Introduction / Context:
This is a straight line problem that connects the general form of a line with its slope. Knowing how to express a line in slope intercept form is fundamental in coordinate geometry and is often tested in aptitude questions.
Given Data / Assumptions:
- The equation of the line is ax + 5y = 8.
- The slope m of this line is −4/3.
- We must find the value of a.
Concept / Approach:
A line written as Ax + By + C = 0 can be rearranged to y = mx + c, where m is the slope. For the equation ax + 5y = 8, we isolate y and express it in the form y = m x + c. We then compare the coefficient of x with the given slope −4/3 to determine a.
Step-by-Step Solution:
Start with ax + 5y = 8.
Rearrange to express y in terms of x: 5y = −ax + 8.
Divide both sides by 5: y = (−a / 5)x + 8 / 5.
This is now in the form y = m x + c with slope m = −a / 5.
We are told that the slope m is −4 / 3.
So set −a / 5 = −4 / 3.
Cancel the negative signs: a / 5 = 4 / 3.
Multiply both sides by 5: a = 5 * (4 / 3) = 20 / 3.
Verification / Alternative check:
Substitute a = 20 / 3 back into the equation: (20 / 3)x + 5y = 8. Solving for y gives y = −(20 / 3) / 5 x + 8 / 5 = −4 / 3 x + 8 / 5. This shows the slope is −4 / 3 as required, confirming a = 20 / 3 is correct.
Why Other Options Are Wrong:
Options b and d are reciprocals of the correct magnitude and would give slopes of −3 / 20 or 3 / 20, not −4 / 3. Option c (−20 / 3) would produce a slope of 4 / 3, with the wrong sign. Option e (4 / 3) leads to a slope of −4 / 15, which is also incorrect.
Common Pitfalls:
A common mistake is to mix up the roles of A and B in Ax + By + C = 0, or to forget that the slope is −A / B. Another error is to mis-handle negatives while equating slopes. Working carefully through the steps to isolate y and then comparing coefficients prevents such errors.
Final Answer:
The required value of the coefficient is
a = 20 / 3.
Discussion & Comments