Math Problem Statement

The equation of line AB is given as y=3x+4. Determine which of the following lines are parallel to line AB. Select all suitable options. Option A: 3y=12x+6 Option B: 4y=12x+5 Option C: y=4x+4 Option D: y=6x+8 Option E: 6x-2y=10

Solution

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

Step 1: Identify the slope of line ABAB

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

For line ABAB:

  • Slope (mm) = 3

Step 2: Find the slope of each option

  • Option A: 3y=12x+63y = 12x + 6 Divide both sides by 3 to get the equation in slope-intercept form:\text{Divide both sides by 3 to get the equation in slope-intercept form:} y=4x+2y = 4x + 2

    • Slope (mm) = 4
  • Option B: 4y=12x+54y = 12x + 5 Divide both sides by 4 to get the equation in slope-intercept form:\text{Divide both sides by 4 to get the equation in slope-intercept form:} y=3x+54y = 3x + \frac{5}{4}

    • Slope (mm) = 3
  • Option C: y=4x+4y = 4x + 4

    • Slope (mm) = 4
  • Option D: y=6x+8y = 6x + 8

    • Slope (mm) = 6
  • Option E: 6x2y=106x - 2y = 10 \text{Rewrite the equation in slope-intercept form by solving for y:} 2y=6x+10-2y = -6x + 10 y=3x5y = 3x - 5

    • Slope (mm) = 3

Step 3: Compare the slopes

  • A line is parallel to line ABAB 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 ABAB 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.

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Math Problem Analysis

Mathematical Concepts

Linear equations
Slope
Parallel lines

Formulas

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Theorems

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Suitable Grade Level

Grades 7-9