1 Lesson 1.2.3 Solving One-Step Equations Solving One-Step Equations

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Lesson 1.2.3Lesson 1.2.3

Solving One-StepEquations

Solving One-StepEquations

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Lesson

1.2.3Solving One-Step EquationsSolving One-Step Equations

California Standard:Algebra and Functions 4.1

What it means for you:

Key Words:

Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.

You’ll learn how to solve an equation to find out the value of an unknown variable.

• solve• isolate• inverse

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Lesson


Solving an equation containing a variable means finding the value of the variable.

5 + x = 12 x = 7

It’s all about changing the equation around to get the variable on its own.

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Do the Same to Both Sides and Equations Stay True

Lesson


The equals sign in an equation tells you that the two sides of the equation are of exactly equal value.

So if you do the same thing to both sides of the equation, like add five or take away three, they will still have the same value as each other.

All three are balanced equations.

4 + 6 = 9 + 1

Add 5 to both sides.

Then simplify.

Original, balanced equation.

4 + 6 + 5 = 9 + 1 + 5

15 = 15

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• If a variable has been multiplied by a number, divide both sides by the same number. × ÷

• If a variable has been divided by a number, multiply both sides by the same number. ÷ ×

You Can Use This to Find the Value of a Variable

Lesson


To get a variable in an equation on its own you need to do the inverse operation to the operation that has already been performed on it.

• If a variable has had a number added to it, subtract the same number from both sides. + –

• If a variable has had a number subtracted from it, add the same number to both sides. – +

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Lesson


For example, solve the equation y – 5 = 33.

y has had 5 subtracted from it.

You’ve got the variable alone on one side of the equation,

so now you know its value.

y – 5 = 33

y – 5 + 5 = 33 + 5

y + 0 = 38

y = 38

Do the additions to simplify both sides.

So add 5 to both sides.

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Reverse Addition by Subtracting

Lesson


When a variable has had something added to it, you can undo the addition using subtraction.

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Lesson


Example 1

Find the value of x when x + 15 = 45.

Solution

x + 15 = 45

x + 15 – 15 = 45 – 15 Subtract 15 from both sides

Simplify to find x

Write out the equation

x = 30

Solution follows…

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Reverse Subtraction by Adding

Lesson


When a variable has had something taken away from it, you can undo the subtraction using addition.

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Lesson


Example 2

Find the value of k when k – 17 = 10.

Solution

k – 17 = 10

k – 17 + 17 = 10 + 17 Add 17 to both sides

Simplify to find k


k = 27

Solution follows…

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Example 3

Solution follows…

Lesson


Find the value of g when –10 = g – 9.

Solution

–10 = g – 9

–10 + 9 = g – 9 + 9 Add 9 to both sides

Simplify to find g


–1 = g

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Guided Practice

Solution follows…

Lesson


Find the value of the variable in Exercises 1–8.

1. x – 7 = 14

3. f + 13 = 9

5. y – 14 = 30

7. 4.5 = 9 + v

2. 70 = t + 41

4. g – 3 = –54

6. 22 = 14 + d

8. –6 = b – 4

x = 21

f = –4

y = 44

v = –4.5

t = 29

g = –51

d = 8

b = –2

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Reverse Multiplication by Dividing

Lesson


When a variable in an equation has been multiplied by a number, you can undo the multiplication by dividing both sides of the equation by the same number.

y has been multiplied by 2. 2y = 18

2y ÷ 2 = 18 ÷ 2

y = 9Do the divisions to simplify both sides.

So divide both sides by 2.

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Example 4

Solution follows…

Lesson


Find the value of b when 20b = 100.

Solution

20b = 100

20b ÷ 20 = 100 ÷ 20 Divide both sides by 20

Simplify to find b


b = 5

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Reverse Division by Multiplying

Lesson


When a variable in an equation has been divided by a number, you can undo the division by multiplying both sides of the equation by the same number.

d has been divided by 2. = 50

• 2 = 50 • 2

d = 100Do the multiplications to simplify both sides.

So multiply both sides by 2.

d

2

d

2

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Example 5

Solution follows…

Lesson


Find the value of t when t ÷ 4 = 6.

Solution

t ÷ 4 = 6

t ÷ 4 • 4 = 6 • 4 Multiply both sides by 4

Simplify to find t


t = 24

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Guided Practice

Solution follows…

Lesson


Find the value of the variables in Exercises 9–16.

9. 3k = 18

11. h ÷ 5 = –3

13. q ÷ 8 =

15. d ÷ –2 = –4

10. b ÷ 3 = 4

12. –9y = 99

14. 10t = –55

16. 240 = 8m

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k = 6

h = –15

q = 4

d = 8

b = 12

y = –11

t = –5.5

m = 30

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Independent Practice

Solution follows…

Lesson


1. The Sears Tower in Chicago is 1451 feet tall, which is 405 feet taller than the Chrysler Building in New York. Use the equation C + 405 = 1451 to find the height of the Chrysler Building.


2. k + 7 = 10

4. s + 4 = –7

6. h + 0 = 14

3. c + 10 = –27

5. 70 = 5 + b

7. 32 = 11 + a

1046 feet

k = 3

s = –11

h = 14

c = –37

b = 65

a = 21

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Solution follows…

Lesson


8. The Holland Tunnel in New York is 342 feet longer than the 8216-foot-long Lincoln Tunnel. Use the equation H – 342 = 8216 to find the length of the Holland Tunnel.


9. x – 7 = 13

11. p – 13 = –82

13. 100 = g – 18

10. 41 = m – 35

12. t – 27 = 37

14. –7 = y – 2

8558 feet

x = 20

p = –69

g = 118

m = 76

t = 64

y = –5

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Solution follows…

Lesson


15. Marlon buys a sweater for $28 that has $17 off its usual price in a sale. Write an equation to describe the cost of the sweater in the sale compared with its usual price. Then solve the equation to find the usual price of the sweater.


16. 5c = 80

18. 22x = –374

20. –3k = –24

17. v ÷ 7 = 3

19. h ÷ –2 = 4

21. –27 = f ÷ 3

x – 17 = 28, x = $45

c = 16

x = –17

k = 8

v = 21

h = –8

f = –81

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Solution follows…

Lesson


22. The tallest geyser in Yellowstone Park is the Steamboat Geyser. Reaching a height of 380 feet, it is twice as high as the Old Faithful Geyser. Use the equation 2F = 380 to find the height reached by the Old Faithful Geyser. 190 feet

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Round UpRound Up

Lesson


Solving an equation tells you the value of the unknown number — the variable.

To solve an equation all you need to do is the reverse of what’s already been done to the variable. That way you can isolate the variable.

Just remember that you need to do the same thing to both sides. That’s what keeps the equation balanced.

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1 Lesson 1.2.3 Solving One-Step Equations Solving One-Step Equations