Exercises: Point-gradient equation of a line

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.

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 $Q(-4, 1)$.

A plumber charges $600$ pesos for a call-out fee and $400$ pesos for each hour of work. Given that for a $2$-hour job he charged a total of $1\,400$ pesos, 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.

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.

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 $280$ pesos and a rental of $5$ hours costs $400$ pesos. Determine the gradient of the linear cost function and write its point-gradient equation using the data from the first customer.

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 $A$ and $B$, and then check whether the coordinates of $C$ satisfy that equation.

A car travels along a straight road at a constant speed. At $10:00$ hours it is at kilometer $45$ and at $12:30$ hours it passes kilometer $245$. Write the point-slope equation modeling the car's position as a function of time elapsed since $10:00$ hours, expressing time in hours.

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.

The following graph shows the profile of descent at constant speed of a submarine. From the indicated points, determine the point-gradient equation of its trajectory.

The value of computer equipment depreciates linearly over time. If it was purchased for $24\,000$ pesos and after $4$ years its book value is $8\,000$ pesos, write the point-gradient equation relating the value of the equipment to the years elapsed, using the point corresponding to the fourth year.

Obtain the point-gradient equation of the line passing through the point $P(-3, 5)$ parallel to the line joining the points $M(1, -2)$ and $N(4, 7)$.

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 $s: x - 3y + 3 = 0$.

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 $C$.

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 $37^\circ C$.

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.

In a simple economic model, the quantity demanded of a product decreases linearly as its price increases. At a price of $400$ pesos, $500$ units are demanded, and it has been observed that for every increase of $40$ pesos in price, demand drops by $50$ units. Write the point-gradient equation for demand using the first set of data as a reference, and calculate at what price demand will be zero.

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-slope equation of the reflected ray using the point of impact on the mirror as a reference.

Write the point-gradient equation of the line passing through the point $P(2, -1)$ that, when intersecting the semi-axes (positive x-axis and negative y-axis), forms a right-angled triangle of area $4$ square units.