Problems with systems of equations
You are queueing at the till in a hardware shop in Athlone: one customer buys 4 screws and 3 nuts, another buys 2 screws and 5 nuts, and each total is on the receipt. Can you work out the price of one screw? 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 Junior Cycle, or three, when the problem combines three different quantities.
Setting up: as many equations as unknowns
A seaside hotel has double and single rooms. In total, there are $50$ rooms and $87$ beds. Assuming that all rooms are fully occupied, how many rooms of each type does…
A seaside hotel has double and single rooms. In total, there are $50$ rooms and $87$ beds. Assuming that all rooms are fully occupied, how many rooms of each type does the hotel have?
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.
On a farm school, there are chickens and cows. Counting the heads of the animals gives a total of $45$; counting the legs gives $140$. How many animals of each type are…
On a farm school, there are chickens and cows. Counting the heads of the animals gives a total of $45$; counting the legs gives $140$. How many animals of each type are there on the farm?
Prices and mixtures
When the problem gives a total and a percentage change, the percentage turns into a multiplier.
For a mixture, one equation counts the amount (litres) and one counts the value or the quantity of pure substance.
When three items are involved instead of two, the setting up is the same, but the resulting system has three equations.
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.
In a winery, two types of wine are mixed—one at $4$ €/litre and another at $5.50$ €/litre—to obtain a barrel of $300$ litres of mixture that will be sold at $4.60$…
In a winery, two types of wine are mixed—one at $4$ €/litre and another at $5.50$ €/litre—to obtain a barrel of $300$ litres of mixture that will be sold at $4.60$ €/litre. How many litres of each type of wine were used to make the mixture?
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.
Ages, digits and geometry
For age problems, translate each time carefully: "in $10$ years" adds $10$ to both ages.
For a digits problem, write the number as $100a+10b+c$ and its reverse as $100c+10b+a$.
When one of the relationships involves the area or the diagonal (rather than the perimeter), the equation is no longer linear: use one linear equation to substitute into the quadratic one. For example, with perimeter $20$ m and area $21$ m², $x+y=10$ and $xy=21$ give $x(10-x)=21$, so $x^2-10x+21=0$, hence $x=7$ or $x=3$: the plot is $7$ m by $3$ m. For a diagonal, use Pythagoras' theorem: $x^2+y^2=d^2$.
In a right-angled triangle with legs $a$, $b$ and hypotenuse $c$, the area is $\frac12ab$ and also $\frac12 c h$, where $h$ is the altitude to the hypotenuse. So $ab=ch$. Example: legs $3$ and $4$, hypotenuse $5$ give $h=\frac{12}{5}=2.4$. Together with $a+b+c=P$ and $a^2+b^2=c^2$ this gives three equations for $a$, $b$, $c$.
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.
Determine the lengths of the sides of a right-angled triangle, knowing that its perimeter is $60$ cm and that the altitude to the hypotenuse is $12$ cm. Set up a…
Determine the lengths of the sides of a right-angled triangle, knowing that its perimeter is $60$ cm and that the altitude to the hypotenuse is $12$ cm. Set up a non-linear system of equations with three unknowns (legs and hypotenuse) to solve it.
Motion and sharing
For rates of work, add rates, not times: a tap that fills a tank in $t$ hours fills $\frac1t$ of it per hour.
A cyclist makes a return trip between two towns separated by a distance $d$. On the outward journey, with a tailwind, they ride at a constant speed of $30$ km/h. On the…
A cyclist makes a return trip between two towns separated by a distance $d$. On the outward journey, with a tailwind, they ride at a constant speed of $30$ km/h. On the return journey, with a headwind, they ride at $20$ km/h. If the total time taken for the complete return trip is $2.5$ hours, set up a system of equations and find the distance between the two towns and the time taken on each leg of the journey.
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
To discuss, eliminate one variable and look at the last equation: it has the form $(\text{something in }k)\,z=\text{number}$. If the coefficient is non-zero there is a unique solution; if it is zero and the right side is non-zero there is no solution; if both are zero there are infinitely many.
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 last-mile urban delivery company operates with a total fleet of $24$ electric vans of three cargo capacities: small ($1\text{ m}^3$), medium ($2\text{ m}^3$), and…
A last-mile urban delivery company operates with a total fleet of $24$ electric vans of three cargo capacities: small ($1\text{ m}^3$), medium ($2\text{ m}^3$), and large ($4\text{ m}^3$). The total simultaneous cargo volume of the entire fleet is $56\text{ m}^3$. Logistics management is testing an organizational criterion depending on a real parameter $k$: the number of small vans ($x$) plus $k$ times that of medium vans ($y$) must equal that of large vans ($z$), that is, $x + k\times y - z = 0$.
- a) Write the system of three equations with three unknowns and apply Gaussian elimination to the augmented matrix to row-reduce it in terms of the parameter $k$.
- b) Discuss with reasoning the number of solutions according to the values of $k$, indicating for which value the system is inconsistent and for which ones it is consistent with a unique solution.
- c) In actual practice, the number of vehicles of each type must be a non-negative integer ($x, y, z \in \mathbb{N}$). Solve the system for $k = 1$ and for $k = -1$, and reason in detail which of the two cases is logistically feasible in reality and which one turns out to be physically impossible despite being mathematically consistent.
Back to the till in Athlone: with $4x+3y=2.10$ and $2x+5y=1.75$ you found a screw costs 37.5c and a nut 20c.