In coordinate geometry, find the equation of the straight line that intercepts the x-axis at 3/4 and the y-axis at 2/3. Choose the correct equation of this line from the options.

Introduction / Context:
This question tests your understanding of the intercept form of the equation of a straight line in coordinate geometry. When a line cuts the x-axis and y-axis at known intercepts, there is a convenient form of the equation that uses those intercepts directly. Being able to move between intercept form and standard form is useful in graphing, analytic geometry, and many exam questions involving lines.

Given Data / Assumptions:

• The line intercepts the x-axis at x = 3/4, so it passes through the point (3/4, 0).
• The line intercepts the y-axis at y = 2/3, so it passes through the point (0, 2/3).
• We must find the equation of this line in standard form.
• The coordinate system is the usual Cartesian plane.

Concept / Approach:
The intercept form of the equation of a straight line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. Substituting a = 3/4 and b = 2/3 gives us an equation in terms of x and y. From there, we can clear denominators to obtain an equation in the standard general form Ax + By = C. We then compare this equation with the options provided and select the matching one.

Step-by-Step Solution:
1) Use the intercept form x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. 2) Substitute a = 3/4 and b = 2/3 to get x / (3/4) + y / (2/3) = 1. 3) Simplify each fraction: x / (3/4) = x * (4/3) = 4x / 3, and y / (2/3) = y * (3/2) = 3y / 2. 4) The equation becomes 4x / 3 + 3y / 2 = 1. 5) Clear denominators by multiplying every term by 6 (the least common multiple of 3 and 2): 6 * (4x / 3) + 6 * (3y / 2) = 6 * 1. 6) Compute each term: 6 * (4x / 3) = 8x, 6 * (3y / 2) = 9y, and 6 * 1 = 6. 7) The equation simplifies to 8x + 9y = 6.

Verification / Alternative check:
Verify that the intercepts match the given values. To find the x-intercept, set y = 0 in 8x + 9y = 6, giving 8x = 6, so x = 6/8 = 3/4. This matches the required x-intercept. To find the y-intercept, set x = 0, yielding 9y = 6, so y = 6/9 = 2/3, which matches the required y-intercept. Therefore, the line with equation 8x + 9y = 6 has exactly the intercepts specified in the problem.

Why Other Options Are Wrong:
Option b, 9x + 8y = 12, gives an x-intercept of 4/3 and a y-intercept of 3/2, both different from the required intercepts. Option c, 8x − 9y = 6, does not even cross the y-axis at a positive intercept. Option d, 9x + 8y = −12, has both intercepts negative. Option e, 6x + 6y = 5, produces intercepts 5/6 and 5/6, which are not 3/4 and 2/3. Only option a yields the correct intercept pair.

Common Pitfalls:
A common error is to invert the intercept form as a/x + b/y = 1 instead of x/a + y/b = 1. Others may forget to multiply through by the least common multiple to clear denominators, leading to fractional coefficients that do not match the options directly. Some learners also mix up x- and y-intercepts when checking options. Carefully applying the intercept form and systematically verifying both intercepts prevents these mistakes.

Final Answer:
The equation of the line with x-intercept 3/4 and y-intercept 2/3 is 8x + 9y = 6.