### Key Concepts

• Solve two -step equations using models

• Solve two -step Equations algebraically

• Compare algebraic and arithmetic solutions

**5.2 Solve Two-Step Equations**

**How to Solve Two-Step Equations.**

- First, add or subtract both sides of the linear equation by the same number.

- Secondly, multiply or divide both sides of the linear equation by the same number.

- * Instead of step #2, always multiply both sides of the equation by the reciprocal of the coefficient of the variable.

**Example 1**: Solve the two-step equation below.

5m +6 = 16

**Sol:**

5m + 6 = 16

5m + 6 – 6 = 16 – 6

5m = 10

m = 10

÷ 5 ÷ 𝟓

m = 2

**5.2.1. Solve Two–Step equations Using Models **

**Example 1:**

After 4 hours shift, Sarah earned a total of $45, including a 5 tip. How much money did she earn per hour?

**Sol:**

**Step 1: **

Use a bar diagram to represent the situation.

Let *w* represent the hourly wage 4w +5 =45

**Step 2:**

**Use the subtraction property of equality to isolate the term containing the variable.**

4w +5 -5 = 45 -5

4w = 40

**Step 2:**

**Use the division property of equality to isolate the variable by itself on one side of the equation.**

4w =40

4w44w4

=

404404 Partitive division.

W =10.

So, she earns $10 per hour.

**5.2.2. Solve Two –Step Equations Algebraically**

**Algebraic** is a branch of mathematics that deals with variables and numbers for solving problems.

**Example1:**

Lucy has a history test in a week. The test will cover 120 pages in her textbook. She has read only 36 pages. If she plans to read an equal amount every day of the week to finish the 120 pages, how much must she read each day?

**Sol:**

Let p= the number of pages she reads per day.

36 pages plus seven times *p* pages is 120 pages.

36 + 7p = 120

36 – 36 +7p = 120 -36 Subtract 36 from each side.

7p = 84 Divided each side by 7.

P =12

Each day, she must read 12 pages.

**Example2:**

Solve two-step equation 2x + 4 =10

**Sol:**

2x + 4 =10

2x + 4 – 4 = 10 -4 Subtract 4 from each side.

2x = 6 Divided each side by 2.

x = 3

**5.2.3. Compare Algebraic and Arithmetic solutions**

**Arithmetic,** being the most basic of all branches of mathematics, deals with the basic counting of numbers by using operations like addition, multiplication, division, and subtraction on them.

**Algebraic** is a branch of mathematics that deals with variables and numbers for solving problems.

**Example1:**

A salesperson is paid $50 base pay per week and $3 commission per sale. If she needs to make 100 this week, how many sales must she make?

**Sol:**

**Algebraically (with equation):**

Let x =sales

3x +50 = 100

3x +50 -50 = 100 -50 Subtract 50 from each side.

3x = 50 Divided each side by 3.

x = 16.6

17 sales.

**Arithmetically (without equation):**

100 – 50 = 50

50 ÷ 3 = 16.6

17 sales.

She must make 17 sales.

## Exercise:

1. Solve the equation 0.4p – 2.45 = -1.3.

2. Solve the equation n/6 +8 = 10.

3. A group of four people ate dinner at a restaurant.They divided the bill equally, and every person left a $2 tip. Each member paid $14. Write and solve an equation to determine the amount of the total bill.

4. Anadi saves S20 every month in her savings account. She withdrew $60 one time to go shopping. At present, she has $280 in her account. Write and solve an equation to determine how many months she has been saving.

5. Charmin is a server in a restaurant. On a weekend night. Charmin earned $4 an hour and $65 in tips. She made a total of $93. Write and solve an equation to determine how many hours she worked.

6. Ken spent half of his money. The next day he earned $12. At present, he has $32. How much money did Ken have before he spent his money?

7. Circle the equation that does not match the bar model below.

8. Solve the equation 5x +3 =23.

9. Solve the equation to find the value of the variable.

12e +8= -40.

10. How can we show this using algebra?

### Concept Map

### What have we learned:

• Solve Two – Step equations

• Understand how to Solve Two -Step equations Using Models

• Solve Two -Step Equations Algebraically

• Compare Algebraic and Arithmetic solutions

• Understand how to isolate the term and variable.

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