Point-slope equation of a line
The explicit form $y=mx+n$ requires us to know the y-intercept, a piece of information that is not always available. The point-slope equation lets us write the equation of a line from any point it passes through and its gradient, without first having to work out where it crosses the $Y$ axis. It is the central algebraic tool for determining lines in coordinate geometry.
Derivation of the point-slope equation
If a line has gradient $m$ and passes through $P(x_1,y_1)$, then for any other point $(x,y)$ on the line, by the definition of gradient, $$m = \frac{y-y_1}{x-x_1}$$ Multiplying both sides by $(x-x_1)$ gives the point-slope form.
Write the point-gradient equation of the line passing through the point $P(3, -2)$ with gradient $m = 4$.
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 $A$.
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.
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.
From point-slope to explicit form and to general form
Transform the point-gradient equation $y + 3 = \frac{1}{2}(x - 4)$ into its explicit form $y = mx + n$.
Transform the point-gradient equation $y + 3 = \frac{1}{2}(x - 4)$ into its explicit form $y = mx + n$.
Write the point-gradient equation of the line parallel to $y = 3x - 7$ passing through the point $Q(-4, 1)$.
Write the point-gradient equation of the line parallel to $y = 3x - 7$ passing through the point $Q(-4, 1)$.
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…
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.
A line determined by two points
When two points $A(x_1,y_1)$ and $B(x_2,y_2)$ on a line are known ($x_1 \neq x_2$), the procedure is always the same: first calculate the gradient, then apply the point-slope equation with either of the two points.
Find the point-gradient equation of the line perpendicular to the line $y - 2 = \frac{3}{4}(x + 1)$ passing through the origin.
Find the point-gradient equation of the line perpendicular to the line $y - 2 = \frac{3}{4}(x + 1)$ passing through the origin.
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…
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.
Determine the value of the parameter $k$ such that the line with equation $y - 3 = k(x + 2)$ passes through the point of intersection of the lines $r: 2x + y - 8 = 0$…
Determine the value of the parameter $k$ such that the line with equation $y - 3 = k(x + 2)$ passes through the point of intersection of the lines $r: 2x + y - 8 = 0$ and $s: x - 3y + 3 = 0$.
Parallel and perpendicular lines
A car travels along a straight road at a constant speed. At $10:00$ hours it is at kilometre $45$ and at $12:30$ hours it passes kilometre $245$. Write the…
A car travels along a straight road at a constant speed. At $10:00$ hours it is at kilometre $45$ and at $12:30$ hours it passes kilometre $245$. Write the point-gradient equation modelling the car's position as a function of time elapsed since $10:00$ hours, expressing time in hours.
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…
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.
Applications: collinear points and perpendicular bisector
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 $C$.
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 $C$.
The Celsius ($^\circ C$) and Fahrenheit ($^\circ F$) temperature scales have a linear relationship. Knowing that water freezes at $0^\circ C$ ($32^\circ F$) and boils at…
The Celsius ($^\circ C$) and Fahrenheit ($^\circ F$) temperature scales have a linear relationship. Knowing that water freezes at $0^\circ C$ ($32^\circ F$) and boils at $100^\circ C$ ($212^\circ F$), deduce the point-gradient equation to convert degrees Celsius to Fahrenheit using the freezing point. Then, use it to calculate the Fahrenheit temperature equivalent to a normal human body temperature of $37^\circ C$.