Exercises: Linear equations with two unknowns

Check whether the pairs of values $(1, 2)$, $(3, -1)$ and $(0, 4)$ are solutions to the linear equation $3x + 2y = 7$.

Given the linear equation with two unknowns $5x - 2y = 10$, express the unknown $y$ in terms of $x$.

Find three distinct solutions to the linear equation $-2x + 4y = 8$.

Find the value that coefficient $c$ must take so that the pair of values $(-2, 5)$ is a solution to the equation $4x - 3y = c$.

Translate the following statement into algebraic language using a linear equation with two unknowns: "The price of three adult cinema tickets plus the price of two child tickets totals 34 euros".

State with reasons which of the following equations are linear equations with two unknowns:

• a) $x^2 + y = 7$

• b) $3x - \frac{y}{4} = 1$

• c) $x \cdot y = 12$

• d) $-x + 6y = 0$

Given the equation $7x + 3y = 21$, find the value of $y$ when $x = 0$, and the value of $x$ when $y = 0$.

Write two equations that are equivalent to $6x - 9y = 15$, obtaining one by multiplying all its terms by an integer and the other by dividing them.

In urban bus transport in a Spanish city, a single ticket costs $1,50$ € and a journey paid with a multi-trip card costs $0,80$ €. A group of friends spent exactly $12$ € in total by combining both payment methods during an afternoon.

1. a) Write the linear equation in two variables ($x$ = number of single tickets, $y$ = number of card journeys) representing the total expenditure.
2. b) Before doing exact calculations, estimate mentally whether it is possible that they purchased $10$ single tickets with that budget and justify your answer.
3. c) Check whether the pair of values with $x = 4$ and $y = 7,5$ is a numerical solution of the equation, and explain with reasoning why it cannot be a valid solution in the real-world context of this problem.

Mateo wants to simplify the linear equation in two variables $2x + 4y = 12$ to find solution pairs more easily. In his notebook he writes down the following:

"I divide by $2$ so the numbers are smaller: $x + 2y = 12$.

Now I choose the value $y = 3$ and substitute: $x + 2 \cdot 3 = 12 \implies x + 6 = 12 \implies x = 6$.

Therefore, the pair $(6, 3)$ is a solution to the original equation."

1. a) Identify the error Mateo made when looking for an equivalent equation and explain why his procedure violates the balance rule.
2. b) Check, by substituting into the original equation $2x + 4y = 12$, whether the pair $(6, 3)$ obtained by Mateo is truly a valid solution.
3. c) Write the correct simplified equation and find a pair $(x, y)$ that is indeed a real solution.

In the linear equation $4x - y = 18$, find the solution in which both unknowns have exactly the same value (that is, $x = y$).

An exam consists of multiple-choice questions worth 2 points each and open-ended problems worth 5 points each. Set up a linear equation that relates the number of questions of each type to obtain a maximum score of 30 points. Find a solution in which the number of multiple-choice questions is exactly five times the number of open-ended problems.

Find a linear equation of the form $ax + by = c$ that has the pairs of values $(2, 3)$ and $(4, -1)$ as solutions.

Observe the straight line represented in the following Cartesian plane.

Determine the linear equation with two unknowns corresponding to this line, expressing it in the form $ax + by = c$ with integer coefficients. From the equation obtained, find the value of $y$ when $x = 12$.

A bicycle rental company charges a fixed insurance fee plus an amount for each hour of use. If renting a bicycle for 3 hours costs 14 euros and for 5 hours costs 22 euros, set up a linear equation relating the total cost ($y$) to the hours of use ($x$). What is the value of the fixed fee?

Given the linear equation with fractional coefficients $\frac{x - 1}{2} + \frac{2y + 3}{5} = 1$, express it in its general form $ax + by = c$ by clearing the denominators. Then, find a solution in which $x$ is a negative integer.

Calculate the area of the triangle formed by the graphical representation of the equation $8x + 5y = 40$ when it intersects the coordinate axes $X$ and $Y$.

Determine the values of the coefficients $a$ and $b$ of the linear equation $ax + by = 12$ knowing that the line passes through the point $(2, -1)$ and that, when rearranging for the unknown $y$, the gradient of the resulting line is $4$.

A line drawn in a Cartesian plane contains all the solutions of a linear equation in two variables of the form $ax + by = c$, where the coefficients $a$, $b$, and $c$ are integers with no common factors.

1. a) Determine the values of $a$, $b$, and $c$ knowing that the line passes through the point $(2, -1)$ and crosses the horizontal axis at the point $(4, 0)$. Write the resulting linear equation.
2. b) Using your equation, find a solution $(x, y)$ different from the previous two in which the value of $y$ is a positive integer.
3. c) Justify how many solutions this equation has in total and explain why they geometrically correspond to the points on a straight line.

In a Spanish municipality, a discount on the waste collection tax is applied to reward waste sorting. According to the municipal program, each kilogram of recycled light packaging (yellow bin, $x$) generates a savings of $0,30$ €, and each kilogram of paper and cardboard (blue bin, $y$) represents a savings of $0,20$ €. A community of neighbours has set a goal to achieve an exact total savings of $60$ € in one month.

1. a) Write the linear equation in two variables that describes the relationship between the recycled kilograms of each waste type and the target savings.
2. b) Calculate the two intercepts with the axes: when no cardboard is recycled ($y = 0$) and when no packaging is recycled ($x = 0$). Explain the meaning of each of these two cases in the daily life of the community.
3. c) A spokesperson for the neighbours claims that by contributing $100\text{ kg}$ of packaging and $150\text{ kg}$ of cardboard they will achieve exactly the $60$ € discount. Justify using calculations whether the statement is true or false.

The following figure shows three lines corresponding to three linear equations with two unknowns of the form $ax + by = c$.

Match each line (A, B and C) to one of the following conditions on its coefficients, justifying your answer mathematically:

• Condition 1: $a > 0$, $b > 0$, $c > 0$

• Condition 2: $a > 0$, $b < 0$, $c = 0$

• Condition 3: $a = 0$, $b > 0$, $c > 0$

An electronics shop sells USB flash drives for €14 and SD cards for €9. A school has spent exactly €195 on these devices. Write the corresponding linear equation and find ALL possible combinations of flash drives and cards they could have bought, knowing that they have purchased at least one of each type.

Show with clear reasoning that the linear equation $15x + 21y = 43$ cannot have any solution in which $x$ and $y$ are both integers. (Hint: look at the divisibility of the coefficients).

Two water pumps fill a tank. The first pumps $x$ litres per minute and the second $y$ litres per minute. If we turn on the first for 12 minutes and the second for 18 minutes, the tank receives 420 litres. If we turn on both simultaneously for 15 minutes, the tank receives 375 litres. Set up a linear equation for each situation, simplify them, and determine whether they represent equivalent equations.

The relationship between temperature in degrees Celsius ($C$) and degrees Fahrenheit ($F$) is given by a linear equation with two unknowns. Knowing that water freezes at $0^\circ \text{C}$ (which is equivalent to $32^\circ \text{F}$) and boils at $100^\circ \text{C}$ (which is equivalent to $212^\circ \text{F}$), find the general equation relating them. Is there any temperature at which both scales show exactly the same numerical value? If so, calculate it.

Given the parametric linear equation $(k-1)x + (k+2)y = 3k$, find the value of the parameter $k$ so that the corresponding line is parallel to the y-axis ($Y$). For that value of $k$, what is the resulting equation and what feature do all its solutions share?

A farmer wants to bottle exactly 173 litres of olive oil using 5-litre containers and 3-litre bottles. Express this situation using a linear equation with two unknowns. If he wants to use the smallest total number of containers possible without having any oil left over or missing, how many containers of each type should he use?

Consider the linear equation $ax + by = c$ with integer coefficients. If we know that the point $(p, q)$ is an integer solution of the equation (that is, $a \cdot p + b \cdot q = c$), prove algebraically that any point of the form $(p + n \cdot b, q - n \cdot a)$, where $n$ is any integer, is also a solution of the equation.

To plot the linear equation $3x - 5y = 30$ on a grid and find integer solutions, two students propose solving for a different variable:

• Lucía's method: solve for $x$ in terms of $y$, obtaining $x = \frac{30 + 5y}{3}$.

• Carmen's method: solve for $y$ in terms of $x$, obtaining $y = \frac{3x - 30}{5}$.

1. a) Analyse the denominators of both expressions and justify which of the two methods is simpler and faster for generating integer values without decimals appearing, relying on the divisibility rules for 3 and 5.
2. b) Using the method you selected as more efficient, calculate three pairs of integer solutions (positive, negative, or zero) that are not the intercepts with the axes.
3. c) Using the expression from Carmen's method, reason mathematically why every time the variable $x$ increases by $5$ units, the variable $y$ must necessarily increase by $3$ units for the equality to hold.

According to data from the Ministry for the Ecological Transition on emissions in urban transport, a petrol car emits an average of $0,12\text{ kg}$ of $\text{CO}_2$ per kilometre travelled, while a diesel delivery van emits $0,18\text{ kg}$ of $\text{CO}_2$ per kilometre. A local courier company wants to set a combined daily limit of exactly $36\text{ kg}$ of $\text{CO}_2$ between the kilometres travelled by its car ($x$) and those of its van ($y$).

1. a) Write the linear equation in two variables that models this environmental situation.
2. b) Transform the equation above into an equivalent equation with the smallest possible integer coefficients. Explain why both equations share exactly the same solution pairs.
3. c) If during one working day the car has travelled $150\text{ km}$, use the equivalent equation to calculate how many kilometres the van can travel to meet precisely the set emissions quota.
4. d) The company is evaluating a working day in which the van always covers twice the distance of the car ($y = 2x$). Calculate how many kilometres each vehicle should travel to exhaust the set quota, and reason whether these distances are reasonable and feasible for an 8-hour urban delivery day.