Exercises: Solving systems of equations

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

$\begin{cases} 3x - y = 5 \\ 2x + 3y = 18 \end{cases}$

Solve the following system of equations using the equating method:

$\begin{cases} x + 4y = 1 \\ x - 2y = 13 \end{cases}$

Solve the following system of equations using the elimination method:

$\begin{cases} 5x + 2y = 11 \\ 3x - 2y = 5 \end{cases}$

Expand the brackets and solve the following linear system:

$\begin{cases} 2(x + 1) - 3(y - 2) = 16 \\ 5x + y = 3 \end{cases}$

Solve the following non-linear system, formed by the equation of a parabola and a line:

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

Find all real solutions to the following system of quadratic equations:

$\begin{cases} x^2 + y^2 = 25 \\ x^2 - y^2 = 7 \end{cases}$

Solve the following triangular system of three equations with three unknowns:

$\begin{cases} 2x - y + 3z = 14 \\ 5y - 2z = -3 \\ 4z = 16 \end{cases}$

The sum of two numbers is $45$ and their difference is $11$. Set up a system of linear equations and find these numbers.

Look closely at the following two systems of linear equations:

System A: $\begin{cases} x = 3y - 2 \\ 2x + 5y = 18 \end{cases}$

System B: $\begin{cases} 4x + 3y = 11 \\ 4x - 5y = -5 \end{cases}$

1. a) Without solving them yet, analyze the structure of both and state, giving reasons, which of the three basic algebraic methods (substitution, elimination by equating, or elimination by addition/subtraction) is the fastest and most direct for each of them, explaining which visual feature of the equations justifies your choice.
2. b) Choose one of the two systems and solve it using the method you selected in the previous section, checking at the end whether the solution obtained satisfies both equalities.

A youth association from Toledo organizes an environmental volunteering trip and buys two types of interurban transport tickets at the ticket counter: single tickets for short trips at $1,80$ € each and multi-journey cards at $9,50$ € each. In total they purchase $25$ transport tickets and the bill amounts to exactly $137,40$ €.

1. a) Clearly define what quantity each unknown represents and write the system of two linear equations that models this purchase.
2. b) Solve the system using the method you consider most convenient and interpret the result, explaining how many single tickets and how many multi-journey cards the association bought.

Solve the following linear system, clearing the denominators first to obtain simpler equivalent equations:

$\begin{cases} \frac{x - 1}{3} + \frac{y + 2}{4} = 3 \\ \frac{2x + y}{7} - \frac{x - y}{2} = 3 \end{cases}$

Find the points of intersection between the hyperbola and the line given by the following non-linear system:

$\begin{cases} x \cdot y = 12 \\ 2x + y = 10 \end{cases}$

Solve the following system of three equations with three unknowns using Gaussian elimination or back-substitution:

$\begin{cases} x + 2y - z = 2 \\ 2x - y + 3z = 9 \\ 3x + y + z = 8 \end{cases}$

Find the solutions to the following system of radical equations:

$\begin{cases} x - y = 5 \\ \sqrt{x} + \sqrt{y} = 5 \end{cases}$

Solve the following system of exponential equations:

$\begin{cases} 2^x + 3^y = 17 \\ 2^{x+1} - 3^{y-1} = 13 \end{cases}$

A grandfather tells his grandson: "$5$ years ago, my age was five times yours, but in $10$ years' time it will only be double." Set up a system of linear equations and determine the current age of each.

The perimeter of a rectangular plot is $140$ metres. If the base were decreased by $5$ metres and the height were increased by $5$ metres, the plot would become a square. Calculate the current dimensions of the plot.

In a chemistry laboratory, two acid solutions have been mixed. The first has a concentration of $20\%$ and the second of $50\%$. If they want to obtain $60$ litres of a new solution at $30\%$, how many litres of each type should be mixed?

Lucía tried to solve the following system of equations with denominators on the board:

$\begin{cases} \frac{x+1}{2} - \frac{y-2}{3} = 2 \\ 2x - y = 1 \end{cases}$

In her notebook she wrote this working:

• Step 1 (clear denominators by multiplying the first equation by $6$): $3 \cdot (x + 1) - 2 \cdot (y - 2) = 12 \implies 3x + 3 - 2y - 4 = 12 \implies 3x - 2y = 13$.

• Step 2 (isolate $y$ in the second equation): $y = 2x - 1$.

• Step 3 (substitute into the simplified first equation): $3x - 2 \cdot (2x - 1) = 13 \implies 3x - 4x - 2 = 13 \implies -x = 15 \implies x = -15$.

• Step 4 (calculate the remaining unknown): $y = 2 \cdot (-15) - 1 = -31$.

1. a) Lucía made two common sign errors when expanding parentheses. Identify in which steps they are located and explain precisely what mathematical rule or property she did not follow in each one.
2. b) Correct the procedure and properly solve the system to find the correct values of $x$ and $y$.

A family from Valencia with a regulated electricity tariff (PVPC) analyzes the energy consumption charge of their last two bimonthly bills. In the first one, they consumed $250\text{ kWh}$ during peak hours and $180\text{ kWh}$ during off-peak hours, with a total energy cost of $81,60$ €. In the second bill, they consumed $150\text{ kWh}$ during peak hours and $220\text{ kWh}$ during off-peak hours, resulting in a cost of $62,40$ €. The unit price per kWh during peak hours ($p$) and off-peak hours ($v$) remained constant in both months.

1. a) Without solving the system yet, reason which of the two prices ($p$ or $v$) must be higher and estimate a plausible price range in euros per kWh (or euro cents) for each period, taking into account the typical cost of electricity in Spanish households.
2. b) Set up the system of linear equations that describes the situation and solve it by elimination to find the exact prices of $p$ and $v$ in €/kWh.
3. c) Compare the exact results with your previous estimate and critically assess whether the figures obtained confirm your initial prediction.

Solve the following system of equations with algebraic fractions by applying a suitable change of variable:

$\begin{cases} \frac{3}{x} + \frac{2}{y} = 2 \\ \frac{4}{x} - \frac{1}{y} = \frac{5}{6} \end{cases}$

Find the solution to the following system of logarithmic equations:

$\begin{cases} \log(x) + \log(y) = 2 \\ x - 4y = 0 \end{cases}$

Solve the following exponential system:

$\begin{cases} 3^x + 2^y = 17 \\ 3^{x+1} - 2^{y+2} = -5 \end{cases}$

A transport company has three types of vans: A, B and C. Van A can carry $2$ tonnes, B $3$ tonnes and C $5$ tonnes. On a certain day, $12$ vans were used in total to transport $38$ tonnes of goods. If the number of type A vans used is equal to the sum of type B and C vans, set up and solve a system of three equations to find out how many vans of each type were used.

Solve the following symmetric non-linear system:

$\begin{cases} x^2 + y^2 = 58 \\ x \cdot y = 21 \end{cases}$

Two cyclists set off at the same time from two towns, A and B, $90$ km apart, travelling towards each other. If they travel at constant speeds, they meet after $3$ hours. If the cyclist departing from A had left one hour before the cyclist from B, they would have met $2$ hours and $40$ minutes after the cyclist from B set off. At what speed is each cyclist travelling?

Determine the value of the parameter $k$ so that the following linear system is inconsistent, and justify the relative position of the two lines forming it in the Cartesian plane:

$\begin{cases} 3x - (k+1)y = 5 \\ 6x - 8y = 12 \end{cases}$

An athletics stadium has a track formed by a central rectangle and two semicircles at the ends. The total perimeter of the inner line of the track is $400$ metres and the area of the central rectangle is $5600$ square metres. Set up a system of non-linear equations to find the radius $R$ of the semicircles and the length $L$ of the straight section. Take $\pi \approx 3$.

Imagine you are part of the team designing the diagnostic assessment tests for 4th year of ESO and you need to design a system of two linear equations with two unknowns ($x$ and $y$) that simultaneously satisfies the following three conditions:

1. Its unique solution must be exactly $x = 3$, $y = -2$.

2. None of the coefficients of the unknowns nor the constant terms can be $0$, $1$, or $-1$.

3. When applying the elimination method by directly adding the two equations side by side (without needing to previously multiply either equation by another number), one of the unknowns must be completely eliminated.

1. a) Write the system you have constructed and justify step by step why it satisfies each of the three conditions.
2. b) Modify the second equation of your system so that the resulting new system is inconsistent (has no solution). Explain with sound reasoning the algebraic justification and the geometric interpretation (relative position of the lines) of why that change prevents there from being a solution.

In a regional waste sorting plant in Galicia, three types of recyclable materials are managed weekly: plastic, paper/cardboard, and glass. During a sampling week, the following data were collected:

1. Across all three materials, a total of $180\text{ tonnes}$ were processed.

2. The sum of the tonnes of paper/cardboard and glass was exactly twice the amount of plastic processed.

3. The logistics cost of recycling is $40$ € per tonne of plastic, $20$ € per tonne of paper/cardboard, and $10$ € per tonne of glass, bringing the total processing cost for that week to $4300$ €.

1. a) Assign an unknown to each material and formulate a canonical system of three linear equations in three unknowns representing the situation.
2. b) Apply the Gaussian elimination method to transform the system into an equivalent row-echelon form, clearly stating the elementary row operations performed.
3. c) Solve the row-echelon system from bottom to top (back-substitution) to determine how many tonnes of each waste material were processed and analyze whether the quantities maintain a logical proportion in the context of urban selective waste collection.