Exercises: Equation of a line in point-slope form

Write the equation of the line in point-slope form that passes through the point $A(3, -2)$ and has a gradient of $m = 4$.

Given the line with equation $y - 5 = -2(x + 1)$, state an exact point it passes through and its gradient. Then, rearrange the expression to write it in explicit form.

Calculate the gradient of the line passing through the points $P(-1, 4)$ and $Q(2, 10)$ and write its equation in point-slope form using the coordinates of point $P$.

Look at the line represented on the Cartesian plane. Identify its gradient and an exact point with integer coordinates through which it passes, and write its equation in point-slope form.

Write the equation in point-slope form of the line parallel to $y = 3x - 7$ passing through the point $B(-4, 1)$.

A line has the point-slope equation $y + 3 = \frac{1}{2}(x - 6)$. Check mathematically whether the point $C(8, -2)$ lies on this line.

Determine the equation in point-slope form of the angle bisector of the first and third quadrants, taking the origin as the reference point.

The water level of a reservoir increases steadily at a rate of $15$ cm per day. If on the $10$ day of measurement the level was $420$ cm, write the point-slope equation relating the water level $y$ to the number of days elapsed $x$.

To calculate the equation of an accessible ramp starting at point $A(-3, 1)$ with a downward slope of $5\,\%$ ($m = -0,05$), David writes in his notebook:

1. a) Identify the two conceptual mistakes David made when substituting the coordinates of the point into the point-slope formula $y - y_1 = m(x - x_1)$.
2. b) Write the correct equation in point-slope form.
3. c) Convert the equation to slope-intercept form ($y = mx + n$) and algebraically verify that the coordinates of the starting point $A(-3, 1)$ satisfy the equation.

In the urban taxi service of Zaragoza (weekday daytime fare), the taximeter applies a fixed cost per kilometer traveled of $m = 1,10$ €/km. At the end of a $9$ km ride from Plaza del Pilar to an industrial park, the taximeter shows a total fare of $13,05$ €.

1. a) Identify the independent variable ($x$) and the dependent variable ($y$), indicating their units. Express the trip data as a known point $P(x_1, y_1)$ and the slope $m$.
2. b) Write the equation that models the total cost of the trip in point-slope form.
3. c) Convert the equation to its slope-intercept form ($y = mx + n$) and interpret the real economic meaning of the y-intercept $n$ within the taxi service.

An HVAC technician notes that $2$ hours after turning on an industrial system, the room temperature is $18^\circ$C, and after $5$ hours it is $10,5^\circ$C. Assuming that the temperature drop follows a linear model, write the point-slope equation determining the temperature $y$ as a function of time $x$.

Find the equation in point-slope form of the line that is perpendicular to $y = -\frac{1}{3}x + 5$ and passes through the point of intersection of the lines $y = 2x - 1$ and $y = x + 3$.

A van hire company charges a fixed amount plus a constant rate per kilometre driven. If for a journey of $50$ km they charge $65$ € and for one of $120$ km they charge $107$ €, determine the gradient of the cost function and write its equation in point-slope form.

Given the line $y - k = 3(x + 2)$, find the value of the parameter $k$ so that the line passes exactly through the point $A(1, 4)$.

Write the equation in point-slope form of the perpendicular bisector of the segment with endpoints $A(-2, 5)$ and $B(4, 1)$. Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint.

A cyclist maintains a constant speed during his training session. At minute $15$ of his ride he is at the $6$ km marker on the road, and at minute $40$ he reaches kilometre $16$. Express the cyclist's position in point-slope form and calculate which kilometre marker he will be at in minute $60$.

Write the point-slope equation of the line that intercepts the x-axis at $x = -3$ and the y-axis at $y = 6$. Express the result using only the x-intercept as a reference point.

Determine the equation in point-slope form of the line parallel to $2x - 5y + 10 = 0$ passing through the midpoint of the segment with endpoints $P(-1, -4)$ and $Q(7, 2)$.

On a sunny spring day in Granada, the AEMET station records at 14:00 h a temperature of $22,4^\circ$C at an altitude of $680$ m. According to the usual vertical lapse rate in the troposphere, the temperature decreases linearly at a constant rate of $0,65^\circ$C for every $100$ m of altitude gained (that is, with a slope $m = -0,0065^\circ$C/m). A group of hikers ascends to the Poqueira mountain refuge, located at an altitude of $2500$ m.

1. a) Estimate mentally before calculating: knowing that the climb is about $1800$ m and the temperature drops by about $0,65^\circ$C every $100$ m, do you expect the temperature at the refuge to be around $20^\circ$C, $10^\circ$C, or $0^\circ$C? Briefly justify your estimate.
2. b) Take the reference point $A(680; 22,4)$ and write the equation in point-slope form expressing the temperature $y$ (in $^\circ$C) as a function of the altitude $x$ (in meters).
3. c) Using the equation, calculate the exact temperature at $2500$ m and compare the result with your initial estimate to judge whether it was reasonable.

On the layout plan of a logistics zone, a straight service road must be perpendicular to the main road connecting points $A(1, 2)$ and $B(5, 4)$. Additionally, the auxiliary road must satisfy two technical conditions: it must pass through an access point $P(3, y_1)$ and it must intersect the x-axis (where $y = 0$) at a point with horizontal coordinate $x \ge 4$.

1. a) Find the slope $m_{AB}$ of the main road and determine the exact slope $m$ that the perpendicular service road must have.
2. b) Write the equation of the service road in point-slope form, keeping the y-coordinate $y_1$ of point $P$ as an unknown.
3. c) Algebraically deduce which values the y-coordinate $y_1$ can take so that the x-intercept satisfies the condition $x \ge 4$. Choose a specific valid value for $y_1$ and write the final point-slope equation of the road.

An access ramp must have an exact gradient of $8\%$ to comply with building accessibility standards. If the ramp starts at ground level at point $A(0, 0)$ and must negotiate an incline to reach a platform located at point $B(x_0; 1,2)$ (with coordinates measured in metres), write the equation of the line of the ramp in point-slope form and determine the horizontal coordinate $x_0$.

In the triangle with vertices $A(-2, -1)$, $B(4, 1)$ and $C(1, 5)$, find the equation in point-slope form of the line containing the altitude corresponding to vertex $C$.

Given the lines $r_1: y - 2 = a(x + 1)$ and $r_2: y - b = 3(x - 2)$, find algebraically the values of the parameters $a$ and $b$ such that both expressions represent coincident lines.

A linear economic supply model shows that at a price of $10$ €, $1000$ units of a product are supplied, and for every euro the price increases, supply increases by $150$ units. On the other hand, the demand for the same product is given by the point-slope equation $y - 1200 = -100(x - 15)$, where $y$ represents the number of units and $x$ is the price in euros. Write the point-slope equation for supply and find the market equilibrium point (price and quantity).

A laser light ray starts at point $P(-4, 5)$, strikes a flat mirror perfectly aligned with the x-axis, and is reflected passing through point $Q(6, 3)$. Write the equation in point-slope form of the line describing the reflected ray.

Write the point-slope equation of the line passing through the points $(a, 0)$ and $(0, b)$ (with $a \neq 0$ and $b \neq 0$). Then, transform it algebraically step by step to obtain the intercept form equation of the line $\frac{x}{a} + \frac{y}{b} = 1$.

Find the point-slope equation of the line that forms an angle of $135^\circ$ with the positive x-axis and which, together with the coordinate axes, determines a triangle of area $8\text{ u}^2$ located entirely in the first quadrant.

An industrial tank initially contains $500$ litres of a liquid chemical compound. A valve is opened that drains the tank at a constant rate. Knowing that the equation modelling the volume $V$ (in litres) as a function of time $t$ (in minutes) can be expressed in point-slope form as $V - 200 = m(t - 15)$, calculate the gradient $m$ and determine how long it will take for the tank to empty completely.

In a water supply network, a secondary pipe is planned to run perpendicular to the main pipeline passing through points $A(1, 3)$ and $B(7, 6)$. The new pipe must necessarily start from an inspection chamber located at $C(3, 8)$. Two engineers consider two methods to obtain its equation:

• Method 1 (Point-slope equation): Calculate the slope $m_1$ of the main pipeline, deduce the perpendicular slope $m_2 = -1/m_1$, and substitute the coordinates of chamber $C(3, 8)$ directly into the formula $y - y_1 = m_2(x - x_1)$.

• Method 2 (Determining the y-intercept): Calculate $m_2$, write the theoretical slope-intercept form $y = m_2 x + n$, substitute $x = 3$ and $y = 8$ to set up and solve a linear equation for $n$, and reconstruct the final equation.

1. a) Apply both methods step by step and verify that they both lead to the exact same equation in slope-intercept form.
2. b) If during civil works the chamber is moved $2$ meters north (now located at $C'(3, 10)$) while keeping the same perpendicular slope, justify which of the two methods allows updating the mathematical model more quickly and directly.
3. c) During a routine inspection, an illegal tap is found at point $D(5, 4)$. Algebraically justify whether this tap is connected to the calculated secondary pipe or not.

On the Madrid-Valencia high-speed railway line, an AVE train travels at a constant cruising speed on open track. The telemetry system records that at 12:15 h ($x_1 = 15$ minutes elapsed since 12:00 h) the train passes kilometer mark $65$ of the line ($y_1 = 65$ km), and at 12:45 h ($x_2 = 45$ minutes) it crosses kilometer mark $185$ ($y_2 = 185$ km).

1. a) Write the equation in point-slope form that models the kilometer position $y$ as a function of elapsed time $x$ (in minutes after 12:00 h), using the first telemetry record as the reference point.
2. b) Interpret the physical meaning of the slope $m$ of the model: calculate its value in km/min, convert it to km/h by multiplying by $60$, and assess whether this speed is consistent with commercial limits for the AVE in Spain (legal maximum of $300$ km/h).
3. c) Convert the equation to slope-intercept form ($y = mx + n$) and calculate the value of $y$ for $x = 0$ (at 12:00 h). Explain why this mathematical value does not describe the train's actual position at the station of origin and what this implies regarding the domain of validity of a linear model.
4. d) Knowing that the Valencia Joaquín Sorolla terminus station is at kilometer mark $385$, calculate at what time the train will arrive if it maintains this cruising speed without interruption until its final approach.

Topic challenge. How much do you really pay for electricity? Analyze your household bill.

1. Data collection: Take a real electricity bill from your home (or check the standing charge and the average price per $kWh$ on the Red Eléctrica de España, REE website). Identify the fixed cost in euros ($n$) and the variable price for each $kWh$ consumed ($m$, the slope). Reason whether it is an affine, linear, or constant function.

2. Point-slope modeling: Take the actual consumption for the month to set a working point $P(x_0, y_0)$, where $x_0$ is the $kWh$ used and $y_0$ is the resulting total cost. Write the equation of the line in point-slope form: $y - y_0 = m(x - x_0)$. Use it to predict the bill if your household manages to reduce consumption by $20\,\%$.

3. Graphing and report: Enter the function into GeoGebra or a spreadsheet to plot the graph. Prepare a short report (one page) explaining the real-world meaning of the slope and proposing two justified measures to lower the bill.

Assessment criteria:

• Correct identification of real data ($m$ and $n$) and classification of the function type.

• Proper application of the point-slope form and accuracy in calculations.

• Skill in digital graphical representation (GeoGebra or spreadsheet) and use of units (€ and $kWh$).

• Clarity and realism of the report's conclusions and proposals.