Difficulty: Easy
Correct Answer: 5x - 3y = 15
Explanation:
Introduction / Context:
This question checks your understanding of the intercept form and standard form of a straight line in coordinate geometry. Knowing how to convert between these two forms helps you move quickly from graphical information, such as intercepts on the axes, to algebraic equations that can be used in further calculations.
Given Data / Assumptions:
Concept / Approach:
First, we write the line in intercept form, which is x/a + y/b = 1 where a is the x-intercept and b is the y-intercept. Then we substitute a = 3 and b = −5. Finally, we clear denominators and rearrange the equation into the standard form Ax + By = C with integer coefficients and a positive constant on the right side.
Step-by-Step Solution:
Verification / Alternative check:
Check the intercepts from the equation 5x − 3y = 15. For the x-intercept, put y = 0 to get 5x = 15, so x = 3. For the y-intercept, put x = 0 to get −3y = 15, so y = −5. These match the given intercepts, confirming the correctness of the equation.
Why Other Options Are Wrong:
The equation 3x − 5y = 15 gives x-intercept 5 and y-intercept −3, which are reversed. The equation 5x + 3y = 15 gives intercepts (3, 5), both positive. The equation 3x + 5y = 15 gives both intercepts positive as well. The equation 3x − 5y = −15 has intercepts (−5, 3), which do not match the problem statement.
Common Pitfalls:
Students often confuse which intercept goes in the denominator of which term or forget that a negative y-intercept leads to a negative denominator. Another common mistake is not multiplying by the correct common multiple, which leaves fractions in the final answer. Being systematic with intercept form avoids these errors.
Final Answer:
The equation of the line with x-intercept 3 and y-intercept −5 is 5x − 3y = 15.
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