Equation of a line: y − y₁ = m(x − x₁)

Write the point-gradient equation of the line passing through the point $P(3, -2)$ with gradient $m = 4$.

Calculate the gradient of the line passing through the points $A(-1, 5)$ and $B(2, 11)$. Then, write its equation in point-gradient form using the point $A$.

Given the line with equation $y - 5 = -2(x + 1)$, state its gradient and obtain directly from the formula the coordinates of a point through which it passes.

Write the point-gradient equation of the angle bisector of the first and third quadrants. Then, write the point-gradient equation of the angle bisector of the second and fourth quadrants, using the origin in both cases.

Transform the point-gradient equation $y + 3 = \frac{1}{2}(x - 4)$ into its explicit form $y = mx + n$.

From the general equation of the line $3x - 5y + 15 = 0$, find a point on it whose $x$-coordinate is an integer, calculate its gradient and write the corresponding equation in point-gradient form.

Calculate the area of the triangle formed by the line with equation $y - 4 = -\frac{2}{3}(x + 3)$ and the coordinate axes. To do so, first find the points of intersection of the line with the axes.

A plumber charges $30$ € for a call-out fee and $20$ € for each hour of work. Given that for a $2$-hour job he charged a total of $70$ €, write the point-gradient equation relating the total cost to the hours worked, taking this known point as a reference.

On the graph of a line, the points $(0, 0)$ and $P(2, 3)$ are highlighted. Using the highlighted points, calculate the slope of the line and write its equation in point-slope form taking point $P$ as the reference.

A bicycle rental company charges a fixed amount for the service plus a constant price per hour of use. It is known that a rental of $3$ hours costs $14$ € and a rental of $5$ hours costs $20$ €. Determine the gradient of the linear cost function and write its point-gradient equation using the data from the first customer.

Write the point-gradient equation of the line parallel to $y = 3x - 7$ passing through the point $Q(-4, 1)$.

Oisín has to find the general equation of the line perpendicular to $r: 3x - 6y + 2 = 0$ that passes through the point $A(1, -2)$. On his exam, he writes the following:

"Step 1: The slope of line $r$ is $m_1 = -\dfrac{A}{B} = -\dfrac{3}{-6} = \dfrac{1}{2}$.

Step 2: Since perpendicular lines have opposite slopes, the slope of the required line must be $m_2 = -\dfrac{1}{2}$.

Step 3: I apply the point-slope form: $y - (-2) = -\dfrac{1}{2}(x - 1)$, which upon expanding gives $y + 2 = -\dfrac{1}{2}x + \dfrac{1}{2}$, and in general form becomes $x + 2y + 3 = 0$".

1. a) Identify precisely in which step he makes the mistake and explain the conceptual error Oisín makes regarding the condition of perpendicularity.
2. b) Correct the exercise from that step onward and write the correct general equation ($Ax + By + C = 0$) of the perpendicular line.

Find the point-gradient equation of the line perpendicular to the line $y - 2 = \frac{3}{4}(x + 1)$ passing through the origin.

A light ray follows a straight-line path given by the equation $y - 5 = -\frac{1}{2}(x - 2)$. On reaching the x-axis (acting as a perfect flat mirror), the ray is reflected. Knowing that the angle of incidence equals the angle of reflection, find the point-gradient equation of the reflected ray using the point of impact on the mirror as a reference.

Check algebraically whether the points $A(-2, 4)$, $B(1, -2)$ and $C(5, -10)$ are collinear. To do so, first find the point-gradient equation of the line passing through $A$ and $B$, and then check whether the coordinates of $C$ satisfy that equation.

Find the point-gradient equation of the perpendicular bisector of the segment with endpoints $A(-4, 2)$ and $B(2, 6)$. Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint.

Given the triangle with vertices $A(1, 1)$, $B(7, 4)$ and $C(3, 7)$, find the point-gradient equation of the altitude from vertex $C$.