At what point does the line 2x - 3y = 6 cut the y axis?

Introduction / Context:
This is a basic coordinate geometry question about finding where a straight line intersects the y axis. The y intercept is a key feature of a line and is often used when graphing or analysing linear equations.

Given Data / Assumptions:

• The equation of the line is 2x - 3y = 6.
• We must find the point where this line cuts the y axis.
• On the y axis, the x coordinate is always 0.

Concept / Approach:
To find the y intercept of a line, set x = 0 in the equation and solve for y. This gives the coordinates (0, y). This works because any point on the y axis has x equal to zero by definition.

Step-by-Step Solution:
Step 1: Start with the line equation 2x - 3y = 6. Step 2: At the y axis, x = 0. Substitute x = 0 into the equation. Step 3: The equation becomes 2 * 0 - 3y = 6. Step 4: Simplify: 0 - 3y = 6, so -3y = 6. Step 5: Divide by -3: y = 6 / (-3) = -2. Step 6: Therefore, the line cuts the y axis at the point (0, -2).

Verification / Alternative check:
We can rewrite the line in slope intercept form y = mx + c. From 2x - 3y = 6, rearrange to -3y = -2x + 6, so y = (2 / 3)x - 2. Here, the y intercept is clearly -2, confirming the point (0, -2). This is in full agreement with our earlier calculation.

Why Other Options Are Wrong:
(0, 2): Substituting x = 0, y = 2 gives 2 * 0 - 3 * 2 = -6, not 6.
(-2, 0): This point lies on the x axis and not on the y axis; also it does not satisfy the equation correctly for an intercept on y.
(2, 0): Also an x axis point, not a y axis point.
(0, 3): Substituting gives 2 * 0 - 3 * 3 = -9, which does not equal 6.

Common Pitfalls:
One common mistake is to set y = 0 when looking for the y intercept instead of setting x = 0. That actually finds the x intercept. Another error is incorrect algebra when isolating y. Keeping the definition of the axes in mind and working step by step avoids these issues.

Final Answer:
The line cuts the y axis at the point (0, -2).