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Find the two equations that represent the same line

Let's analyze the five equations shown in the image and find which two represent the same line.

Here are the equations:

1. \( 4x + 5y - 5 = 0 \)
2. \( y = -\frac{4}{5}x + 3 \)
3. \( y + 3 = -\frac{4}{5}(x - 5) \)
4. \( 2x - 3y = 0 \)
5. \( y = \frac{2}{3}x - 1 \)

### Step 1: Transform all equations into the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. **Equation 1: \( 4x + 5y - 5 = 0 \)**
   \[
   5y = -4x + 5
   \]
   \[
   y = -\frac{4}{5}x + 1
   \]

2. **Equation 2: \( y = -\frac{4}{5}x + 3 \)** (already in slope-intercept form)

3. **Equation 3: \( y + 3 = -\frac{4}{5}(x - 5) \)**
   \[
   y + 3 = -\frac{4}{5}x + 4
   \]
   \[
   y = -\frac{4}{5}x + 1
   \]

4. **Equation 4: \( 2x - 3y = 0 \)**
   \[
   3y = \frac{2}{3}x
   \]
   \[
   y = \frac{2}{3}x
   \]

5. **Equation 5: \( y = \frac{2}{3}x - 1 \)** (already in slope-intercept form)

### Step 2: Compare the slope and intercept of the equations.

- **Equation 1:** \( y = -\frac{4}{5}x + 1 \)
- **Equation 2:** \( y = -\frac{4}{5}x + 3 \)
- **Equation 3:** \( y = -\frac{4}{5}x + 1 \)
- **Equation 4:** \( y = \frac{2}{3}x \)
- **Equation 5:** \( y = \frac{2}{3}x - 1 \)

### Conclusion:
- **Equations 1 and 3** both simplify to \( y = -\frac{4}{5}x + 1 \), so they represent the same line.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the geometric meaning of two equations representing the same line?
2. How can you check if two different forms of linear equations represent the same line?
3. What are other forms of linear equations besides slope-intercept form?
4. How do parallel and perpendicular lines differ in their slopes?
5. Can two lines with different slopes ever represent the same line?

**Tip:** The slope-intercept form \( y = mx + b \) is useful to quickly identify if two lines are parallel or the same!

Analyze linear equations to determine which two represent the same line. Using the slope-intercept form (y = mx + b), we compare the slope and intercept of different equations. Learn step-by-step how to identify identical lines. Suitable for Grades 9-12.

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