Problems with systems of equations
Setting up a system from a word problem is often harder than solving it. The difficulty lies in turning sentences into equations and deciding how many unknowns are needed: two, as in 4th ESO, or three, when the problem combines three different quantities.
Setting up: as many equations as unknowns
The sum of two integers is $54$ and their difference is $12$. Set up a system of equations and calculate the value of both numbers.
The sum of two integers is $54$ and their difference is $12$. Set up a system of equations and calculate the value of both numbers.
Prices and mixtures
When three items are involved instead of two, the setting up is the same, but the resulting system needs Gaussian elimination.
In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $2,10$ €. Another customer buys $2$ screws and $5$ nuts for $1,75$ €. Set up a system of…
In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $2,10$ €. Another customer buys $2$ screws and $5$ nuts for $1,75$ €. Set up a system of linear equations and calculate the price of each item.
A sports shop sells footballs, basketballs and volleyballs. The price of a football is $25$ €, a basketball is $30$ € and a volleyball is $20$ €. A club has bought $40$…
A sports shop sells footballs, basketballs and volleyballs. The price of a football is $25$ €, a basketball is $30$ € and a volleyball is $20$ €. A club has bought $40$ balls for a total of $1030$ €. Knowing that the number of footballs bought is equal to the sum of the basketballs and volleyballs, set up a system of three equations and determine how many balls of each sport were purchased.
A company distributes its annual profits among its three partners A, B and C. Partner A receives twice as much as partner B, and partner C receives $4000$ € more than…
A company distributes its annual profits among its three partners A, B and C. Partner A receives twice as much as partner B, and partner C receives $4000$ € more than partner B. If the total profit to be distributed is $52000$ €, set up a system of three linear equations and calculate the amount each partner receives.
Ages and geometry
When one of the relationships involves the area or the diagonal (rather than the perimeter), the equation is no longer linear: it is set up in the same way, but it is solved using the techniques for non-linear systems in the next section.
A father's current age is three times that of his son. In $10$ years' time, the father's age will be exactly twice that of the son. Determine their current ages.
A father's current age is three times that of his son. In $10$ years' time, the father's age will be exactly twice that of the son. Determine their current ages.
The perimeter of a rectangular plot is $130$ metres. The length of the plot is known to be $15$ metres greater than its width. Set up a system of equations and determine…
The perimeter of a rectangular plot is $130$ metres. The length of the plot is known to be $15$ metres greater than its width. Set up a system of equations and determine the dimensions of the plot.
The dimensions of a rectangular playing field are such that its area is $6000$ m² and the length of its diagonal is $130$ m. Set up a non-linear system of equations to…
The dimensions of a rectangular playing field are such that its area is $6000$ m² and the length of its diagonal is $130$ m. Set up a non-linear system of equations to calculate the length and width of the field.
Motion and sharing
A mathematics exam consists of $30$ questions. For each correct answer, $5$ points are added, and for each incorrect or unanswered question, $2$ points are deducted. A…
A mathematics exam consists of $30$ questions. For each correct answer, $5$ points are added, and for each incorrect or unanswered question, $2$ points are deducted. A student scored $94$ points in total. How many questions did the student answer correctly?
Two taps fill a tank in $6$ hours when opened together. If the first tap takes $5$ hours less than the second to fill the tank individually, set up a system of equations…
Two taps fill a tank in $6$ hours when opened together. If the first tap takes $5$ hours less than the second to fill the tank individually, set up a system of equations (which will include algebraic fractions) to determine how long each tap would take to fill it individually.
Two trains set off simultaneously from two stations A and B separated by $450$ km, travelling along parallel tracks in opposite directions. They pass each other after…
Two trains set off simultaneously from two stations A and B separated by $450$ km, travelling along parallel tracks in opposite directions. They pass each other after $3$ hours. If the train leaving A had left $1$ hour earlier than the train from B, they would have passed each other $306$ km from station A. Set up a system of equations to calculate the constant speeds of both trains.
Discussing the consistency of a problem with a parameter
Some real-life problems (sharing traffic between routes, distributing resources) depend on a parameter that you do not know in advance. Discussing the system tells you whether the problem has a unique solution, several possible solutions or none at all, before you solve it completely.
A system of linear equations depending on a parameter $k$ models the distribution of vehicles on three alternative routes $x$, $y$, $z$:…
A system of linear equations depending on a parameter $k$ models the distribution of vehicles on three alternative routes $x$, $y$, $z$:
$ \begin{cases} x + y + z = 100 \\ 2x + 3y + kz = 250 \\ x + 2y + 3z = 160 \end{cases} $
Determine for which values of the parameter $k$ the model has a unique solution, infinitely many solutions, or no solution, and interpret what it physically means for the system to be inconsistent.
A farmer wants to fence off a rectangular field bordering a straight river, so no fence is needed along the riverbank. He has exactly $400$ metres of fencing. Knowing…
A farmer wants to fence off a rectangular field bordering a straight river, so no fence is needed along the riverbank. He has exactly $400$ metres of fencing. Knowing that the enclosed area must be $20000$ m², set up a system of equations to find the dimensions of the field and discuss with justification whether there is a unique solution to the problem.