Exercises: Systems of linear equations

Check whether the pair of values $(x, y) = (2, -1)$ is a solution to the following system of linear equations:

$3x + 2y = 4$

$x - 4y = 6$

Solve the following system of equations using the substitution method:

$x = 2y + 5$

$3x - 4y = 11$

Solve the following system of equations using the equating method:

$y = 3x - 2$

$y = -x + 6$

Solve the following system of equations using the elimination method:

$5x + 3y = 21$

$2x - 3y = 0$

Look at the graphical representation of two lines that form a system of linear equations. State the solution to the system and classify it according to its number of solutions (consistent independent, consistent dependent, or inconsistent).

The sum of two numbers is $54$ and their difference is $12$. Set up a system of linear equations and find the value of both numbers.

In a café, $3$ coffees and $2$ croissants cost $7,50$ €. If we buy $1$ coffee and $4$ croissants of the same type, the price is $6,50$ €. How much does each coffee and each croissant cost?

Solve the following system of equations using whichever algebraic method you consider most appropriate:

$4x - y = 7$

$2x + y = 11$

Marcos tries to solve the following system of linear equations using the elimination method:

$\begin{cases} 3x - 2y = 8 \\ 5x + 4y = 6 \end{cases}$

To eliminate the unknown $y$, he decides to multiply the first equation by $2$ and writes in his notebook:

$6x - 4y = 8$

Then, he adds both equations term by term, obtaining $11x = 14$, from which he concludes that $x = \frac{14}{11}$.

1. a) Accurately identify what mistake Marcos made and explain why his step invalidates the equivalence of the system based on the algebraic properties of equalities.
2. b) Correct his work and find the exact solution $(x, y)$ of the original system.

Carefully analyze the following two systems of linear equations:

System 1: $\begin{cases} x = 2y - 5 \\ 3x + 4y = 5 \end{cases}$

System 2: $\begin{cases} 5x + 3y = 19 \\ 2x - 3y = 2 \end{cases}$

1. a) Without solving them yet, reasonably justify which of the three usual algebraic methods (substitution, equating coefficients, or elimination) is faster and more efficient for System 1, and which for System 2, considering the coefficients of the unknowns.
2. b) Solve System 2 using the method you selected in the previous section and verify that the solution obtained satisfies both equalities.

Solve the following system of linear equations, first expanding to remove the brackets:

$2(x - 3) + 3(y + 1) = 4$

$x - 4(y + 2) = -10$

Solve the following system, first clearing the denominators:

$\frac{x}{2} + \frac{y}{3} = 4$

$\frac{x}{3} - \frac{y}{2} = \frac{1}{2}$

$5$ years ago, a mother was three times as old as her son. In $10$ years, the mother's age will be double the son's age. Set up a system of linear equations to calculate their current ages.

The perimeter of a rectangular plot is $120$ metres. If the length of the plot is $10$ metres longer than the width, what are the dimensions of the plot?

A merchant mixes olive oil at $4$ €/litre with sunflower oil at $2$ €/litre to obtain $60$ litres of a mixture to sell at $3,50$ €/litre. Set up a system of equations to find out how many litres of each type of oil were used.

In a money box there are $40$ coins, some of $0,20$ € and others of $0,50$ €. If the total amount is $12,50$ €, how many coins of each type are in the money box?

Plot the graph of the following system of linear equations and state its solution from the graph:

$x + y = 5$

$2x - y = 4$

You can do it on paper and send a photo with your answer.

In a vehicle repair shop there are $35$ vehicles in total, including cars and motorbikes. If we count all their wheels (excluding spare tyres), there is a total of $110$ wheels. How many cars and how many motorbikes are in the shop?

In an auditorium in Zaragoza, all 150 available tickets have been sold for a charity play. There were two types of tickets: general admission at $18\text{ €}$ and student discounted tickets at $10\text{ €}$. The final box office revenue was $2220\text{ €}$.

1. a) Before setting up any equation, make a reasoned estimate: were more general admission or discounted tickets sold? To justify it, calculate what the revenue would have been if all 150 tickets had been $10\text{ €}$ or all $18\text{ €}$.
2. b) Set up a system of two linear equations with two unknowns that models the situation and solve it using whichever method you consider most convenient.
3. c) Check whether the number of tickets found for each type matches your previous qualitative deduction and explain why the solution is consistent in the real context.

Consider the following system with unknown coefficients:

$\begin{cases} 2x + ky = 6 \\ 4x - 6y = c \end{cases}$

1. a) Determine what numerical values the parameters $k$ and $c$ must take so that the system is linearly dependent / consistent indeterminate (infinitely many solutions). Explain what geometric relationship the two lines have in this case.
2. b) Keeping the value of $k$ obtained in the previous section, state what condition $c$ must satisfy for the system to be inconsistent. What relative position do the lines have on the Cartesian plane?
3. c) Construct a different second linear equation so that, combined with $2x - 3y = 6$, it forms a new consistent independent system whose unique solution is the pair $(x, y) = (3, 0)$.

Solve the following system of linear equations, first simplifying the expressions with fractions and brackets:

$\frac{3(x - 1)}{2} + \frac{y + 2}{4} = \frac{11}{4}$

$\frac{x + y}{5} - \frac{x - y}{2} = \frac{3}{2}$

Determine the value of the parameter $k$ so that the following system of equations is inconsistent (that is, the lines representing it are parallel):

$kx + 3y = 5$

$4x - 2y = 7$

In an electronics shop, a laptop and a tablet cost a total of $1500$ €. During sale week, a discount of $15\,\%$ is applied to the laptop and $10\,\%$ to the tablet, bringing the combined price to $1305$ €. Set up a system of equations to find the original price of each item.

Determine the value of the parameter $m$ so that the following system has infinitely many solutions (is a consistent dependent system):

$3x - my = 12$

$-6x + 8y = -24$

Two cyclists leave at the same time from two towns, A and B, $78$ km apart, travelling towards each other in a straight line. They take exactly $3$ hours to meet. If the speed of the first cyclist is $4$ km/h faster than that of the second, at what speed is each cyclist travelling?

In a laboratory, two solutions of the same acid are available, one at a concentration of $10\,\%$ and the other at $25\,\%$. We wish to obtain $45$ litres of a new solution at a concentration of $16\,\%$. How many litres of each original solution should be mixed?

Look at the following graphical representation of a system formed by two lines, $r_1$ and $r_2$, whose equations are of the form $ax+by=c$ and $a'x+b'y=c'$. From the graph, classify the system according to its number of solutions and explain what algebraic relation must exist between the coefficients of both equations for this situation to occur.

A person invests a total of $25000$ € in two different investment funds. The first fund offers an annual interest rate of $4\,\%$, while the second offers $6\,\%$. If after one year the total profit obtained from both funds amounts to $1320$ €, how much money was invested in each fund?

A municipal electric bike-sharing service in Sevilla offers a subscription plan that includes a fixed monthly maintenance fee ($x$) and a fixed cost for each trip lasting more than 30 minutes ($y$).

During the month of April, Pablo made 16 billable trips and his total bill was $23,20\text{ €}$. For her part, Carmen made 28 billable trips under the same plan and paid $34,00\text{ €}$.

1. a) Model the situation using a system of linear equations and calculate the value of the fixed monthly fee ($x$) and the price of each trip ($y$).
2. b) A new user has a maximum monthly budget of $20\text{ €}$. What is the maximum number of trips of more than 30 minutes they can afford to take per month with this plan?
3. c) Obtain the expression for the average cost per trip as a function of the total number of trips $n$ taken in the month. Reason from how many monthly trips the average cost per trip becomes less than $1,30\text{ €}$ and interpret what happens to the weight of the fixed fee as the use of the service increases.

Consider the following system of linear equations depending on the parameters $a$ and $b$:

$\begin{cases} ax + 2y = 4 \\ 3x - y = b \end{cases}$

1. a) Without solving it using traditional algebraic methods, classify the type of system (consistent independent, consistent indeterminate, or inconsistent) according to the possible values of $a$ and $b$, using the ratios between the coefficients and the constant terms.
2. b) If we set $a = -6$ and $b = -2$, prove algebraically that the system has infinitely many solutions, express the general solution in terms of one of the unknowns, and provide three distinct particular numerical solutions.
3. c) From a geometric point of view, describe the relative position of both lines in each of the cases analyzed in section a) and rigorously justify why it is impossible for a system of two linear equations with two unknowns to have exactly two distinct solutions.