Math Problem Statement

Fill in the table to determine the ending and starting numbers by working backward through the mathematical operations.

Solution

I see you've uploaded an image that involves working backward through a problem with a table and equations. Let me analyze and break down the steps for solving it:


Table Analysis:

The table has four columns:

  1. Starting value
  2. Multiply by 7 (resulting in 7x7x)
  3. Add 10 (resulting in 7x+107x + 10)
  4. Multiply by 2 (resulting in 2(7x+10)2(7x + 10))

We are tasked to:

  1. Fill in missing values in the table based on the operations.
  2. Use the given values (e.g., starting at 2-2 or ending at 9090) to work backward where necessary.

Step-by-Step Solution:

First Row:

  • Starting value: xx
  • Multiply by 7: 7x7x
  • Add 10: 7x+107x + 10
  • Multiply by 2: 2(7x+10)2(7x + 10)

Since we are working symbolically, this row is already complete.


Second Row:

  • Starting value: 2-2

Steps:

  1. Multiply 2-2 by 7: 7(2)=147(-2) = -14
  2. Add 10: 14+10=4-14 + 10 = -4
  3. Multiply by 2: 2(4)=82(-4) = -8

Complete row:

  • Multiply by 7: 14-14
  • Add 10: 4-4
  • Multiply by 2: 8-8

Third Row:

  • The final value is 9090.

We need to work backward:

  1. Divide 9090 by 2 (reverse of multiplying by 2): 90÷2=4590 \div 2 = 45
  2. Subtract 10 (reverse of adding 10): 4510=3545 - 10 = 35
  3. Divide by 7 (reverse of multiplying by 7): 35÷7=535 \div 7 = 5

Complete row:

  • Starting value: 55
  • Multiply by 7: 3535
  • Add 10: 4545
  • Multiply by 2: 9090

Completed Table:

StartingMultiply by 7Add 10Multiply by 2
xx7x7x7x+107x + 102(7x+10)2(7x + 10)
2-214-144-48-8
55353545459090

Would you like further explanation of any specific step, or help with similar problems?


Related Questions:

  1. How do you reverse each operation in an equation?
  2. Why is dividing the last value by 2 the first step when working backward?
  3. How can you generalize this process for any function or equation?
  4. What is the importance of inverse operations in solving equations?
  5. Can you create a similar problem using different operations?

Tip:

Always apply inverse operations in reverse order when working backward to ensure accuracy!

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Math Problem Analysis

Mathematical Concepts

Algebra
Inverse Operations
Equations

Formulas

y = 7x
y = 7x + 10
y = 2(7x + 10)

Theorems

Inverse Operations Principle

Suitable Grade Level

Grades 6-8