Select the correct answers to fill the gaps.The gradient of the line passing through points with coordinates (2a, Blank 1 Question 19[select: , 0, -3, -3a, 1, 3a] ) and (4a, Blank 2 Question 19[select: , 0, -3, -3a, 1, 3a] ) is -3.

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BlackTom AI · Accepted Answer

We start by restating the problem: we need to fill Blank 1 and Blank 2 with values from the given options so that the gradient of the line through the points (2a, Blank 1) and (4a, Blank 2) equals -3. The available options for each blank are: 0, -3, -3a, 1, 3a.

Step-by-step analysis for Blank 1 (the y-coordinate at x = 2a):
- Let Blank 1 = x1 and Blank 2 = x2. The slope m between points (2a, x1) and (4a, x2) is m = (x2 − x1) / (4a − 2a) = (x2 − x1) / (2a).
- We are told m = −3, so (x2 − x1) / (2a) = −3, which gives x2 − x1 = −6a.
- For each candidate value of x1 from the options, we can compute the required x2 = x1 − 6a and check if that x2 is also among the allowed options.
  - If x1 = 0, then x2 would need to be −6a, which is not listed among the options for Blank 2.
  - If x1 = −3, then x2 would need to be −3 − 6a = −6a − 3, which is not among the options.
  - If x1 = −3a, then x2 would need to be −3a − 6a = −9a, not among the options.
  - If x1 = 1, then x2 would need to be 1 − 6a, not among the options (and would depend on a, which is not a listed option).
  - If x1 = 3a, then x2 would need to be 3a − 6a = −3a, which is indeed among the options.
- The only pairing where both coordinates come from the provided options and satisfy the slope condition is x1 = 3a with x2 = −3a.

Step-by-step analysis for Blank 2 (the y-coordinate at x = 4a):
- Using the pair from Blank 1 above, x2 = −3a. This directly satisfies the required difference x2 − x1 = (−3a) − (3a) = −6a, which yields the slope m = (−6a)/(2a) = −3, matching the given gradient.
- If we try any other candidate for Blank 2 with the same Blank 1 = 3a, the difference would be different from −6a, leading to a slope other than −3, which contradicts the problem’s condition.

Summary of why the other options fail: for Blank 1, choosing any value other than 3a forces the needed Blank 2 to be a value not present in the option list (or makes the slope calculation inconsistent with the given gradient). For Blank 2, with any Blank 1 other than 3a, the resulting x2 would have to take a value not in the option list, or would not produce the required difference of −6a, thus invalidating the slope requirement.

Thus, the combination that satisfies all conditions uses Blank 1 = 3a and Blank 2 = −3a, consistent with the slope formula and the given answer choices.

MCD1550 / MCD2140 - T1 - 2025 TEST 1 (Week 4)

Select the correct answers to fill the gaps.The gradient of the line passing through points with coordinates (2a, Blank 1 Question 19[select: , 0, -3, -3a, 1, 3a] ) and (4a, Blank 2 Question 19[select: , 0, -3, -3a, 1, 3a] ) is -3.

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