Exercises: Point-gradient equation of the line

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

Calculate the slope of the line that passes through the points $A(-1, 5)$ and $B(2, 11)$. Then, write its equation in point-slope form using the point $A$.

Given the line with equation $y - 5 = -2(x + 1)$, state its slope and obtain the coordinates of a point it passes through directly from the formula.

Transform the point-slope equation $y + 3 = \frac{1}{2}(x - 4)$ into its slope-intercept form $y = mx + n$.

Write the point-slope equation of the line parallel to $y = 3x - 7$ that passes through the point $Q(-4, 1)$.

A plumber charges $\$30$ for the service call fee and $\$20$ for each hour of work. If we know that for a job of $2$ hours they charged a total of $\$70$, write the point-slope equation that relates the total cost to the hours worked, using this known point as a reference.

Look at the following graph. From the highlighted points, calculate the slope of the line and write its equation in point-slope form using the point $P$ as a reference.

Write the point-slope equation of the angle bisector of the first and third quadrants. Then, write the point-slope 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 for a rental of $3$ hours, $\$14$ is paid, and for a rental of $5$ hours, $\$20$ is paid. Determine the slope of the linear cost function and write its point-slope equation using the data from the first customer.

Find the point-slope equation of the line that is perpendicular to the line $y - 2 = \frac{3}{4}(x + 1)$ and passes through the origin.

Check analytically if the points $A(-2, 4)$, $B(1, -2)$ and $C(5, -10)$ are collinear. To do this, first find the point-slope equation of the line that passes through $A$ and $B$, and then verify if the coordinates of $C$ satisfy this equation.

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

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

The value of computer equipment depreciates linearly over time. If it was bought for $\$1200$ and after $4$ years its book value is $\$400$, write the point-slope equation that relates the value of the equipment to the years elapsed, using the point corresponding to the fourth year.

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

Determine the value of the parameter $k$ so 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-slope equation of the perpendicular bisector of the line 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-slope equation of the altitude corresponding to the 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-slope equation that allows converting degrees Celsius to Fahrenheit using the freezing point. Then, use it to calculate the temperature in Fahrenheit equivalent to the 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 this, 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, the demand falls by $50$ units. Write the point-slope equation of the demand using the first piece of data as a reference, and calculate at what price the demand will be zero.

A ray of light follows a straight-line trajectory given by the equation $y - 5 = -\frac{1}{2}(x - 2)$. Upon reaching the x-axis (which acts as a perfect plane mirror), the ray is reflected. Knowing that the angle of incidence is equal to 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-slope equation of the line that passes through the point $P(2, -1)$ and which, when intersecting the coordinate semi-axes (positive x-axis and negative y-axis), forms a right triangle of area $4$ square units.