Word Problems with Systems
At a corner store, Yaritza pays $\text{RD}\$270$ for 3 juices and 2 rolls. Her brother pays $\text{RD}\$255$ for 2 juices and 3 rolls. How much does each item cost? With just one purchase you can't tell, but with both you can. The hard part is almost never solving the system: it is translating the problem into equations. Here you will learn to do it with a method.
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.
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?
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
In price problems, each equation says: (quantity) × (price) + … = total paid.
Mixtures
In a mixture, two things are conserved: the total amount and the total value (money, or amount of pure substance).
Percentages
If a quantity $x$ increases by $5\%$, the new quantity is $1.05x$; if $y$ decreases by $10\%$, it becomes $0.9y$.
In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $126$ pesos. Another customer buys $2$ screws and $5$ nuts for $105$ pesos. Set up a system of…
In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $126$ pesos. Another customer buys $2$ screws and $5$ nuts for $105$ pesos. 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 $240$ pesos/liter and another at $330$ pesos/liter—to obtain a barrel of $300$ liters of mixture that will be sold at…
In a winery, two types of wine are mixed—one at $240$ pesos/liter and another at $330$ pesos/liter—to obtain a barrel of $300$ liters of mixture that will be sold at $276$ pesos/liter. How many liters of each type of wine were used to make the mixture?
A family allocates a portion of their monthly income to mortgage payments and another portion to food. Last month, the sum of these two expenses was $72\,000$ pesos.…
A family allocates a portion of their monthly income to mortgage payments and another portion to food. Last month, the sum of these two expenses was $72\,000$ pesos. This month, the mortgage payment rose by $5\%$ and the food expenditure decreased by $10\%$, making the total expenditure on these two items $70\,200$ pesos. Set up a system of equations to calculate how much they spent on each item last month.
Ages, digits, and scores
Numbers with several digits
A three-digit number with hundreds digit $c$, tens digit $d$, and units digit $u$ is worth $100c+10d+u$. If the digits are reversed, it is worth $100u+10d+c$.
Scores
On an exam or in a competition, each type of correct answer adds or subtracts points: $(\text{correct answers})\cdot(\text{points}) + \dots = \text{total}$, together with the equation for the total number of questions.
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.
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?
The sum of the three digits of a number is $12$. The tens digit is the arithmetic mean of the other two. If the order of the digits is reversed, the resulting number is…
The sum of the three digits of a number is $12$. The tens digit is the arithmetic mean of the other two. If the order of the digits is reversed, the resulting number is $198$ units greater than the original. Set up a system of equations and find the number.
Geometry: perimeter, area, and diagonal
Typical data and their equations:
| Data | Equation | Type |
|---|---|---|
| Perimeter $P$ | $2x+2y=P$ | linear |
| Area $A$ | $xy=A$ | nonlinear |
| Diagonal $d$ | $x^2+y^2=d^2$ (Pythagoras) | nonlinear |
| Right triangle with legs $a$, $b$, hypotenuse $c$ | $a^2+b^2=c^2$ (Pythagoras) | nonlinear |
| Height $h$ to the hypotenuse | $ab=ch$ (the area is computed in two ways: $\frac{ab}{2}=\frac{ch}{2}$) | nonlinear |
If all the relationships are linear, you use elimination or substitution. When an area or a diagonal appears, solve for one unknown in the linear equation and substitute it into the other: you get a quadratic equation.
The perimeter of a rectangular plot is $130$ meters. The length of the plot is known to be $15$ meters greater than its width. Set up a system of equations and determine…
The perimeter of a rectangular plot is $130$ meters. The length of the plot is known to be $15$ meters 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.
A single quadratic equation: fence and box
Sometimes the problem reduces to one unknown after substitution. Strategy:
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Choose the unknown (the width, for example) and express the others in terms of it.
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Write the final condition (area, volume) with that unknown.
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Bring the equation to the form $ax^2+bx+c=0$ and solve it.
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Discard the impossible solutions.
Boxes without a lid
If squares of side $c$ are cut from the corners of a rectangular sheet of width $a$ and length $l$, the resulting box has a base of $(l-2c)$ by $(a-2c)$ and a height of $c$. Its volume is $V=(l-2c)(a-2c)\cdot c$.
A farmer wants to fence off a rectangular field bordering a straight river, so no fence is needed along the riverbank. He has exactly $400$ meters 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 $400$ meters 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.
An open-topped box is to be constructed from a rectangular sheet of cardboard by cutting squares of side $5$ cm from each corner and folding up the sides. The length of…
An open-topped box is to be constructed from a rectangular sheet of cardboard by cutting squares of side $5$ cm from each corner and folding up the sides. The length of the original sheet is twice its width. If the volume of the resulting box must be $1500$ cm³, set up the necessary equation or system and determine the dimensions of the original sheet of cardboard.
Motion and work
The basic relationship of uniform motion is $d=v\cdot t$. In each problem, write a $d=vt$ equation for each vehicle.
Working together
If a faucet fills a tank in $x$ hours, in one hour it fills $\dfrac1x$ of the tank. If two faucets work together, their parts per hour add up.
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.
Three unknowns
When there are three unknown quantities, you need three equations. To solve, eliminate one unknown and you go back to a $2\times2$ system.
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 $240\,000$ pesos more…
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 $240\,000$ pesos more than partner B. If the total profit to be distributed is $3\,120\,000$ pesos, set up a system of three linear equations and calculate the amount each partner receives.
A sporting goods store sells soccer balls, basketballs, and volleyballs. The price of a soccer ball is $1\,500$ pesos, a basketball is $1\,800$ pesos, and a volleyball…
A sporting goods store sells soccer balls, basketballs, and volleyballs. The price of a soccer ball is $1\,500$ pesos, a basketball is $1\,800$ pesos, and a volleyball is $1\,200$ pesos. A club bought $40$ balls for a total of $61\,800$ pesos. Knowing that the number of soccer balls 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.
In a sports competition, gold, silver and bronze medals award $5$, $3$ and $1$ points respectively. A team has won a total of $25$ medals and $75$ points. Knowing that…
In a sports competition, gold, silver and bronze medals award $5$, $3$ and $1$ points respectively. A team has won a total of $25$ medals and $75$ points. Knowing that the number of silver medals exceeds the number of gold medals by $1$, set up a system of equations and find how many medals of each type the team won.
Mixtures, alloys, and production with three components
Here the ideas of mixtures, percentages, and three unknowns are combined. One equation is usually the total amount and another is the value, the concentration, or the total cost.
A chemistry laboratory has three solutions of the same acid with concentrations of $10\%$, $20\%$ and $40\%$. They want to obtain $100$ ml of a new solution at $25\%$.…
A chemistry laboratory has three solutions of the same acid with concentrations of $10\%$, $20\%$ and $40\%$. They want to obtain $100$ ml of a new solution at $25\%$. If the same volume of the $10\%$ solution is used as the $20\%$ solution, what volume of each solution should be mixed?
A company manufactures three computer models: basic, intermediate and advanced. The total daily production is $400$ units. The manufacturing cost is $18\,000$ pesos for…
A company manufactures three computer models: basic, intermediate and advanced. The total daily production is $400$ units. The manufacturing cost is $18\,000$ pesos for the basic model, $30\,000$ pesos for the intermediate model and $48\,000$ pesos for the advanced model, with a total daily cost of $11\,700\,000$ pesos. If the company decides to increase production of the basic model by $20\%$ and reduce that of the advanced model by $10\%$, total production would increase by $15$ units. Set up a system of equations with three unknowns and determine the initial production of each model.
An alloy is composed of gold, silver and copper. The total weight of a piece of this alloy is $120$ grams. It is known that the combined weight of gold and silver…
An alloy is composed of gold, silver and copper. The total weight of a piece of this alloy is $120$ grams. It is known that the combined weight of gold and silver exceeds the weight of copper by $40$ grams. When submerged in water, the piece loses $10$ grams of its apparent weight due to upthrust. Given that the weight loss in water for pure gold is $1/20$ of its weight, for silver $1/10$, and for copper $1/8$, set up a system of equations and calculate how many grams of each metal the piece contains.
A system with a parameter
Some models, such as distributing vehicles along three routes, come with a parameter $k$ that is not known in advance. The goal is to see for which values of $k$ the system has one solution, infinitely many, or none. The idea: eliminate unknowns until you reach an equation of the form $(\text{something with }k)\cdot z=\text{number}$.
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.
Summary
Back to the beginning: with $3x+2y=270$ and $2x+3y=255$, a juice costs $\text{RD}\$60$ and a roll costs $\text{RD}\$45$.