Exercises: Systems of equations

Look at the graphical representation of the two lines forming a system of equations. Without doing algebraic calculations, determine the solution to the system and classify it as Consistent Independent (unique solution), Consistent Dependent (infinitely many solutions), or Inconsistent (no solution).

Translate the following situation into algebraic language and set up a system of two equations in two unknowns, clearly stating what each variable represents:

"In a café, a group of friends paid $\text{€}\ 11$ for $2$ soft drinks and $3$ bags of crisps. Another group paid $\text{€}\ 13$ for $4$ soft drinks and $1$ bag of crisps."

Solve the following system using the substitution method, taking advantage of the fact that one of the unknowns is already isolated in the first equation:

$\begin{cases} y = 3x \\ 2x + y = 10 \end{cases}$

Solve the following system of equations using the equating method. Notice that the unknown $x$ is already isolated in both expressions:

$\begin{cases} x = 5 - y \\ x = 2y - 1 \end{cases}$

Solve the following system using the elimination method. Notice that the coefficients of the unknown $y$ are equal in magnitude but opposite in sign, so you can add the equations directly:

$\begin{cases} 4x + y = 7 \\ 2x - y = 5 \end{cases}$

Copy and complete the following table of values for the two equations of the system in your notebook. Then, state the common solution by looking at the results:

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

In a video game shop, second-hand controllers and games are sold. Cian bought one controller and one game for $€50$. If the price of the game is three times that of the controller, how much does each item cost?

Set up a system of equations and solve it using whichever method you prefer.

Two friends, Aoife and Seán, are accumulating points in their favourite video game to unlock a new reward. Stealth missions are represented by the unknown $x$ and assault missions by the unknown $y$. The system of equations describing the objectives they must complete today is as follows:

Check numerically whether the pair of values $(x = 3, y = 2)$ is the solution to the system. Looking at the graph below, state whether the friends have managed to find the exact point where both conditions (total missions and accumulated points) are met.

Look at the following two systems of linear equations without solving them yet:

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

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

1. a) For System A, justify why the substitution method is the most direct approach compared to elimination.
2. b) For System B, justify why the elimination method is the most convenient option at first glance.
3. c) Choose one of the two systems and solve it step by step applying the method you justified.

Cian tries to solve the following system of equations in his notebook:

To do so, he writes the following steps:

1. I solve for $x$ in the first equation: $x = 8 - 2y$.

2. I substitute that expression into the first equation: $(8 - 2y) + 2y = 8$.

3. I simplify and obtain: $8 = 8$.

4. Conclusion: Since $8 = 8$ is always true, the system is consistent dependent and has infinitely many solutions.

1. a) Explain what conceptual mistake Cian made in step 2 and clarify why his final conclusion is false.
2. b) Correct the exercise by substituting into the appropriate equation and calculate the actual solution $(x, y)$ of the system.

Look at the following graphical representation of a system of two linear equations with two unknowns and answer the questions:

1. a) State the coordinates of the intersection point $P$ of both lines.
2. b) What does that point represent in relation to the system of equations?
3. c) Classify the system according to the number of solutions it has.
4. d) Is the point $P$ a solution of the following system?

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

Briefly explain why it is or why it is not, checking both equalities.

When organising an e-sports event, a total of 250 tickets were sold. There are two types of seats: "Gamer" tickets, which cost $\text{€}\ 15$, and "Pro" tickets, which cost $\text{€}\ 25$. If the total amount raised was $\text{€}\ 4\ 700$, set up a system of equations and calculate how many tickets of each type were sold using the substitution method.

A new rectangular play area is being designed in the school playground. The architect has established that the total perimetre must be $56$ metres. Furthermore, to leave enough space for the basketball hoops, the length of the area must be $4$ metres less than three times the width.

Calculate the dimensions of the play area by setting up a system of equations and solving it using the equating method.

Two well-known streamers are comparing their years of experience on the platform. The sum of their current ages is $44$ years. In $10$ years' time, the age of the older one will be exactly twice the present age of the younger one.

Set up a system of linear equations and find out the present age of each one.

In a bulk food shop, we want to prepare a special $20\ kg$ mix of nuts to sell at a price of $\text{€}\ 8.50/kg$. To do this, we will use cashews costing $\text{€}\ 10/kg$ and peanuts costing $\text{€}\ 6/kg$.

Use the elimination method to find out how many kilograms of cashews and how many of peanuts we should mix.

Look at the following graphical representation of two linear functions that form a system of equations.

1. a) Identify the equations of both lines from their intercepts and gradient.
2. b) Determine the solution of the system graphically.
3. c) Classify the system according to its number of solutions (SCD, SCI, or SI).

A two-digit number has a curious property: the tens digit is twice the units digit. If we subtract $27$ from this number, we get the number obtained by reversing the order of its digits.

Translate the statement into a system of equations (remember to express the number in expanded form: $10x + y$) and find the number.

Two cyclists, Aoife and Cian, ride around a circular track of $400$ metres. If they ride in opposite directions, they pass each other every $20$ seconds. If they ride in the same direction, Aoife (who is faster) overtakes Cian every $100$ seconds.

Knowing that distance travelled is equal to speed multiplied by time ($e = v \times t$), set up a system of equations to calculate the speed of each cyclist in metres per second.

A tourist office puts two types of urban transit passes on sale: the 'Basic' pass for $8.50$ € and the 'Plus' pass for $15.00$ €. At the end of the day, a total of $50$ passes have been sold and the total revenue has been $542.00$ €.

1. a) Accurately identify the variables and set up the system of two linear equations representing the situation.
2. b) Solve the system using the algebraic method you consider most appropriate to determine how many passes of each type were sold.
3. c) If, due to a computer typo, the recorded revenue had been $540.00$ €, show algebraically why that amount would not be possible in reality with those prices and that number of passes.

Design a system of two linear equations with two variables ($x$ and $y$) that simultaneously satisfies these three conditions:

1. Its only solution must be the ordered pair $(x, y) = (3, -2)$.

2. None of the coefficients of the variables or the constant terms can be equal to $0$, $1$, or $-1$.

3. The coefficients of the variable $y$ in both equations must be opposite integers (for example, $4$ and $-4$), so that the system is ready to eliminate it by directly adding both equations using elimination.

Write down the system you have constructed and formally verify that $(3, -2)$ satisfies both equalities.

Solve the following linear equation, paying special attention to simplifying the fractions and to the negative sign preceding the brackets:

Study the discriminant of the following quadratic equation and determine for which value of the parameter $k$ the equation has a single real solution (repeated root). Once $k$ is found, solve the equation:

A courier company uses drones and vans for its deliveries. We know that twice the number of drones exceeds three times the number of vans by 2 units. Furthermore, if they were to buy 2 more drones and lose one van, they would have three times as many drones as vans. Set up a system of linear equations and find out how many vehicles of each type the company has.

In a chemistry laboratory, two saline solutions are to be mixed. The first has a salt concentration of $15\%$ and the second of $40\%$. If exactly $20$ litres of a mixture with a concentration of $30\%$ are required, how many litres of each solution should be used? Set up and solve the system of equations.

Find the dimensions of a rectangular wooden frame, knowing that its perimetre is $60\text{ cm}$ and its area is $216\text{ cm}^2$. To solve it, set up a system in which one of the equations is quadratic, or solve it using a quadratic equation by applying the relationship between the sum and product of the roots.

Two cyclists, Ioana and Boris, take part in a long-distance race. Boris sets off from the starting line at a constant speed of $24\text{ km/h}$. Half an hour later, Ioana sets off in pursuit at a speed of $32\text{ km/h}$.

1. a) How long will it take Ioana to catch up with Boris?
2. b) At what distance from the starting line will they meet?

Hint: Use the system of equations based on the formula $e = v \times t$.

Find a two-digit number, knowing that the sum of its digits is $12$ and that, if the order of its digits is reversed, the resulting number is $18$ units greater than the original number.

Note: Remember that a number with digits $xy$ is written algebraically as $10x + y$.

Look at the graphical representation of the lines $r$ and $s$ forming a system of equations.

1. a) Identify the equations of both lines using their axis intercepts.
2. b) Determine, algebraically and graphically, whether the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent.
3. c) Find the point of intersection $(x, y)$.

An agricultural cooperative bottles $5$-liter jugs by blending two types of oil: extra virgin olive oil (EVOO), with a production cost of $9.00$ € per liter, and sunflower oil, with a cost of $2.00$ € per liter. The desired average cost of the liter of blend in each jug is exactly $4.80$ € (which represents a total cost of $24.00$ € per jug).

1. a) Before calculating anything, give a reasoned estimate of whether the jug will contain more than $2.5$ liters of EVOO or more than $2.5$ liters of sunflower oil. To do this, compare the closeness of the average price ($4.80$ €) to the prices of each of the components.
2. b) Set up and solve the system of linear equations to calculate exactly how many liters of EVOO and how many of sunflower oil each jug contains. Does the result confirm your previous estimate?
3. c) If the cooperative intended to obtain a $5$-liter blend whose final cost was $48.00$ € using exclusively these two types of oil, analyze algebraically what would happen when solving the system and explain what physical meaning that result has in practice.

Two courier delivery companies charge the following rates for package delivery:

• Swift Couriers: charges a flat fee of $12.00$ € plus $0.80$ € for each kilometer traveled.

• Shamrock Express: charges a flat fee of $k$ € plus a rate per kilometer of $m$ €/km.

Let $x$ be the distance of the trip in kilometers and $y$ the final cost in euros.

1. a) Write the equation that models the cost of Swift Couriers.
2. b) Determine with justification what values the parameters $k$ and $m$ must have for the system formed by the two rates to be:

• Inconsistent (the two lines are parallel and never match in price for any mileage). Explain the consequence for the customer.

• Consistent dependent (both rates turn out to be identical for any distance).

1. c) If Shamrock Express sets $k = 18.00$ € and $m = 0.50$ €/km, set up the system of equations, find the intersection point, and interpret beyond how many hired kilometers Shamrock Express is cheaper than Swift Couriers.