In coordinate geometry (straight line), find the slope m of the line 2x - 5y = 12 by first rewriting it in slope–intercept form and then choosing the correct value of the slope from the options given below.

Introduction / Context:
In coordinate geometry, a straight line written in standard form Ax + By + C = 0 can be converted into slope–intercept form y = mx + c. The coefficient m represents the slope of the line and indicates how steeply the line rises or falls as x increases. In this question we are asked to determine the slope of the line 2x - 5y = 12 and then choose the correct numerical value from the given options.

Given Data / Assumptions:

• The line is given in standard form: 2x - 5y = 12.
• We assume a Cartesian coordinate plane with x as the horizontal axis and y as the vertical axis.
• The slope m is defined as the coefficient of x when the line is written as y = mx + c.
• All arithmetic is performed using real numbers.

Concept / Approach:
To find the slope, we convert the given line into the slope–intercept form. This means we solve the equation for y in terms of x. The resulting coefficient of x is the slope m. Rearranging carefully and paying attention to signs is the key idea.

Step-by-Step Solution:
Start with the standard form: 2x - 5y = 12. Isolate the y term by subtracting 2x from both sides: -5y = -2x + 12. Divide every term by -5 to solve for y: y = ( -2x + 12 ) / -5. Simplify the expression: y = (2/5)x - 12/5, so the slope m = 2/5.
Verification / Alternative check:
Another way to understand the slope is to rewrite the equation as y = (2/5)x - 12/5 and then note that for every increase of 5 units in x, y increases by 2 units, which confirms a slope of 2/5. You could also pick two points on the line, compute the change in y divided by the change in x, and you would again obtain 2/5 as the slope.

Why Other Options Are Wrong:
Option a ( -5/2 ) corresponds to incorrectly taking the ratio -5/2 from the original coefficients without proper rearrangement. Option b ( 5/2 ) is the reciprocal of the correct slope and arises from confusion between rise over run and run over rise. Option c ( -2/5 ) results from a sign error when dividing by -5. Option e ( 0 ) would represent a horizontal line, which does not match the given equation because it clearly depends on x.

Common Pitfalls:
Many learners forget to change the sign of each term when dividing by a negative coefficient of y. Another frequent mistake is to assume that the slope is simply -A/B from Ax + By + C = 0 without actually rearranging the expression, which can lead to sign errors.

Final Answer:
Therefore, the correct value of the slope m of the line 2x - 5y = 12 is 2/5.