Find the equation of the straight line that passes through the point (-1, 3) and has an x-intercept of 4 units, and then choose the correct equation from the options given.

Introduction / Context:
This problem combines concepts of intercepts and the general equation of a straight line. You are given one point through which the line passes and the x-intercept. From this information you must determine the line equation that satisfies both conditions. Such questions are standard in coordinate geometry sections of aptitude tests.

Given Data / Assumptions:

• The line passes through the point (-1, 3).
• The line has an x-intercept equal to 4 units, so it passes through (4, 0).
• We must find the equation of the line that passes through both (-1, 3) and (4, 0).

Concept / Approach:
A line is uniquely determined by two distinct points. We first compute the slope using the two points, then use the point slope form of the line equation and transform it into a standard linear form. Finally, we compare this equation with the options provided. Alternatively, we can derive the equation directly in intercept form and verify with one point.

Step-by-Step Solution:
Let the two points be P(-1, 3) and Q(4, 0).Compute the slope m: m = (y2 − y1) / (x2 − x1) = (0 − 3) / (4 − (-1)) = -3 / 5.Use the point slope form with point Q(4, 0): y − 0 = m(x − 4) so y = (-3/5)(x − 4).Expand: y = (-3/5)x + (12/5).Multiply both sides by 5 to remove the denominator: 5y = -3x + 12, or rearranged as 3x + 5y = 12.

Verification / Alternative check:
Check that both given points satisfy the equation 3x + 5y = 12. For (4, 0), we have 3 * 4 + 5 * 0 = 12, which is correct. For (-1, 3), we have 3 * (-1) + 5 * 3 = -3 + 15 = 12, which also works. Therefore, 3x + 5y = 12 is the correct line. None of the other options satisfy both points simultaneously.

Why Other Options Are Wrong:

• 3x - 5y = 12 fails for the point (-1, 3) because 3 * (-1) - 5 * 3 = -3 - 15 = -18, not 12.
• 3x + 5y = -12 gives negative values when the points are substituted, so it does not pass through the given points.
• 3x - 5y = -12 and 5x + 3y = 12 also fail to satisfy both points, indicating they represent different lines.

Common Pitfalls:

• Making sign errors when computing the slope from the two points.
• Incorrectly using only one condition, such as the x-intercept, and not enforcing the point (-1, 3).
• Errors while clearing fractions when converting the equation to standard form.

Final Answer:
3x + 5y = 12