Find the equation of a straight line passing through the point (1, -5) and having gradient of $\frac{3}{4}$

Analyze the question: We need the equation of a line through (1, -5) with gradient 3/4.

Concept: Use the point-gradient form:

y - y₁ = m(x - x₁)

Solution:

1. Substitute m = 3/4, x₁ = 1, y₁ = -5:

y - (-5) = (3/4)(x - 1) y + 5 = (3/4)(x - 1)

2. Multiply both sides by 4 to remove the fraction:

4(y + 5) = 3(x - 1) 4y + 20 = 3x - 3

3. Move everything to one side:

0 = 3x - 4y - 3 - 20 3x - 4y - 23 = 0

Check: Put (1, -5) into 3x - 4y - 23: 3(1) - 4(-5) - 23 = 3 + 20 - 23 = 0 ✓

The gradient is also correct. Rewriting as y = (3/4)x - 23/4 gives a gradient of 3/4 ✓

Answer: The equation is 3x - 4y - 23 = 0, which is option D.

Tip: You can always test the options by substituting the given point. Here, only D gives 0 when you put in (1, -5). Also check the sign pattern: a positive gradient with the form ax - by + c means x and y have opposite signs, which eliminates A and B right away.