To determine which lines are parallel to the line $AB$ given by the equation $y = 3x + 4$, we need to compare the slopes of the given lines with the slope of line $AB$.

Step 1: Identify the slope of line $AB$

The equation of line $AB$ is $y = 3x + 4$. The slope-intercept form of a line is given by $y = mx + b$, where $m$ is the slope.

For line $AB$:

• Slope ($m$) = 3

Step 2: Find the slope of each option

• Option A: $3y = 12x + 6$ $\text{Divide both sides by 3 to get the equation in slope-intercept form:}$ $y = 4x + 2$

  • Slope ($m$) = 4

• Option B: $4y = 12x + 5$ $\text{Divide both sides by 4 to get the equation in slope-intercept form:}$ $y = 3x + \frac{5}{4}$

  • Slope ($m$) = 3

• Option C: $y = 4x + 4$

  • Slope ($m$) = 4

• Option D: $y = 6x + 8$

  • Slope ($m$) = 6

• Option E: $6x - 2y = 10$ \text{Rewrite the equation in slope-intercept form by solving for y:} $-2y = -6x + 10$ $y = 3x - 5$

  • Slope ($m$) = 3

Step 3: Compare the slopes

• A line is parallel to line $AB$ if it has the same slope, i.e., a slope of 3.

• Option A: Slope = 4 (Not parallel)

• Option B: Slope = 3 (Parallel)

• Option C: Slope = 4 (Not parallel)

• Option D: Slope = 6 (Not parallel)

• Option E: Slope = 3 (Parallel)

Conclusion:

The lines parallel to line $AB$ are:

• Option B
• Option E

Would you like further details or have any questions?

Here are five related questions to explore:

1. How would you determine if two lines are perpendicular instead of parallel?
2. What is the relationship between the slopes of perpendicular lines?
3. Can two lines be parallel if their slopes are negative reciprocals of each other?
4. How does the y-intercept affect the parallelism of two lines?
5. What is the significance of the slope-intercept form in graphing linear equations?

Tip: When determining parallelism, always compare the slopes of the lines. Parallel lines will always have identical slopes.