any of the numerical values in the equation can be represented by a variable.

Examples:

20 + c = 50

x + 30 = 50

20 + 30 = y


Why are we Solving Equations?

First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to.

Now, we are told what it simplifies to and we need to find the value of the variable.

When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).


In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation.

Let's review the inverses of each operation:

Addition Subtraction

Multiplication Division


There are four properties of equality that we will use to solve equations. They are as follows:

Addition Property

If a=b, then a + c=b + c for all real numbers a, b, and c. The same number can be added to each side of the equation without changing the solution of the equation.

Subtraction Property

If a=b, then a-c=b-c for all real numbers a, b, and c. The same number can be subtracted from each side of the equation without changing the solution of the equation.

Multiplication Property

If a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation.

Division Property

If a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.


To solve for "x" in the following equation...

x + 7 = 32

Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides.

x + 7 = 32

- 7-7

x = 25

In the original equation, replace x with 25 and see if it makes the equation true.

x + 7 = 32

25 + 7 = 32

32 = 32


For each equation, write the inverse operation needed to solve for the variable.

a.) y +7 = 14 subtract 7b.) a - 21 = 10 add 21

c.) 5s = 25 divide by 5d.) x = 5 multiply by 12

12

move

move

move

move


Think about this...

To solve c - 3 = 12

Which method is better? Why?

Kendra

Added 3 to each side of the equation

c - 3 = 12

+3 +3

c = 15

Ted

Subtracted 12 from each side, then added 15.

c - 3 = 12

-12 -12

c - 15 = 0

+15 +15

c = 15


Think about this...

In the expression

To which does the "-" belong?

Does it belong to the x? The 5? Both?

The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable.

So:

×


What is the inverse operation needed to solve this equation?

7x = 49

1

Addition

A

Subtraction

B

Multiplication

C

D

Division


2

What is the inverse operation needed to solve this equation?

x - 3 = -12

Addition

A

Subtraction

B

Multiplication

C

D

Division


One Step Equations

Return to Table

of Contents


To solve equations, you must work backwards through the order of operations to find the value of the variable.

Remember to use inverse operations in order to isolate the variable on one side of the equation.

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


Examples:

y + 9 = 16

- 9 -9The inverse of adding 9 is subtracting 9

y = 7

6m = 72

6 6The inverse of multiplying by 6 is dividing by 6

m = 12

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

×


One Step Equations

Solve each equation then click the box to see work & solution.

click to show

inverse operation

click to show

inverse operation

x - 8 = -2

+8 +8

x = 6

2 = x - 6

+6 +6

8 = x

click to show

inverse operation

click to show

inverse operation

x + 2 = -14

-2 -2

x = -16

7 = x + 3

-3 -3

4 = x

click to show

inverse operation

click to show

inverse operation

15 = x + 17

-17 -17

-2 = x

x + 5 = 3

-5 -5

x = -2


One Step Equations

3x = 15

3 3

x = 5

-4x = -12

-4 -4

x = 3

-25 = 5x

5 5

-5 = x

click to show

inverse operation

click to show

inverse operation

x

(2)

= 10

(2)

2

x = 20

click to show

inverse operation

x

(-6)

click to show

inverse operation

= 36

(-6)

-6

x = -216

click to show

inverse operation


Solve.

x - 6 = -11

3


4

Solve.

j + 15 = -17


5

Solve.

-115 = -5x


6

Solve.

= 12

x

9


7

Solve.

51 = 17y


8

Solve.

w - 17 = 37


9

Solve.

-3 =

x

7


10

Solve.

23 + t = 11


11

Solve.

108 = 12r


Two-Step Equations

Return to Table

of Contents


Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.

This means that you undo in the opposite order (PEMDAS):

1st: Addition & Subtraction

2nd: Multiplication & Division

3rd: Exponents

4th: Parentheses

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


Examples:

3x + 4 = 10

- 4 - 4Undo addition first

3x = 6

3 3 Undo multiplication second

x = 2

-4y - 11 = -23

+ 11 +11 Undo subtraction first

-4y = -12

-4___-4 Undo multiplication second

y = 3

Remember - whatever you do to one side

of an equation, you MUST do to the other!!!

×


Two Step Equations

Solve each equation then click the box to see work & solution.

3x + 10 = 46

- 10 -10

3x = 36

3 3

x = 12

-4x - 3 = 25

+3 +3

-4x = 28

-4 -4

x = -7

6-7x = 83

-6 -6

-7x = 77

-7 -7

x = -11

-2x + 3 = -1

- 3 -3

-2x = -4

-2 -2

x = 2

9 + 2x = 23

-9 -9

2x = 14

2 2

x = 7

8 - 2x = -8

-8 -8

-2x = -16

-2 -2

x = 8


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?


12

Solve the equation.

5x - 6 = -56


13

Solve the equation.

16 = 3m - 8


Solve the equation.

x

2

14

- 6 = 30


15

Solve the equation.

5r - 2 = -12


16

Solve the equation.

12 = -2n - 4


17

Solve the equation.

- 7 = 13

x

4


18

Solve the equation.

+ 3 = -12

x

5

-


19

What is the value of n in the equation

0.6(n + 10) = 3.6?

-0.4

A

5

B

-4

C

D

4


20

In the equation

, n is equal to

8

A

2

B

1/2

C

D

1/8


Which value of x is the solution of the equation

21

1/2

A

2

B

2/3

C

D

3/2


22

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?


Multi-Step Equations

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of Contents


Steps for Solving Multiple Step Equations

As equations become more complex, you should:

1. Simplify each side of the equation.

(Combining like terms and the distributive property)

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!


Examples:

-15 = -2x - 9 + 4x

-15 = 2x - 9 Combine Like Terms

+9 +9 Undo Subtraction first

-6 = 2x

2 2 Undo Multiplication second

-3 = x

7x - 3x - 8 = 24

4x - 8 = 24Combine Like Terms

+ 8 +8 Undo Subtraction first

4x = 32

4___4Undo Multiplication second

x = 8

×


Now try an example. Each term is infinitely cloned

so you can pull them down as you solve.

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

-7x

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 3

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

+ 6x

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

-6

x = 9


Now try another example. Each term is infinitely cloned

so you can pull them down as you solve.

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

6x

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

- 5

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

+ x

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

44

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

x = -9


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

Sometimes, you need to use the distributive property in order to simplify part of the equation.


Distributive Property

For all real numbers a, b, c

a(b + c) = ab + ac

a(b - c) = ab - ac


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)


Examples:

5(1 + 6x) = 185

5 + 30x = 185Distribute the 5 on the left side

-5 -5 Undo addition first

30x = 180

30 30 Undo multiplication second

x = 6

2x + 6(x - 3) = 14

2x + 6x - 18 = 14Distribute the 6 through (x - 3)

8x - 18 = 14Combine Like Terms

+18 +18 Undo subtraction

8x = 32

8 8 Undo multiplication

x = 4

×


Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down)

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

7x

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

95

95

95

95

95

95

95

95

95

95

95

95

95

95

95

95

95

95

95

95

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

x = 3

answer


Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

)

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

-2x

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102

6

9

102

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

x = -4

answer


23

Solve.

3 + 2t + 4t = -63


24

Solve.

19 = 1 + 4 - x


25

Solve.

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


26

Solve.

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


27

Solve.

9 - 4y + 16 + 11y = 4


28

Solve.

6(-8 + 3b) = 78


29

Solve.

18 = -6(1 - 1k)


30

Solve.

2w + 8(w + 3) = 34


31

Solve.

4 = 4x - 2(x + 6)


32

Solve.

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


33

What is the value of p in the equation

2(3p - 4) = 10?

1

A

2 1/3

B

3

C

D

1/3


Variables on Both Sides

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of Contents


Remember...

1. Simplify both sides of the equation.

2. Collect the variable terms on one side of the equation. (Add or subtract one of the terms from both sides of the equation)

3. Solve the equation.

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


Example:

4x + 8 = 2x + 26

-2x -2x Subtract 2x from both sides

2x + 8 = 26

- 8 -8Undo Addition

2x = 18

2 2Undo Multiplication

x = 9

What if you did it a little differently?

4x + 8 = 2x + 26

-4x -4x Subtract 4x from both sides

8 = -2x + 26

-26 - 26Undo Addition

-18 = -2x

-2 -2Undo Multiplication

9 = x

Recommendation: Cancel the smaller amount of the variable!

×


Example:

6r - 5 = 7r + 7 - 2r

6r - 5 = 5r + 7 Simplify Each Side of Equation

-5r -5r Subtract 5r from both sides (smaller than 6r)

r - 5 = 7

+ 5 +5Undo Subtraction

r = 12

×


Try these:

6x - 2 = x + 13 4(x + 1) = 2x -2 5t - 8 = 9t - 10

-x -x 4x + 4 = 2x -2 -5t -5t

5x - 2 = 13 -2x -2x -8 = 4t - 10

+ 2 +2 2x + 4 = -2+10 +10

5x = 15 -4 -4 2 = 4t

5 5 2x = -6 4 4

x = 32 2 = t

x = -3

1

2


Sometimes, you get an interesting answer.

What do you think about this?

What is the value of x?

3x - 1 = 3x + 1

Since the equation is false, there is "no solution"!

No value will make this equation true.

move this


Sometimes, you get an interesting answer.

What do you think about this?

What is the value of x?

3(x - 1) = 3x - 3

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

Any value will make this equation true.

move this


Try these:

4y = 2(y + 1) + 3(y - 1)14 - (2x + 5) = -2x + 99m - 8 = 9m + 4

4y = 2y + 2 + 3y - 3 14 - 2x - 5 = -2x + 9- 9m - 9m

4y = 5y - 1 9 - 2x = -2x + 9 -8 = 4

-5y -5y +2x +2x No Solution

-y = -1 9 = 9

y = 1 Identity


Mary and Jocelyn left school at 3:00 p.m. and bicycled home along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?

Define

t = Mary's time in hours

t + 0.25 = Jocelyn's time in hours

Jocelyn's distance

(rate time)

Mary's distance

(rate time)

equals

Relate

12t = 9(t+0.25)

Write


12t = 9(t + 0.25)

12t = 9t + 2.25

-9t -9t

3t = 2.25

3 3

t = 0.75

It took Mary 0.75h, or 45 min, to get home.

×


34

Solve.

7f + 7 = 3f + 39


35

Solve.

h - 4 = -5h + 26


36

Solve.

w - 2 + 3w = 6 + 5w


37

Solve.

5(x - 5) = 5x + 19


38

Solve.

-4m + 8 - 2(m + 3) = 4m - 8


39

Solve.

28 - 7r = 7(4 - r)


40

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.

A

B

M

4x + 4

3x

N

17

D

5x – 3

C

O


More Equations

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of Contents


Remember...

1. Simplify each side of the equation.

Collect the variable terms on one side of the equation.

(Add or subtract one of the terms from both sides of the equation)

3. Solve the equation.

(Undo addition and subtraction first, multiplication and division second)

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


Examples:

x = 6

x = 6Multiply both sides by the reciprocal of

x =

x = 10

2x - 3 = + x

-x - xSubtract x from both sides

x - 3 =

+3 +3Undo Subtraction

x =

3

5

5

3

5

3

3

5

5

3

×

30

3

-14

5

-14

5

1

5


There is more than one way to solve an equation with distribution.

Multiply by the reciprocal Multiply by the LCM

72

5

3

5

(-3 + 3x) =

72

5

3

5

3

5

72

5

(-3 + 3x) =

(-3 + 3x) =

5

3

3

5

72

5

5

3

3

5

72

5

5

(-3 + 3x) =

5

(-3 + 3x) =

3(-3 + 3x) = 72

-9 + 9x = 72

+9 +9

9x = 81

9 9

x = 9

-3 + 3x = 24

+3 +3

3x = 27

3 3

x = 9


41

Solve

1

10

-3

5

1

2

x + =


42

Solve

1

5

-68

35

- 2b + 5b =


43

Solve

2

3

x + 8 = 7 + x


44

Solve

2

3

16

3

(8 - 3c) =


45

Solve

-6(7 - 3y) + 4y = 10(2y - 4)


46

Solve

1

2

9

4

1

8

(6 - 2z) = - z - (-4z + 6)


47

Solve Round to the nearest hundredth

9.47x = 7.45x - 8.81


48

Solve Round to the nearest hundredth

13.19 - 8.54x = 7.94x - 1.82


49

Solve

-3(8 - 2m) + 8m = 4(4 + m)


50

Solve

1

2

(2y - 4) = 3(y + 2) - 3y


Transforming Formulas

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Formulas show relationships between two or more variables.

You can transform a formula to describe one quantity in terms of the others by following the same steps as solving an equation.


Example:

Transform the formula d = r t to find a formula

for time in terms of distance and rate.

What does "time in terms of distance and rate" mean?

×

d = r t

r r

= t

Divide both sides by r

d

r


Examples

V = l whSolve for w

V = w

l h

P = 2l + 2wSolve for l

-2w -2w

P - 2w = 2l

2 2

P - 2w = l

2

×


Example:

To convert Fahrenheit temperature to Celsius, you use the formula:

C = (F - 32)

Transform this formula to find Fahrenheit temperature in terms of Celsius temperature. (see next page)

5

9


Solve the formula for F

5

9

C = (F - 32)

C = F -

+ +

C + = F

C + 32 = F

5

9

160

9

160

9

160

9

9

5

(

)

5

9

9

5

160

9

9

5


Transform the formula for area of a circle to find radius when given Area.

A = r2

= r2

A

A

= r


Solve the equation for the given variable.

m p

n q

m p

n q

mq p

n

= for p

= (q)

=

2(t + r) = 5 for t

2(t + r) = 5

2 2

t + r =

- r - r

t = - r

(q)

5

2

5

2


51

The formula I = prtgives the amount of simple interest, I, earned by the principal, p, at an annual interest rate, r, over t years.

Solve this formula for p.

Irt

p =

A

Ir

t

p =

B

I

rt

p =

C

It

r

D

p =


Gm

r

v2 =

A satellite's speed as it orbits the Earth is found using the formula . In this formula, m stands for the mass of the Earth. Transform this formula to find the mass of the Earth.

52

v2

G

- r

v2 – r

G

A

m =

m =

B

rv2 - G

m =

C

rv2

G

m =

D


53

Solve for t in terms of s

4(t - s) = 7

7

4

t = + s

A

t = 28 + s

B

7

4

t = - s

C

7 + s

4

D

t =


54

Solve for w

A = lw

w = Al

A

A

l

w =

B

l

A

w =

C


55

Solve for h

V= r2h

V r2

h = V - r2

A

V

r2

h =

B

C

h =

V

r

h =

D


56

Which equation is equivalent to 3x + 4y = 15?

A

y = 15 − 3x

y = 3x − 15

B

y = 15 – 3x

4

C

y = 3x – 15

4

D


57

If, b ≠ 0, then x is equal to

- a

4b

A

a

4b

B

- 4a

b

C

4a

b

D


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