Equations of straight lines; gradient, intercepts practice questions.

Determine the equation of the straight line that passes through the point (2, 5) and has gradient -3.

• y = -3x - 1
• y = 3x - 1
• y = -3x + 11
• y = -3x + 5

Answer: y = -3x + 11

Using y - y₁ = m(x - x₁): y - 5 = -3(x - 2), so y = -3x + 6 + 5 = -3x + 11.

Which of the following lines is perpendicular to the line y = (2/5)x + 1?

• y = -(5/2)x + 7
• y = (5/2)x - 3
• y = -(2/5)x + 4
• y = (2/5)x - 2

Answer: y = -(5/2)x + 7

Perpendicular gradients multiply to -1, so the required gradient is the negative reciprocal of 2/5, which is -5/2.

State the coordinates of the point where the line 3x + 4y = 24 crosses the x-axis.

• (0, 6)
• (6, 0)
• (0, 8)
• (8, 0)

Answer: (8, 0)

On the x-axis y = 0, so 3x = 24 gives x = 8; the intercept is (8, 0). Setting x = 0 instead gives the y-intercept (0, 6).

Determine the gradient of the line with equation 2x - 5y + 10 = 0.

• -2/5
• 2/5
• 5/2
• 2

Answer: 2/5

Rearranging: 5y = 2x + 10, so y = (2/5)x + 2 and the gradient is 2/5. A common error is forgetting to divide by the coefficient of y or dropping a sign.

A line passes through the points (-1, 4) and (3, -4). Determine its equation.

• y = -2x + 2
• y = -2x - 2
• y = 2x + 6
• y = -(1/2)x + 7/2

Answer: y = -2x + 2

Gradient = (-4 - 4)/(3 - (-1)) = -8/4 = -2. Substituting (-1, 4): 4 = -2(-1) + c gives c = 2, so y = -2x + 2.

Consider the lines y = 2x + 3 and y = 2x - 5. Which statement correctly describes them?

• They are perpendicular
• They intersect at exactly one point
• They are parallel and never meet
• They are the same line

Answer: They are parallel and never meet

Both lines have gradient 2 but different y-intercepts, so they are distinct parallel lines with no point of intersection.