Systems of equations

Classify the following systems of equations without solving them, simply by comparing the ratios of their coefficients and constant terms:

1. a) $\begin{cases} 2x - 4y = 6 \\ -x + 2y = -3 \end{cases}$

1. b) $\begin{cases} 5x + y = 10 \\ 5x + y = 15 \end{cases}$

Observe the graphical representation of the two lines that form the following system of equations. Determine, without doing any algebraic calculations, the solution of the system and check whether the system is consistent and independent (has a unique solution).

$\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$

Solve the following system of linear equations using whichever method you consider most appropriate (substitution, equating expressions, or elimination) and state whether it is a system with a unique solution (SCD), infinitely many solutions (SCI), or no solution (SIN):

$\begin{cases} 3x - 2y = 1 \\ x + 4y = 11 \end{cases}$

Solve the following system of linear equations with fractional coefficients, simplifying the expressions beforehand:

Determine the consistency of the following system of equations depending on the value of the parameter $k$. For what value of $k$ is the system consistent with infinitely many solutions?

Two friends, Lucía and Marcos, went to a café. Lucía paid $110$ pesos for 2 coffees and 1 orange juice. Marcos paid $120$ pesos for 1 coffee and 2 orange juices. Set up a system of equations and calculate the price of a coffee and of an orange juice.

The perimeter of a rectangular venue for events is $50$ meters. It is known that the length of the venue is $5$ meters more than the width. Calculate the dimensions of the venue by setting up a system of two equations with two unknowns ($x$ for the length and $y$ for the width).

To organise an end-of-year trip, Year 12 students are selling two types of T-shirts: the "Eco" model for $240$ pesos and the "Premium" model for $360$ pesos. By the end of the week they have sold a total of $150$ T-shirts and raised $44\,400$ pesos. Set up a system of equations and calculate how many T-shirts of each type they have sold.

In systems of three equations with three unknowns, sometimes the structure allows us to solve it immediately. Find the values of $x$, $y$, and $z$ in the following triangular system using back-substitution (starting with the last unknown):

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

Apply the first step of Gaussian elimination to eliminate the unknown $x$ from the second and third rows in the following system. You do not need to finish solving it, just write down the resulting system after performing the appropriate elementary operations:

$\begin{cases} x + y + z = 6 \\ 2x - y + 3z = 9 \\ -x + 2y + z = 3 \end{cases}$

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

In a downtown café, three friends order different combinations of items.

• The first has a coffee, a soft drink, and a slice of cake for $150$ pesos.

• The second has two coffees and a soft drink for $120$ pesos.

• The third has a coffee and two slices of cake for $170$ pesos.

Calculate the price of each item by setting up a system of three equations and solving it using Gaussian elimination.

Analyse and solve the following system of linear equations in terms of the real parameter $k$ using Gaussian elimination:

$\begin{cases} x + y + kz = 1 \\ x + ky + z = 1 \\ kx + y + z = 1 \end{cases}$

Determine for which values of $k$ the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent. Explain the geometric interpretation of each case (planes intersecting at a point, along a line, or having no common points).

An organic chemistry laboratory wants to prepare $10$ liters of a special solution with an alcohol concentration of $35\%$. To do so, three different mixtures are available:

• Mixture A: $20\%$ alcohol.

• Mixture B: $30\%$ alcohol.

• Mixture C: $50\%$ alcohol.

The laboratory manager imposes the condition that the volume of Mixture C used must be twice the volume of Mixture A used. How many liters of each mixture are needed to obtain the desired combination?

An investment fund management firm for young entrepreneurs distributes capital of $300,000$ pesos among three tech projects: Applied AI, Green Energy, and Cybersecurity.

It is known that:

1. The total annual profit was $23,500$ pesos.

2. The return on investment for each project was $5\%$ for AI, $10\%$ for Green Energy, and $8\%$ for Cybersecurity.

3. The capital invested in Cybersecurity is equal to the sum of the capital invested in AI and half of that invested in Green Energy.

Set up a system of three equations with three unknowns and use Gaussian elimination to calculate how much money was allocated to each project.

A non-linear system is one in which at least one of the equations is not of first degree. Find the points of intersection between the line and the parabola given by the following system:

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

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

A mountain rescue team uses an antenna to locate hikers. The antenna's radio signal, located at the origin of coordinates, defines a circular coverage area given by the equation $x^2 + y^2 = 25$. A rescue helicopter moves along a straight path described by the equation $y - x = 1$. The figure shows the coverage area and the path (each square of the scale equals 1 unit on both axes). Calculate the intersection points where the helicopter enters and leaves the antenna's coverage range by solving the system:

A rectangular plot intended for school gardens needs to be fenced. It is known that the total perimeter of the plot is $50$ meters and its area is $150$ $m^2$. Find the dimensions of the plot by setting up and solving a system of non-linear equations.

Solve the following system of nonlinear equations. Keep in mind that you will need to simplify the algebraic expressions before applying the substitution method and check the validity of all the solutions obtained (remember that denominators cannot be zero, that is, $x\neq y$ and $x\neq -y$):

$\begin{cases} \displaystyle \frac{x+y}{x-y} + \frac{x-y}{x+y} = \frac{10}{3} \\ x^2 + y^2 = 20 \end{cases}$

Find the values of $x$ and $y$ that satisfy the following system of logarithmic equations. Remember to apply logarithmic laws to convert the system into an algebraic one and check that the solutions do not invalidate the arguments of the original logarithms:

$\begin{cases} \log(x) + \log(y) = 2 \\ \log(x^2) - \log(y) = 1 \end{cases}$

An architect is designing a rectangular play area for a school. The total perimeter of the compound is $54$ meters. For safety reasons, a diagonal fence must be installed to divide the rectangle into two triangles. If the length of this diagonal is $15$ meters more than the difference between the length and the width of the rectangle, calculate the dimensions of the play area by setting up a system of quadratic equations.

Solve the following system of exponential equations by using an appropriate substitution ($u = 3^x$, $v = 2^y$):

$\begin{cases} 3^{x+1} - 2^{y+2} = 1 \\ 2 \times 3^x + 2^{y-1} = 10 \end{cases}$

Express the final values of $x$ and $y$ using logarithms if they are not integers.

Given the supply function ($O$) and demand function ($D$) of a financial product in the market:

$D(x) = \displaystyle \frac{40}{x}$

$O(x) = x - 3$

Where $x$ is the price of the asset and $y$ is the quantity of assets.

1. Set up the system of equations to find the market equilibrium point (where supply equals demand).

2. Solve the system analytically.

3. Does the negative solution make economic sense? Justify your answer based on the nature of the variables.