2-2 Solving Two-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California...

2-2 Solving Two-Step Equations

Warm UpWarm Up

Lesson Presentation

California Standards

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Warm UpSolve each equation. 1. 3 + x = 11 2. x – 7 = 19

3. 6x = 15 4.

Find a common denominator.

5.

6.

8 26

12

Possible answer: 6

Possible answer: 20


Preparation for 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

California Standards


equivalent equations

Vocabulary


This equation contains multiplication and addition. Equations that contain two operations require two steps to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse operations to undo them in reverse over one at a time.

Many equations contain more than one operation, such as 2x + 5 = 11.

Operations in the equation

First x is multipliedmultiplied by 2.

Then 5 is addedadded..

To solve

SubtractSubtract 5 from both sides of the equation.Then dividedivide both both sides by 2. sides by 2.


2x + 5 =11–5 –52x = 6

x = 3

Subtract 5 from both sides of the equation.

Divide both sides of the equation by 2.

The solution set is {3}.

Each time you perform an inverse operation, you create an equation that is equivalent to the original equation. Equivalent equations have the same solutions, or the same solution set. In the example above, 2x + 5 = 11, 2x = 6, and x = 3 are all equivalent equations.


Additional Example 1A: Solving Two-Step Equations

Solve 18 = 4a + 10.

18 = 4a + 10 First a is multiplied by 4. Then 10 is added.

–10 –10

8 = 4a

2 = a

Subtract 10 from both sides.

8 = 4a is equivalent to 18 = 4a + 10.

Since a is multiplied by 4, divide both sides by 4 to undo the multiplication.



Additional Example 1B: Solving Two-Step Equations

Solve 5t – 2 = –32.

5t – 2 = –32 First t is multiplied by 5. Then 2 is subtracted.

Add 2 to both sides.

5t = –30 is equivalent to 5t – 2 = –32.

Since t is multiplied by 5, divide both sides by 5 to undo the multiplication.

The solution set is {–6}.

5t = –30

t = –6

+2 +2


Solve the equation. Check your answer.

7x = 7

–4 + 7x = 3

Check It Out! Example 1a

Add 4 to both sides.

7x = 7 is equivalent to –4 + 7x = 3.

Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.


First x is multiplied by 7. Then –4 is added.

–4 + 7x = 3

+ 4 +4

x = 1


Check

–4 + 7(1) 3

3 3

To check your solution, substitute 1 for x in the original equation.

Check It Out! Example 1a Continued


–4 + 7x = 3

–4 + 7 3



7.2 = 1.2y

1.5 = 1.2y – 5.7

Check It Out! Example 1b

Add 5.7 to both sides.

7.2 = 1.2y is equivalent to 1.5 = 1.2y – 5.7.

Since y is multiplied by 1.2, divide both sides by 1.2 to undo the multiplication.


First y is multiplied by 1.2. Then 5.7 is subtracted.

+ 5.7 +5.7

6 = y

1.5 = 1.2y – 5.7


Check It Out! Example 1b Continued


Check

1.5 1.2(6) – 5.7

1.5 1.5

1.5 7.2 – 5.7

1.5 = 1.2y – 5.7 To check your solution, substitute 6 for y in the original equation.



Check It Out! Example 1c

n = 0

Subtract 2 from each side.

Since n is divided by 7, multiply both sides by 7 to undo the division.


First n is divided by 7. Then 2 is added.

–2 –2

= 0 = 0 is equivalent to + 2 = 2.



Check It Out! Example 1c Continued

Check

To check your solution, substitute 0 for n in the original equation.


Additional Example 2A: Solving Two-Step Equations That Contain Fractions

Solve the equation.

Method 1 Use fraction operations.

Since is subtracted from , add to

both sides to undo the subtraction.


Additional Example 2A Continued

y = 16

Since y is divided by 8 multiply both sides by 8.

Simplify.



Additional Example 2A Continued Method 2 Multiply by the least common denominator (LCD) to clear fractions.

y – 6 = 10

+6 +6

y = 16

Multiply both sides by 8, the LCD of the fractions.

Distribute 8 on the left side.

Simplify. Since 6 is subtracted from y, add 6 to both sides to undo the subtraction.



Additional Example 2A Continued

Check your answer.

To check your solution, substitute 16 for y in the original equation.

Check


Additional Example 2B: Solving Two-Step Equations That Contain Fractions

Solve the equation.


Since is added to , subtract

from both sides to undo the addition.


Additional Example 2B Continued

Since r is multiplied by multiply both

sides by , the reciprocal.

Simplify.

The solution set is .



Method 2 Multiply by the least common denominator (LCD) to clear the fractions.





8r + 9 = 7– 9 –9

8r =–2

Simplify. Since 9 is added 8r, subtract 9 from both sides to undo the addition.

Since r is multiplied by 8, divide both sides 8 to undo the multiplication.




Check your answer.

Check

To check your solution, substitute

for r in the original equation.


You can multiply both sides of the equation by any common denominator of the fractions. Using the LCD is the most efficient.

Helpful Hint





Since is subtracted from , add

to both sides to undo the subtraction.



Simplify.


Since x is multiplied by multiply both

sides by , the reciprocal.


Method 2 Multiply by the least common denominator (LCD) to clear the fractions.




4x – 5 = 50


+ 5 +5

4x = 55

Simplify. Since 5 is subtracted from 4x add 5 to both sides to undo the subtraction.

Simplify. Since x is multiplied by 4, divide both sides 4 to undo the multiplication.


4x – 5 = 50





Check

5 5


for x in the original equation.





Since is added to , subtract

from both sides to undo the addition.



Simplify.

Since u is multiplied by multiply both

sides by the reciprocal, .



Method 2 Multiply by the least common denominator (LCD) to clear fractions.




6u + 4 = 7


– 4 – 4

6u = 3

Simplify. Since 4 is added to 6u subtract 4 from both sides to undo the addition.

Simplify. Since u is multiplied by 6, divide both sides 6 to undo the multiplication.



6u + 4 = 7




Check


for u in the original equation.


Check It Out! Example 2c



Since is subtracted from , add

to both sides to undo the subtraction.



Simplify.

The solution set is {15}.n = 15

Since n is multiplied by multiply both

sides by the reciprocal, .



Check

To check your solution, substitute 15 for n in the original equation.


Additional Example 3: Problem-Solving Application

Jan joined the dining club at the local café for a fee of $29.95. Being a member entitles her to save $2.50 every time she buys lunch. Jan calculates that she has saved a total of $12.55 so far by joining the club. Write and solve an equation to find how many times Jan has eaten lunch at the café.

11 Understand the Problem

The answer will be the number of meals Jan had eaten.


List the important information:

• Jan paid a $29.95 dining club fee.

• Jan saved $2.50 on each meal.

• Her total savings was $12.55.

22 Make a Plan

Let m represent the number of meals that Jan had purchased. That means that Jan saved $2.50m. She must also subtract the cost the dining club fee from the amount she saved for the meals. Write an equation to represent this situation.

Additional Example 3 Continued


Savings on meals minus

initial

feeis total

savings

2.50(m) – 29.95 = 12.55

Solve33

2.50(m) – 29.95 = 12.55

+ 29.95 +29.95

2.50m = 42.50

m = 17 meals

Since 29.95 is subtracted from 2.50m, add 29.95 to both sides to undo the subtraction.

Since m is multiplied by 2.50, divide both sides by 2.50 to undo the multiplication.



Look Back44

Check that the answer is reasonable. The savings per meal is $2.50, so if Jan had 17 meals minus the initial cost of joining the dining club which is $29.95, it is equal to a savings of $12.55.




Sara paid $15.95 to become a member at a gym. She then paid a monthly membership fee. Her total cost for 12 months was $735.95. How much was the monthly fee?


The answer will be the monthly fee that Sara had paid during the year.



• Sara paid $15.95 for the membership.

• Sara paid a monthly fee for 12 months.

• Her total cost for 12 months was $735.95.

22 Make a Plan

Let m represent the monthly fee that Sara paid. Sara paid that fee for 12 months. She must also add the cost of the membership. Write an equation to represent this situation.



Monthly fee

is total cost.

Solve33

12m + 15.95 = 735.95

–15.95 –15.95

12m = 720.00

m = $60 month

Since 15.95 is added to 12m, subtract 15.95 from both sides to undo the addition.

Since m is multiplied by 12, divide both sides by 12 to undo the multiplication.

initial fee

plus

12m = 735.9515.95+



Look Back44

Check that the answer is reasonable. The cost per month is $60.00, so if Sara paid for 12 months plus the initial membership fee which is $15.95, it is equal to the total cost of $735.95.




Lynda has 12 records in her collection. She adds the same number of new records to her collection each month. After 7 months Lynda has 26 records. How many records does Lynda add each month?


The answer will be the number of records that Lynda adds each month.



• Lynda started with 12 records.

• Lynda adds records each month.

• At the end of 7 months she had 26 records.

22 Make a Plan

Let r represent the number of records added monthly. Lynda added new records for 7 months to her starting number of records 12. Write an equation to represent this situation.



Months is total records

Solve33

7r + 12 = 26–12 –12

7r = 14

r = 2 records a month

Since 12 is added to 7r, subtract 12 from both sides to undo the addition.

Since r is multiplied by 7, divide both sides by 7 to undo the multiplication.

initial records

plus

7r = 2612+



Look Back44

Check that the answer is reasonable. The number of records added per month is 2, so if Lynda started with 12 records and added the 2 per month for 7 months, the total number is equal to 26.


2-2 Solving Two-Step EquationsLesson Quiz

Solve each equation.

1. 4y + 8 = 2 2. 3 + 2x = 11

3. 4.

4

–8

5. Nancy bought 5 rolls of color film and 6 rolls of black-and-white film. The 5 rolls of color film cost $15, and Nancy’s total was $39. Write and solve an equation to find the cost of one roll of black-and-white film.

6b + 15 = 39; $4

2-2 Solving Two-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation California...

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