Write and Solve Equations using Patterns

Key Concepts

• Find a pattern to write an equation.
• Make and analyze a table to write an equation.

Introduction: 

Use pattern to write an equation: 

You can use patterns in a table to write an equation that relates the independent and dependent variables. 

Example 1: 

Use the pattern in the table below to write an equation. 

Solution: 

In the above table, the dependent variable m is 3 times the independent variable j. 

Hence, the equation is m = 3j. 

4.9.1 Find a pattern to write an equation 

Example 2: 

The table shows the number of yards, y, that a professional rides in s seconds. Find a pattern that relates the variables. If the cyclist maintains this speed, how far would the cyclist ride in 8 seconds? 

Solution: 

Step 1: 

Look for a pattern in the table that relates y, the dependent variable and s, the independent variable. 

Step 2: 

Write an equation to describe the relationship. 

12.2 times the value of s = the value of y 

12.2s = y 

      Or 

y = 12.2s 

Step 3: 

Find the distance covered in 8 seconds. 

y = 12.2s 

y = 12.2(8)  

y = 97.6 

Example 3: 

The table below shows how many candles are in different numbers of boxes. Find a pattern that explains the relationship between the values of c and b. Use words and numbers to describe the pattern. How many candles will there be in 10 boxes? 

Solution: 

Step 1: 

Look for a pattern in the table that relates c, the dependent variable and b, the independent variable. 

Step 2: 

Write an equation to describe the relationship. 

4 times the value of b = the value of c 

4b = c 

     Or 

c = 4b 

Step 3: 

Find the number of candles in 10 boxes. 

c = 4b 

c = 4(10)  

c = 40 

4.9.2 Make and analyze a table to write an equation 

Example 4: 

Kayla owes her mother $70. She repays her mother an amount of $5 each week. How much will Kayla owe her mother after 12 weeks? 

Solution: 

Make a table and look for a pattern that relates the variables. 

Write an equation to describe the relationship. 

Find how much Kayla will owe after 12 weeks. 

a = 70 – 5w 

a = 70 – 5(12) 

a = 70 – 60 

a = 10 

Example 5: 

Lauren went to a pizza shop to eat a small pizza. She read the menu as follows, 

The cost of small pizza is $6.  

Each topping costs $2.  

Help Lauren in finding the total cost of a pizza with 6 toppings. 

Solution: 

Make a table and look for a pattern that relates the variables. 

Write an equation to describe the relationship. 

Find the total cost of a pizza with 6 toppings. 

c = 6 + 2t 

c = 6 + 2(6) 

c = 6 + 12 

c = 18 

Exercise:

1. Use the pattern in the table below to write an equation.

2. Use the equation y = 2x – 7 to complete the following table.

3. Write a rule and an equation that represents the pattern in the following table.

4. Write a rule and an equation that represents the pattern in the following table.

5. The table shows Brenda’s age, b, when Talia’s age, t, is 7, 9, and 10. Find the pattern and then write a rule and an equation that represents the pattern. Then find Brenda’s age when Talia is 12.

6. Write a rule and an equation that represents the pattern in the following table. Then complete the table.

7. Write a rule and an equation that represents the pattern in the following table. Then complete the table.

8. Write an equation that best describes the pattern in the following table.

9. Write an equation that best describes the pattern in the following table.

10. Use the equation t = 5d + 5 to complete the following table.

What have we learned:

• Find a pattern in a table that relates the dependent and independent variable and write an equation.
• Make a table, look for a pattern and write an equation.

Concept map: