What is the condition for the lines 2x + 3y = 6 and 4x + 6y = 12 to be parallel?

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Question: What is the condition for the lines 2x + 3y = 6 and 4x + 6y = 12 to be parallel?

Options:

Correct Answer: They have the same slope

Solution:

Both lines can be rewritten in slope-intercept form. The first line has slope -2/3 and the second line has the same slope, hence they are parallel.

What is the condition for the lines 2x + 3y = 6 and 4x + 6y = 12 to be parallel?

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.

Slope of a Line – Understanding that two lines are parallel if they have the same slope.
Equation of a Line – Recognizing how to convert standard form equations to slope-intercept form to find slopes.