Point-Slope Equation of a Line
A plumber charges $\$600$ for coming to your house and $\$400$ for each hour of work. If you know that for 2 hours the plumber charged $\$1{,}400$, can you write the cost rule without first calculating how much is charged for 0 hours? With the point-slope equation, yes: a point and the slope are enough. At the end of the two-points section we come back to this case.
Deriving the point-slope equation
The form $y=mx+b$ requires knowing the y-intercept $b$, and we don't always have that information. What we are usually given is a point on the line and its steepness.
If a line has slope $m$ and passes through $P(x_1, y_1)$, any other point $(x, y)$ on the line satisfies, by the definition of slope, $$m = \frac{y-y_1}{x-x_1}$$ Multiplying both sides by $(x-x_1)$ gives the point-slope equation.
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 $m = 4$.
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.
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…
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.
Forms of the equation of a line
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 $y = mx + n$.
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…
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.
Intercepts with the axes and triangles
The points where a line crosses the axes help you graph it and calculate areas. The idea is simple: on the $x$-axis the height is zero, and on the $y$-axis the horizontal displacement is zero.
If the intercepts are $(a, 0)$ and $(0, c)$, the legs of the triangle measure $|a|$ and $|c|$ and its area is $\dfrac{|a|\cdot|c|}{2}$. When a problem gives the area and a point on the line, you set up two conditions: the line passes through the given point and the area equals the stated value.
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…
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.
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…
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.
Line determined by two points
If you know $A(x_1, y_1)$ and $B(x_2, y_2)$ with $x_1\neq x_2$, the procedure is always the same: first the slope, then point-slope with either of the two points.
This method also works for real-life models: if something changes at a constant rate (a cost per hour, depreciation, a temperature), the slope is that rate.
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 $A$.
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…
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…
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 $280$ pesos and a…
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.
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…
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.
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 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 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 $37^\circ C$.
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…
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.
Parallel and perpendicular lines
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 $Q(-4, 1)$.
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.
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 $N(4, 7)$.
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 $N(4, 7)$.
Intersection of two lines
Two lines that are not parallel meet at a single point. That point satisfies both equations at once, so its coordinates are the solution of a system of two equations with two unknowns. Here we use the elimination (addition and subtraction) method; in the section on systems you will see all the methods.
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 $s: x - 3y + 3 = 0$.
Applications: distance, collinearity, and perpendicular bisector
The altitude of a triangle follows the same idea: it is perpendicular to the opposite side and passes through the vertex. And if a ray reflects off the $x$-axis, the slope changes sign and the line passes through the point of impact, which is where the path meets the $x$-axis (set $y=0$).
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.
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…
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.
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.
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 $C$.
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…
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.