Solve 5x+2=2x+17 in two ways: with equations and with math drawings of a pan balance. Relate the two methods.

Algebra and Trigonometry (6th Edition)
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ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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### Solving the Equation 5x + 2 = 2x + 17

Solve the equation \( 5x + 2 = 2x + 17 \) in two ways: with equations and with math drawings of a pan balance. Relate the two methods.

#### Method 1: Solving with Equations

1. Start with the given equation:
   \[
   5x + 2 = 2x + 17
   \]

2. Subtract \( 2x \) from both sides to isolate terms with \( x \) on one side:
   \[
   5x - 2x + 2 = 2x - 2x + 17
   \]
   \[
   3x + 2 = 17
   \]

3. Subtract 2 from both sides:
   \[
   3x + 2 - 2 = 17 - 2
   \]
   \[
   3x = 15
   \]

4. Divide both sides by 3 to solve for \( x \):
   \[
   \frac{3x}{3} = \frac{15}{3}
   \]
   \[
   x = 5
   \]

So, the solution to the equation is \( x = 5 \).

#### Method 2: Solving with a Pan Balance Diagram

In a pan balance diagram, each side of the balance represents one side of the equation. 

1. **Create the initial balance:**
   - On the left side of the pan balance, represent \( 5x + 2 \).
   - On the right side of the pan balance, represent \( 2x + 17 \).

2. **Balance the equation:**
   - To balance the representation, remove \( 2x \) from both sides to simplify it, just as in the algebraic solution.
   - This leaves \( 3x + 2 \) on the left pan and \( 17 \) on the right pan.

3. **Adjust for constants:**
   - Subtract 2 units from both sides to maintain balance, resulting in \( 3x \) on the left pan and \( 15 \) on the right pan.

4. **Determine the variable value:**
   - Each group of \( x \) on the left pan collectively balances with
Transcribed Image Text:### Solving the Equation 5x + 2 = 2x + 17 Solve the equation \( 5x + 2 = 2x + 17 \) in two ways: with equations and with math drawings of a pan balance. Relate the two methods. #### Method 1: Solving with Equations 1. Start with the given equation: \[ 5x + 2 = 2x + 17 \] 2. Subtract \( 2x \) from both sides to isolate terms with \( x \) on one side: \[ 5x - 2x + 2 = 2x - 2x + 17 \] \[ 3x + 2 = 17 \] 3. Subtract 2 from both sides: \[ 3x + 2 - 2 = 17 - 2 \] \[ 3x = 15 \] 4. Divide both sides by 3 to solve for \( x \): \[ \frac{3x}{3} = \frac{15}{3} \] \[ x = 5 \] So, the solution to the equation is \( x = 5 \). #### Method 2: Solving with a Pan Balance Diagram In a pan balance diagram, each side of the balance represents one side of the equation. 1. **Create the initial balance:** - On the left side of the pan balance, represent \( 5x + 2 \). - On the right side of the pan balance, represent \( 2x + 17 \). 2. **Balance the equation:** - To balance the representation, remove \( 2x \) from both sides to simplify it, just as in the algebraic solution. - This leaves \( 3x + 2 \) on the left pan and \( 17 \) on the right pan. 3. **Adjust for constants:** - Subtract 2 units from both sides to maintain balance, resulting in \( 3x \) on the left pan and \( 15 \) on the right pan. 4. **Determine the variable value:** - Each group of \( x \) on the left pan collectively balances with
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