What is the condition for the lines 2x + 3y = 6 and 4x + 6y = 12 to be parallel?
Correct Answer: The lines are parallel if their slopes are equal.
- Step 1: Start with the equations of the lines: 2x + 3y = 6 and 4x + 6y = 12.
- Step 2: Rewrite the first equation (2x + 3y = 6) in slope-intercept form (y = mx + b).
- Step 3: To isolate y, subtract 2x from both sides: 3y = -2x + 6.
- Step 4: Divide every term by 3 to solve for y: y = -2/3 x + 2.
- Step 5: Identify the slope (m) of the first line, which is -2/3.
- Step 6: Now, rewrite the second equation (4x + 6y = 12) in slope-intercept form.
- Step 7: Subtract 4x from both sides: 6y = -4x + 12.
- Step 8: Divide every term by 6 to solve for y: y = -2/3 x + 2.
- Step 9: Identify the slope (m) of the second line, which is also -2/3.
- Step 10: Compare the slopes of both lines. Since both slopes are equal (-2/3), the lines are parallel.
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