Review One-Step Equations - Chandler Unified School District€¦ · 1 Review solving one-step equations with integers, fractions, and decimals. One-step equations Vocabulary equation

$Page 1: Review One-Step Equations - Chandler Unified School District€¦ · 1 Review solving one-step equations with integers, fractions, and decimals. One-step equations Vocabulary equation$
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Review solving one-step equations with integers, fractions, and decimals. One-step equations

Vocabularyequationsolvesolutioninverse operationisolate the variableAddition Property of EqualitySubtraction Property of Equality

An equation uses an equal sign to show that two expressions are equal. All of these are equations.

3 + 8 = 11 r + 6 = 14 24 = x – 7 1002

= 50

To solve an equation, find the value of the variable that makes the equation true. This value of the variable is called the solution of the equation.

Determine which value of x is a solution of the equation.

x + 8 = 15; x = 5, 7, or 23

Additional Example 1: Determining Whether a Number is a Solution of an Equation

Substitute each value for x in the equation.

Substitute 5 for x.13= 15?

So 5 is not solution.

x + 8 = 15?

5 + 8 = 15?

Determine which value of x is a solution of the equation.x + 8 = 15; x = 5, 7, or 23

Additional Example 1 Continued



So 7 is a solution.

x + 8 = 15?

7 + 8 = 15?

2

Determine which value of x is a solution of the equation.x + 8 = 15; x = 5, 7, or 23

Additional Example 1 Continued



So 23 is not a solution.

x + 8 = 15?

23 + 8 = 15?

Determine which value of x is a solution of the equation.x – 4 = 13; x = 9, 17, or 27

Try This: Example 1


Substitute 9 for x.5 = 13?


x – 4 = 13?

9 – 4 = 13?


Try This: Example 1 Continued



So 17 is a solution.

x – 4 = 13?

17 – 4 = 13?


Try This: Example 1 Continued




x – 4 = 13?

27 – 4 = 13?

Addition and subtraction are inverseoperations, which means they “undo” each other.

To solve an equation, use inverse operations to isolate the variable. This means getting the variable alone on one side of the equal sign.

To solve a subtraction equation, like y 15 = 7, you would use the Addition Property of Equality.

You can add the same number to both sides of an equation, and the statement will still be true.

2 + 3 = 5+ 4 + 4

2 + 7 = 9

x = yx = y+ z + z

ADDITION PROPERTY OF EQUALITY

Words Numbers Algebra

3

There is a similar property for solving addition equations, like x + 9 = 11. It is called the Subtraction Property of Equality.

You can subtract the same number from both sides of an equation, and the statement will still be true.

4 + 7 = 11 3 3

4 + 4 = 8

x = yx = y z z

SUBTRACTION PROPERTY OF EQUALITY


Solve.

Additional Example 2A: Solving Equations Using Addition and Subtraction Properties

Subtract 10 from both sides.

A. 10 + n = 1810 + n = 18

–10 –100 + n = 8

n = 8 Identity Property of Zero: 0 + n = n.Check

10 + n = 18?10 + 8 = 18

18 = 18?

Solve.

Additional Example 2B: Solving Equations Using Addition and Subtraction Properties

Add 8 to both sides.

B. p – 8 = 9p – 8 = 9

+ 8 + 8

p + 0 = 17p = 17 Identity Property of Zero: p + 0 = p.

Checkp – 8 = 9

?17 – 8 = 9

9 = 9?

Solve.

Additional Example 2C: Solving Equations Using Addition and Subtraction Properties


C. 22 = y – 1122 = y – 11

+ 11 + 11

33 = y + 033 = y Identity Property of Zero: y + 0 = 0.

Check22 = y – 11

?22 = 33 – 11

22 = 22?

Solve.

Try This: Example 2A

Subtract 15 from both sides.

A. 15 + n = 2915 + n = 29

–15 –15

0 + n = 14n = 14 Identity Property of Zero: 0 + n = n.

Check

15 + n = 29?10 + 14 = 29

29 = 29?

Solve.

Try This: Example 2B


B. p – 6 = 7p – 6 = 7

+ 6 + 6

p + 0 = 13p = 13 Identity Property of Zero: p + 0 = p.

Checkp – 6 = 7

?13 – 6 = 7

7 = 7?

4

Solve.

Try This: Example 2C


C. 44 = y – 2344 = y – 23

+ 23 + 23

67 = y + 067 = y Identity Property of Zero: y + 0 = 0.

Check44 = y – 23

?44 = 67 – 23

44 = 44?

Learn to solve equations using

multiplication and division.

Vocabulary

Division Property of EqualityMultiplication Property of Equality

You can solve a multiplication equation using the Division Property of Equality.

You can divide both sides of an equation by the same nonzero number, and the equation will still be true.

4 • 3 = 122 2

x = yx = y

DIVISION PROPERTY OF EQUALITY


z

4 • 3 = 12

12 = 62

z

Solve 8x = 32.

Additional Example 1: Solving Equations Using Division

8x = 32

1x = 4

Divide both sides by 8.

Check

8x = 328 8

x = 4

8x = 328(4) = 32?

32 = 32?Substitute 4 for x.

1 • x = x

Solve 9x = 36.

Try This: Example 1

9x = 36

1x = 4

Divide both sides by 9.

Check

9x = 369 9

x = 4

9x = 369(4) = 36?

36 = 36?Substitute 4 for x.

1 • x = x

5

You can multiply both sides of an equation by the same number, and the statement will still be true.

2 • 3 = 6x = yx = yz z

MULTIPLICATION PROPERTY OF EQUALITY


2 • 3 = 64 • 4 •

8 • 3 = 24

You can solve a division equation using the Multiplication Property of Equality.

Solve = 7.

Additional Example 2: Solving Equations Using Multiplication

n7n7 = 77 • 7 • Multiply both sides by 7.

n = 49

Checkn7 = 7

Substitute 49 for n.497 = 7?

7 = 7?

Solve = 16

Try This: Example 2

n4n4 = 164 • 4 • Multiply both sides by 4.

n = 64Check

n4 = 16

Substitute 64 for n.644 = 16?

16 = 16?

Learn to solve equations with integers.

When you are solving equations with integers, your goal is the same as with whole numbers:isolate the variable on one side of the equation.

0

+ –

–

–

+

+

3 + (–3) = 0

a + (–a) = 0

Recall that the sum of a number and its opposite is 0. When you add the opposite to get 0, you can isolate the variable.

x = – 3

x – 3 = – 6

Add 3 to both sides.x – 3 + 3 = – 6 + 3

–5 + r = 9

r = 14Add 5 to both sides.–5 + 5 + r = 9 + 5

x – 3 + 3 = – 3Commutative Property

0

x – 3 = – 6

–5 + r = 9

Additional Example 1A & 1B: Adding and Subtracting to Solve Equations

Solve.

A.

B.

6

–6 + 8 = n

2 = n

The variable is already isolated.

Add integers.

–6 + 8 = n

z + 6 = –3

z = –9

Add –6 to each side.z + 6 = –3–6 –6

Solve.

C.

D.

Additional Example 1C & 1D: Adding and Subtracting to Solve Equations Continued

p = –2

p – 7 = – 9

Add 7 to both sides.p – 7 + 7 = – 9 + 7

–2 + g = 5

g = 7Add 2 to both sides.–2 + 2 + g = 5 + 2

p – 7 = – 9

–2 + g = 5

Solve.

Try This: Example 1A & 1B

p – 7 + 7 = – 2Commutative Property

0

A.

B.

–1 + 7 = r

6 = r

The variable is already isolated.

Add integers.

–1 + 7 = r

a + 9 = –9

a = –18

Add –9 to each side.

Try This: Example 1C & 1D

a + 9 = –9–9 –9

Solve.

C.

D.x = –7

–5x = 35

Divide both sides by –5.–5x = 35–5 –5

Additional Example 2A: Multiplying and Dividing to Solve Equations Continued

Solve.

A.

Multiply both sides by –4.

z = –20

= 5z –4

–4 = –4 5z –4

Additional Example 2B: Multiplying and Dividing to Solve Equations Continued

Solve.

B.

x = –6

–7x = 42

Divide both sides by –7.


–7x = 42–7 –7

Solve.

A.

7

Multiply both sides by –3.

z = –27

= 9z –3

–3 = –3 9z –3

Solve.

B.

Try This: Example 2

One-Step Equations with Rational Numbers (fractions and decimals)

m + 4.6 = 9

m + 4.6 = 9

Subtract 4.6 from both sides.

Once you have solved and equation it is a good idea to check your answer. To check your answer, substitute your answer for the variable in the original equation.

Remember!

Additional Example 1A: Solving Equations with Decimals.

Solve.

4.4 m =– 4.6= – 4.6

A. 8.2p = –32.8

–4p =

Divide both sides by 8.2–32.8 8.2

8.2p8.2

=

Additional Examples 1B: Solving Equations with Decimals

Solve.B.

= 15

Additional Examples 1C: Solving Equations with Decimals

Solve.

x = 18

Multiply both sides by 1.2

x 1.2x

1.2= 1.2 • 15 1.2 •

C.m + 9.1 = 3m + 9.1 = 3

Subtract 9.1 from both sides.

Try This: Example 1A & 1B

–6.1 m =

–9.1 = –9.1

Solve.A.

B. 5.5b = 75.975.9 5.5

5.55.5 =b

13.8b =

Divide both sides by 5.5

8

= 90

Try This: Examples 1C

y = 405

Multiply both sides by 4.5

y 4.5

y 4.5

= 4.5 • 904.5 •

Solve.

C.

2 7

= –3 7

n +

Additional Examples 2A: Solving Equations with Fractions

n – + = – –2 7

3 7

27

27

n = – 5 7

Subtract from both sides.2 7

Solve.

A.

1 6 = 2

3y –

Additional Examples 2B: Solving Equations with Fractions

Find a common denominator; 6.

y = 5 6

=y 4 6

1 6

+

1 6

= 2 3

–1 6

+ 1 6

+y Add to both sides.1 6

Solve.

B.

Simplify.

5 6 =

5 8x

6 5••

6 5

5 6

= 5 8

x

Simplify.

Additional Examples 2C: Solving Equations with Fractions

Multiply both sides by .6 5

x =3 4

3

4

Solve.

C.

1 9

= –5 9

n +


n – + = – –1 9

5 9

19

19

n = – 2 3

Subtract from both sides.1 9

Simplify – .6 9

Solve.A.

1 2 = 3

4y –

Try This: Example 2B

Find a common denominator; 4.

y = 1 1 4

=y 3 4

2 4

+

1 2

= 3 4

–1 2

+ 1 2

+y Add to both sides.1 2

Solve.B.

Simplify.

9

3 8

= 6 19

x 8 3••8

3

3 8

= 6 19

x

Simplify.

Try This: Examples 2C

Multiply both sides by .8 3

x =16 19

2

1

Solve.

C.

3 8

= 6 19

x

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Review One-Step Equations - Chandler Unified School District€¦ · 1 Review solving one-step equations with integers, fractions, and decimals. One-step equations Vocabulary equation