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 $30$ € 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.

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.

Alejandro 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 Alejandro 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.

Construct the general equations ($Ax + By + C = 0$) of two distinct lines, $r$ and $s$, that simultaneously satisfy the following conditions:

1. Both must be perpendicular to the given line $t: 2x + 5y - 10 = 0$.

2. Line $r$ must intersect the y-axis ($Y$) at a point with a negative $y$-coordinate (choose this value yourself and justify it).

3. Line $s$ must pass exactly through the point $P(4, 1)$.

Show with reasoning how you deduce the common slope of $r$ and $s$ from line $t$, and obtain the general equation of each one.

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.

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 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.

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 $1200$ € and after $4$ years its book value is $400$ €, 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 $N(4, 7)$.

In a region of Castilla-La Mancha, two towns are located on a cartographic map at the coordinates $A(1, 2)$ and $B(7, 4)$ (measured in kilometres). A telecommunications company needs to install a repeater antenna equidistant from both municipalities, which requires placing it on the perpendicular bisector of segment $AB$.

1. a) Model the situation analytically by calculating step by step the general equation of this perpendicular bisector.
2. b) A regional road follows the straight path of the line $r: x - y + 1 = 0$. If the company decides to install the antenna right by the side of this road for ease of access, calculate the exact coordinates of the point where it should be placed.
3. c) Analyse the slopes of road $r$ and segment $AB$ to mathematically justify whether the road crosses perpendicularly, obliquely, or runs parallel to the straight line directly connecting both towns.

Consider the three points in the plane $P(-3, 8)$, $Q(1, 0)$, and $R(4, -6)$. To find out if the three points are collinear, two different procedures are proposed:

• Method 1: Find the explicit equation of the line passing through $P$ and $Q$, and then check whether the coordinates of $R$ satisfy that equation.

• Method 2: Directly calculate the slope of segment $PQ$ ($m_{PQ}$) and that of segment $QR$ ($m_{QR}$), comparing whether both values coincide.

1. a) Reason which of the two methods is more direct and efficient, stating what advantages it presents regarding the number of operations and the risk of algebraic error.
2. b) Apply the chosen method to justify whether $P$, $Q$, and $R$ are collinear or not.
3. c) Use the slope-based approach to determine the value of the unknown coordinate $k$ such that the point $S(k, -10)$ is also collinear with them.

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 $20$ €, $500$ units are demanded, and it has been observed that for every increase of $2$ € 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-gradient 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.

In an operation in the Cantabrian Sea, Salvamento Marítimo detects two drifting boats at positions $A(-6, -2)$ and $B(2, 4)$ (in nautical miles relative to the control centre at $(0,0)$). A rescue helicopter must patrol along the perpendicular bisector of segment $AB$ to remain equidistant from both signals at all times.

1. a) Before performing analytical calculations, make a reasoned estimation: deduce what the sign of the slope of the perpendicular bisector must be and whether its y-intercept will be positive or negative, justifying your prediction from the positions of $A$, $B$, and their midpoint in the Cartesian plane.
2. b) Calculate the explicit equation ($y = mx + n$) of the perpendicular bisector and check whether the obtained values of $m$ and $n$ accurately confirm your deductions from the previous section.
3. c) The helicopter's coastal base is located at $C(6, -8)$. If the aircraft takes off following the straight path given by $4x + 3y = 0$, justify through slope analysis what geometric relationship this departure path has with the line directly connecting the two boats ($AB$) and with the patrol line (the perpendicular bisector).

Consider the fixed line $s: 4x - 6y + 5 = 0$ and a family of lines depending on a real parameter $k$, defined by $r_k: (k - 1)x + 2y - 4 = 0$.

1. a) Reasonably determine the value of $k$ for which the line $r_k$ is parallel to $s$.
2. b) Determine the value of $k$ for which the line $r_k$ is perpendicular to $s$.
3. c) A classmate claims: "If any third line $t$ is perpendicular to $s$, then $t$ must necessarily intersect $r_k$ regardless of the value that parameter $k$ takes". Reason geometrically and algebraically whether your classmate's claim is true or false, analysing what happens to the slope of $t$ compared to the cases calculated in sections a) and b).