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).

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

In a Mathematics exam, Claudia tries to classify the following system of linear equations without actually solving it:

Claudia writes on her sheet: “Since $\dfrac{3}{-2} = -1.5$ and $\dfrac{-6}{4} = -1.5$, the coefficients of the unknowns have the same ratio. This means that the two equations represent the same line and, therefore, the system is consistent dependent, with infinitely many solutions”.

1. a) Find the error in Claudia's reasoning and explain which essential element she forgot to compare.
2. b) Rigorously classify the system by comparing the full ratios of the coefficients: $\dfrac{a}{a'}$, $\dfrac{b}{b'}$, and $\dfrac{c}{c'}$.
3. c) Explain what the actual geometric position of both lines in the Cartesian plane is and how many solutions the system has.

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:

Consider the following two systems of linear equations:

System A:

System B:

1. a) For each system, analyze the arrangement of its coefficients and unknowns and justify which algebraic method (substitution, equating, or elimination) allows you to reach the solution most quickly and with the least risk of working with complex fractions.
2. b) Solve System B using exclusively the method you selected as optimal in the previous section, and verify that the found solution simultaneously satisfies both equations.

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

Two friends, Aoife and Seán, went to a café. Aoife paid $5.50$ € for 2 coffees and 1 orange juice. Seán paid $6.00$ € 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 $50$ metres. It is known that the length of the venue is $5$ metres 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 $12$ € and the "Premium" model for $18$ €. By the end of the week they have sold a total of $150$ T-shirts and raised $2220$ €. 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):

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

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

In a city-centre café, three friends order different combinations of items.

• The first has a coffee, a soft drink, and a slice of cake for $7.50$ €.

• The second has two coffees and a soft drink for $6.00$ €.

• The third has a coffee and two slices of cake for $8.50$ €.

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

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:

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

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.