Exercises: Systems of linear equations

Check whether the pair of values $x = 3$, $y = -2$ is a solution to the following system of equations. Justify your answer by showing your working.

Solve the following system of linear equations using the substitution method:

Apply the equating method to find the solution to the following system:

Use the elimination method to solve the following system of equations:

Look at the following graphical representation of a system of two linear equations with two unknowns. Determine the solution to the system from the graph and classify it according to its number of solutions.

The difference between two numbers is $14$ and twice the larger number plus the smaller number is equal to $55$. Set up a system of equations and calculate the value of both numbers.

On a farm there are cows and chickens. If in total there are $35$ heads and $110$ legs, set up a system of linear equations to find out how many animals of each kind there are.

Calculate the value of the coefficients $a$ and $b$ so that the pair $(x, y) = (2, -1)$ is the solution to the following system of equations:

Marcos wants to classify the following system of linear equations without solving it numerically:

In his notebook he has written: "I check the ratios of the coefficients: $\dfrac{2}{4} = \dfrac{-1}{-2} = \dfrac{1}{2}$. Since the two quotients match, both lines are the same and the system is consistent dependent (it has infinitely many solutions)".

1. a) Find the conceptual error made by Marcos and explain why his analysis is incomplete, relying on the relationship between the ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, and $\dfrac{c_1}{c_2}$.
2. b) Correct the classification: how many solutions does this system actually have and what relative position do the two lines hold in the Cartesian plane?

In a cinema in Valladolid, the regular ticket price is $8,50$ € and the discounted ticket price with a Youth Card (Carné Joven) is $6,00$ €. A group of 10 friends went to the cinema on Friday and paid a total of $67,50$ € at the box office for their tickets.

1. a) Clearly identify the unknowns $x$ and $y$ and write the system of two linear equations that models this purchase situation.
2. b) Without solving the system algebraically, check by substitution whether the group could have consisted of 5 people with regular tickets and 5 with Youth Cards.
3. c) Solve the system using the algebraic method of your choice and determine exactly how many friends bought each type of ticket.

Solve the following system of linear equations, simplifying brackets and denominators first:

$4$ years ago, a father's age was four times his daughter's age. In $8$ years' time, the father's age will be double that of his daughter. Set up a system of equations and determine their current ages.

A perfumery mixes two types of perfume: one costs $12$ €/litre and the other $20$ €/litre. They want to obtain a $50$-litre mixture with a final price of $15$ €/litre. How many litres of each type of perfume should they use?

Classify the following systems as consistent with a unique solution, consistent with infinitely many solutions, or inconsistent, by analysing the ratio of their coefficients, without solving them. Justify your answer in each case.

1. a) $ \begin{cases} 3x - 2y = 5 \\ -6x + 4y = -10 \end{cases} $
2. b) $ \begin{cases} x + 4y = 7 \\ 2x + 8y = 12 \end{cases} $
3. c) $ \begin{cases} 5x - y = 3 \\ 2x + 3y = 8 \end{cases} $

A freight train leaves a station at a constant speed of $80$ km/h. One and a half hours later, a passenger train leaves the same station in the same direction at $120$ km/h. Set up a system of equations to find out how long it will take the passenger train to catch up with the freight train and at what distance from the station this will happen.

Solve the following system of three equations with three unknowns using the elimination method (Gaussian elimination):

In a multiple-choice exam of $50$ questions, $3$ points are awarded for each correct answer and $1$ point is deducted for each incorrect or blank answer. If a student scored $98$ points in total, how many questions did they answer correctly and how many did they get wrong or leave blank?

A rectangle has unknown dimensions. If the base is increased by $3$ cm and the height is decreased by $2$ cm, the area of the rectangle does not change. On the other hand, if the base is decreased by $2$ cm and the height is increased by $4$ cm, the area increases by $16$ cm$^2$. Set up a system of linear equations to calculate the original dimensions of the rectangle.

To solve the system of equations:

three classmates suggest different approaches:

• Lucía proposes solving for $x$ in the first equation and applying the substitution method.

• Sergio suggests directly subtracting the second equation from the first to apply the elimination method.

• Raquel prefers solving for the unknown $y$ in both equations and drawing both lines with a ruler on coordinate axes to locate the point of intersection.

1. a) Compare the three proposals and justify which one is more direct, faster, and carries less risk of calculation errors. Justify your choice by looking at the coefficients of the variables.
2. b) State what technical limitation Raquel's graphical method would have if the solution to the system were a pair of non-integer numbers (for example, with fractions or repeating decimals).
3. c) Solve the system using the algebraic method that you considered most convenient in part a).

According to data from the INE's Hotel Occupancy Survey, rural accommodations experience occupancy peaks during holiday long weekends. A rural hotel in La Palma has 50 rooms in total, divided between 'standard' (at $70$ € per night) and 'superior with terrace' (at $100$ € per night). During a holiday weekend when it had $100\,\%$ occupancy, revenue per night reached $4.100$ €.

1. a) Before performing any algebraic calculations, make a reasoned estimate of whether the hotel has more standard rooms or more superior rooms. Hint: calculate mentally or on scrap paper what the hotel would have made if all rooms were standard and what it would have made if all rooms were superior.
2. b) Set up and solve the system of linear equations to calculate the exact number of rooms of each category the establishment has.
3. c) Check whether the exact result matches the estimate you made in part a) and calculate what percentage of the total is represented by the superior rooms.

Discuss the following system of equations in terms of the parameter $k$. Determine for which values of $k$ the system has no solution and for which it has a unique solution.

The sum of the three digits of a number is $15$. The tens digit is the arithmetic mean of the other two digits. If the order of the digits is reversed, the number obtained exceeds the original by $396$. Set up a system of three equations with three unknowns and find the original number.

Two workers, working together, can finish a job in $12$ days. If the first worker works for $3$ days and the second for $8$ days, they complete exactly $\frac{5}{12}$ of it. Set up a system of equations (hint: use the auxiliary variables $u = \frac{1}{x}$, $v = \frac{1}{y}$) to calculate how many days each worker would take to complete the entire job working alone.

Solve the following non-linear system of equations by performing a suitable change of variable (for example, $A = \frac{1}{x+1}$, $B = \frac{1}{y-2}$):

An investor divides their capital of €$60\,000$ into three investment funds: A, B and C. Fund A yields an annual return of $4\%$, fund B of $6\%$ and fund C of $9\%$. Knowing that they invested the same amount in fund C as in funds A and B combined, and that the total annual profit was €$4\,050$, set up a system of three linear equations and calculate how much money was allocated to each fund.

The perimeter of an isosceles triangle is $40$ cm. If the base were increased by $25\%$ and the equal sides were decreased by $10\%$, the triangle would become equilateral. Set up a system of equations to determine the lengths of the sides of the original triangle.

Solve the following system of three equations with three unknowns, carefully working with the fractions before applying Gaussian elimination or another method of solution:

A boat travels on a river. When travelling downstream, it covers a distance of $60$ km in $3$ hours. When travelling upstream, it takes $5$ hours to cover the same distance. Set up a system of equations to calculate the speed of the boat in still water and the speed of the river current.

Design a system of two linear equations with unknowns $x$ and $y$ yourself, fulfilling the following conditions step by step:

1. a) Write a first linear equation whose coefficients $a_1$ and $b_1$ are non-zero integers different from $1$ and $-1$, such that it holds for the particular solution $x = 3$, $y = -1$.
2. b) Construct a second equation that forms, together with the first, a consistent independent system whose only solution is the pair $(3, -1)$. Prove that the solution is unique by checking the condition of the ratios $\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$.
3. c) Modify only the coefficients of the unknowns ($a_2$ and $b_2$) of your second equation to transform the system into an inconsistent one. Prove that the condition $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$ holds and explain what adjustment you would have to make to the constant term $c_2$ so that the system admits infinitely many solutions.

In Spain, the regulated domestic electricity tariff (2.0TD) divides daily consumption into two main energy periods: peak hours (expensive rate: $0,22$ €/kWh) and off-peak hours (cheap rate: $0,12$ €/kWh). During the month of October, a family from Toledo consumed a total of $280$ kWh and the amount corresponding to the billed energy was $45,60$ €.

1. a) Set up the system of linear equations and calculate how many kWh the family consumed during peak hours and how many during off-peak hours in October.
2. b) In November, the family turned on the storage heaters and their total consumption rose to $300$ kWh (a higher consumption than the previous month). However, by programming their appliances for nighttime hours, the energy charge on their bill dropped to $44,00$ €. Set up the new system and determine the peak and off-peak consumption for November.
3. c) Calculate the percentage of off-peak consumption the household had in October and in November. Based on those percentages, reason mathematically how it is possible that, consuming $20$ kWh more in total in November, they managed to pay less on their bill.