Question

what is the relationship between the three slides
1y=3x+6
2 a tablex=3,1,3,5 /y=6,3,9,15
3f(x)=5x-3

Answer 1

Let's analyze the three slides you've provided to determine their relationships:

### 1. The Equation: \( y = 3x + 6 \)
This is a linear equation in slope-intercept form (\( y = mx + b \)), where:
- The slope (\( m \)) is 3.
- The y-intercept (\( b \)) is 6.

This means that for every unit increase in \( x \), \( y \) increases by 3. The line intersects the y-axis at (0, 6).

### 2. The Table
The table presents pairs of \( x \) and \( y \) values:
- \( x = 3, y = 6 \)
- \( x = 1, y = 3 \)
- \( x = 3, y = 9 \) (note that this has the same x as the first, which might be an error)
- \( x = 5, y = 15 \)

Let's analyze the valid \( (x, y) \) pairs that can correspond with the equation \( y = 3x + 6 \):

For each \( x \):
- For \( x = 3 \): \( y = 3(3) + 6 = 9 \)
- For \( x = 1 \): \( y = 3(1) + 6 = 9 \) (not matching the table)
- For \( x = 5 \): \( y = 3(5) + 6 = 21 \) (not matching the table)

Only the first \( (3, 6) \) actually triggers a match with the y-value from the second table. It seems the table data might not directly correspond with the first equation's output.

### 3. The Function: \( f(x) = 5x - 3 \)
This is another linear function where:
- The slope (\( m \)) is 5.
- The y-intercept (\( b \)) is -3.

This function has a different slope and intercept compared to \( y = 3x + 6 \).

### Summary of Relationships
1. **Between the Equation and the Table:**
- Only the pair \( (3, 9) \) fits \( y = 3x + 6 \). It seems there might be an inconsistency in the table.

2. **Between the First Equation and the Function \( f(x) \):**
- \( y = 3x + 6 \) and \( f(x) = 5x - 3 \) are distinct linear equations with different slopes and intercepts.
- They represent two different lines in a coordinate plane.

In conclusion, there appears to be a mix-up with the data in the table. The relationship between the three slides indicates different linear functions that might intersect or diverge depending on specific values of \( x \).