Difficulty: Medium
Correct Answer: 3x - 2y = -6
Explanation:
Introduction / Context:
This problem involves finding the equation of a line in the coordinate plane that is perpendicular to a given line and has a specified y-intercept. It tests your understanding of slopes of perpendicular lines and converting between slope intercept and standard forms of linear equations.
Given Data / Assumptions:
Concept / Approach:
First, we find the slope of the given line by rewriting it in the form y = mx + c. Then, use the fact that the slope of a line perpendicular to a line with slope m is the negative reciprocal, that is -1 / m. With the perpendicular slope and the given y-intercept, we write the new line as y = m2 * x + 3 and then convert to standard form for comparison with the options.
Step-by-Step Solution:
Step 1: Rewrite 2x + 3y = -6 in slope intercept form: 3y = -2x - 6, so y = (-2 / 3)x - 2.Step 2: The slope of this line is m1 = -2 / 3.Step 3: The slope of a line perpendicular to this is m2 = 3 / 2, the negative reciprocal of -2 / 3.Step 4: Use the y-intercept 3 to write the new line: y = (3 / 2)x + 3.Step 5: Multiply both sides by 2: 2y = 3x + 6, or rearranged as 3x - 2y + 6 = 0.Step 6: Hence the standard form is 3x - 2y = -6.
Verification / Alternative check:
We can verify the y-intercept by setting x = 0 in 3x - 2y = -6, which gives -2y = -6 and y = 3, matching the given intercept. The product of slopes m1 * m2 = (-2 / 3) * (3 / 2) = -1, confirming perpendicularity.
Why Other Options Are Wrong:
The equation 3x - 2y = 6 has the correct slope but gives y-intercept -3. The other equations correspond to slopes that are not perpendicular to -2 / 3 or do not have the correct intercept of 3.
Common Pitfalls:
Students sometimes take the reciprocal without changing the sign or forget to use the correct y-intercept when forming the new line. Another frequent error is rearranging incorrectly when converting between forms.
Final Answer:
The required line is given by 3x - 2y = -6.
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