Exercises: Types of linear functions. First-degree polynomial.

Identify the gradient $m$ and the y-intercept $n$ of the following linear functions:

1. a) $y = -4x + 7$
2. b) $y = \frac{2}{5}x$
3. c) $y = -3$

Write the equation of the line that has a gradient of $5$ and whose y-intercept is $-2$.

Classify the following functions into direct proportion (strictly linear) function, linear function (affine function), or constant function:

1. a) $f(x) = 3x - 1$
2. b) $g(x) = \frac{x}{4}$
3. c) $h(x) = 8$

Find the equation of the line that has gradient $m = -2$ and passes through the point $(1, 4)$.

A taxi fare is given by the function $C(x) = 30x + 60$, where $C(x)$ is the cost in pesos and $x$ is the distance traveled in kilometers. How much will a trip of $8$ kilometers cost?

Calculate the points of intersection with the coordinate axes (x-axis and y-axis) of the line $y = 3x - 12$.

Check algebraically whether the points $A(2, 5)$ and $B(-1, -3)$ lie on the graph of the function $y = 2x + 1$.

From the following graphical representation, determine the gradient, the y-intercept and the equation of the line.

Find the equation of the line passing through the points $A(-2, 5)$ and $B(4, -7)$.

Find the equation of the line that is parallel to $y = 3x - 2$ and passes through the point $(-1, 4)$.

A plumber charges $600$ pesos for a call-out fee, plus $300$ pesos for each hour of work. Write the algebraic expression of the function relating the cost to the time worked and calculate how much he will charge for a repair taking $3.5$ hours.

A water tank initially contains $800$ liters and is being emptied at a constant rate of $20$ liters per minute. Write the function relating the volume of water in the tank to the time elapsed and calculate in which minute the tank will be completely empty.

Find algebraically the point of intersection of the lines $y = 4x - 5$ and $y = -2x + 7$. Check that your result matches the point indicated on the following graph.

Determine the value of the parameter $k$ such that the line with equation $y = kx + 5$ passes through the point $(3, -4)$.

The relationship between temperature in degrees Celsius ($^\circ C$) and degrees Fahrenheit ($^\circ F$) is linear. Water freezes at $0^\circ C$, which is equivalent to $32^\circ F$, and boils at $100^\circ C$, which is equivalent to $212^\circ F$. Find the linear function to convert degrees Celsius ($x$) to degrees Fahrenheit ($y$).

Calculate the area of the triangle formed by the line $y = -\frac{3}{4}x + 3$ intersecting the coordinate axes in the first quadrant.

Find the equation of the line that passes through the point $(0, 5)$ and forms a right-angled triangle with the coordinate axes of area $10$ square units in the first quadrant.

A sneaker factory has monthly fixed costs of $100\,000$ pesos. In addition, producing each pair of sneakers costs $300$ pesos. If the company sells each pair for $800$ pesos, write the cost $C(x)$ and revenue $R(x)$ functions. How many pairs of sneakers must they sell per month to make a profit of at least $40\,000$ pesos?

The line $r$ passes through the points $(1, 2)$ and $(3, 8)$. The line $s$ is parallel to $r$ and its y-intercept is half the y-intercept of $r$. Determine the equation of the line $s$.

A parking lot charges $50$ pesos for the first hour (or fraction thereof). After the first hour, it charges $1$ peso for each additional minute. Write the piecewise function $C(t)$ representing the total cost in pesos as a function of time $t$ in minutes that a vehicle is parked.

The length $L$ of an elastic spring, measured in centimeters, is given by the function $L(p) = 15 + 0.2p$, where $p$ is the suspended weight in grams. Find the expression for the inverse function $p(L)$ and use it to calculate what weight was suspended if the spring measures exactly $22$ cm.

The following graph shows the distance of a cyclist from home as a function of time. Determine the piecewise algebraic expression of the function $d(t)$ and calculate the constant speed of the cyclist during the return journey.

Given the lines $y = (2m - 1)x + 5$ and $y = 3x - n$, determine the values of the parameters $m$ and $n$ such that both equations represent exactly the same line (coincident lines).

Find the values of $a$ and $b$ such that the line $y = ax + b$ simultaneously satisfies two conditions: being parallel to the line $y = \frac{1}{2}x$ and passing through the point of intersection of the lines $y = 2x - 4$ and $y = -x + 5$.