What is the equation of the straight line that is perpendicular to 5x + 3y = 6 and has a y-intercept equal to −3?

Introduction / Context:
Here we use the concept of perpendicular lines in coordinate geometry. Lines that are perpendicular have slopes that are negative reciprocals of one another. This question also requires finding the equation of a line given a slope and a y intercept, and then comparing that with various standard forms.

Given Data / Assumptions:

• Original line: 5x + 3y = 6.
• Required line is perpendicular to this original line.
• Required line has y intercept −3.
• We seek the correct equation from the given options.

Concept / Approach:
First, find the slope of the given line by rewriting it in slope intercept form. If the slope of the original line is m, then the slope of a line perpendicular to it is −1/m. Once we know the new slope and the y intercept, we write y = m2 x + c and then rearrange into an algebraic form to match the options. Matching both slope and intercept ensures that we have the correct perpendicular line.

Step-by-Step Solution:
Start from 5x + 3y = 6. Solve for y: 3y = −5x + 6, so y = (−5/3)x + 2. Thus the slope of the original line is m1 = −5/3. For a perpendicular line, the slope m2 is the negative reciprocal: m2 = 3/5. The y intercept of the required line is −3, so its equation in slope intercept form is y = (3/5)x − 3. Multiply both sides by 5 to remove the denominator: 5y = 3x − 15. Rearrange to standard form: 3x − 5y = 15.

Verification / Alternative check:
Check the y intercept by substituting x = 0 into 3x − 5y = 15, which gives −5y = 15, so y = −3, matching the required intercept. To verify perpendicularity, rewrite 3x − 5y = 15 as y = (3/5)x − 3. Its slope is 3/5. The original slope is −5/3, and the product (−5/3)*(3/5) = −1, confirming that the lines are perpendicular.

Why Other Options Are Wrong:
3x + 5y = 15 has slope −3/5, not 3/5, and would be perpendicular to a line of slope 5/3 rather than to the original line. The equation 3x − 5y = −15 gives y intercept 3, not −3. The equation 3x + 5y = −15 has slope −3/5 and intercept −3, so it does not satisfy the negative reciprocal relation with the original slope. The equation 5x + 3y = −9 has the same slope as the original, so this line is parallel, not perpendicular.

Common Pitfalls:
Students sometimes mix up parallel and perpendicular slopes, or forget to take the negative reciprocal correctly. Another source of error is miscomputing the y intercept when converting between forms. Always verify both slope and intercept at the end to avoid simple algebraic mistakes.

Final Answer:
The required perpendicular line with y intercept −3 is 3x − 5y = 15.