Systems of equations
Two friends order in a café: Lucía pays $5{,}50$ € for 2 coffees and 1 juice; Marcos pays $6{,}50$ € for 1 coffee and 2 juices. Two unknowns (the price of a coffee, the price of a juice), two conditions: a system. In 1st Bachillerato you will set up systems like this all the time —in economics, in statistics, in any problem with two or three unknown quantities— and, besides solving them, you will need to say how many solutions they have before calculating anything.
What a system is and how it is classified
Classify the following systems of equations without solving them, simply by comparing the ratios of their coefficients and constant terms: a)…
Classify the following systems of equations without solving them, simply by comparing the ratios of their coefficients and constant terms:
- a) $\begin{cases} 2x - 4y = 6 \\ -x + 2y = -3 \end{cases}$
- 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…
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).
$\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$
Substitution, equating and elimination are still valid
Solve the following system of linear equations using whichever method you consider most appropriate (substitution, equating expressions, or elimination) and state…
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: $$ \begin{cases} \dfrac{x+2}{3} - \dfrac{y-1}{4} =…
Solve the following system of linear equations with fractional coefficients, simplifying the expressions beforehand:
$$ \begin{cases} \dfrac{x+2}{3} - \dfrac{y-1}{4} = \dfrac{5}{6} \\ 2(x-1) + 3y = 10 \end{cases} $$Non-linear systems: solve for a variable in the easy equation
If any equation has $x^2$, $x\cdot y$ or a root, the system is non-linear. The strategy does not change: solve for a variable in the simplest equation (normally the linear one) and substitute into the other.
A non-linear system is one in which at least one of the equations is not of first degree. Find the points of intersection between the line and the parabola given by the…
A non-linear system is one in which at least one of the equations is not of first degree. Find the points of intersection between the line and the parabola given by the following system:
$\begin{cases} y = x^2 \\ y = x + 2 \end{cases}$
Solve the following system of non-linear equations using the substitution method: $$ \begin{cases} x \cdot y = 12 \\ (x-1)(y+2) = 15 \end{cases} $$
Solve the following system of non-linear equations using the substitution method:
$$ \begin{cases} x \cdot y = 12 \\ (x-1)(y+2) = 15 \end{cases} $$Problems: setting up the system from a worded statement
The most common use of systems in 1st Bachillerato is in problems about prices, sharing and dimensions: translating the text into equations is half the work.
Two friends, Lucía and Marcos, went to a café. Lucía paid $5,50$ € for 2 coffees and 1 orange juice. Marcos paid $6,00$ € for 1 coffee and 2 orange juices. Set up a…
Two friends, Lucía and Marcos, went to a café. Lucía paid $5,50$ € for 2 coffees and 1 orange juice. Marcos 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 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…
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.