Word problems with systems of equations
Ximena and Diego go to the street market. Ximena pays $\$175$ for 3 tortas and 2 fruit waters. Diego pays $\$150$ for 2 tortas and 3 fruit waters. Neither of them saw the prices. How much does one torta cost, and how much does one fruit water cost? You already know how to solve a system. The hard part is translating the word problem into equations, and that is what this page is about.
Setting up: as many equations as unknowns
Step 2 is the hard part. These translations come up again and again:
| In the problem | In algebra |
|---|---|
| The sum of two numbers is 54 | $x + y = 54$ |
| Their difference is 12 | $x - y = 12$ |
| $x$ is three times $y$ | $x = 3y$ |
| $x$ is 15 more than $y$ | $x = y + 15$ |
| In 10 years (ages) | $x + 10$ and $y + 10$ |
| Goes up 5% | $1.05x$ |
| Goes down 10% | $0.90y$ |
| Three-digit number $c\,d\,u$ | $100c + 10d + u$ |
| A fraction $\tfrac{1}{20}$ of the weight $x$ | $\dfrac{x}{20}$ |
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?
Prices and mixtures
Back to the street market. Each purchase is an equation: $t$ is the price of one torta and $a$ is the price of one fruit water.
Each line on the graph is a purchase. The point where they cross is the only pair of prices that satisfies both. Move your eyes to $(45, 20)$.
In a mixture there are always two equations: one for quantity (what the kilos or liters add up to) and one for value (price, or amount of substance, times quantity).
Back to the beginning: the torta costs $\$45$ and the fruit water costs $\$20$.
In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $42$ pesos. Another customer buys $2$ screws and $5$ nuts for $35$ pesos. Set up a system of…
In a hardware shop, a customer buys $4$ screws and $3$ nuts for a total of $42$ pesos. Another customer buys $2$ screws and $5$ nuts for $35$ 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 $80$ pesos/liter and another at $110$ pesos/liter—to obtain a barrel of $300$ liters of mixture that will be sold at $92$…
In a winery, two types of wine are mixed—one at $80$ pesos/liter and another at $110$ pesos/liter—to obtain a barrel of $300$ liters of mixture that will be sold at $92$ 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 $24\,000$ 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 $24\,000$ pesos. This month, the mortgage payment rose by $5\%$ and the food expenditure decreased by $10\%$, making the total expenditure on these two items $23\,400$ pesos. Set up a system of equations to calculate how much they spent on each item last month.
Counts, ages, and scores
Many problems have a "double count": you count the same thing in two ways (heads and legs, tickets and money, correct answers and points).
With ages, the difference between two people never changes, but the ratio between them does. Use $x + n$ for "in $n$ years" and $x - n$ for "$n$ years ago."
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?
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?
Geometry: perimeter, area, and diagonal
The perimeter gives a linear equation. But the area ($xy$) and the diagonal ($x^2 + y^2 = d^2$, by the Pythagorean theorem) give nonlinear equations. They are solved by substitution: solve for one unknown in the linear equation and replace it.
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.
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.
Geometry: boxes and right triangles
Open-top box. When you cut squares of side $c$ from the corners of a sheet of size $L \times A$, the box measures $(L - 2c)$ in length, $(A - 2c)$ in width, and $c$ in height.
Right triangle with three unknowns $a, b, c$ (legs and hypotenuse): usually the perimeter, the area, or a height is given, and always the Pythagorean theorem.
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.
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: $d = vt$
In uniform motion, all the information fits in $d = vt$. Write that relationship for each vehicle or person that moves, and look for what they share: the distance, the time, or the sum of distances.
With a round trip (same distance, different speed), what repeats is the distance: $d = v_1 t_1 = v_2 t_2$. With teamwork (faucets, workers), you add rates: if one faucet fills a tank in $x$ hours, in one hour it fills $\tfrac{1}{x}$ of the tank.
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 the problem has three quantities, you need three equations. The idea is the same as in elimination: eliminate one unknown from two pairs of equations so you end up with a $2 \times 2$ 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 $80\,000$ 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 $80\,000$ pesos more than partner B. If the total profit to be distributed is $1\,040\,000$ 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 $500$ pesos, a basketball is $600$ pesos and a volleyball is $400$…
A sporting goods store sells soccer balls, basketballs and volleyballs. The price of a soccer ball is $500$ pesos, a basketball is $600$ pesos and a volleyball is $400$ pesos. A club has bought $40$ balls for a total of $20\,600$ 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.
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?
Three unknowns with digits, percents, and fractions
For these problems, the table of translations at the beginning is your best tool. Three typical cases:
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Digits. The number is $100c + 10d + u$. Reversed, it is $100u + 10d + c$. "The middle digit is the average of the other two" means $d = \tfrac{c + u}{2}$.
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Percent changes in production. "Increases 20%" is $1.2x$, and "decreases 10%" is $0.9z$. The new total production is the sum of the new amounts.
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Fractions of a weight. "Loses $\tfrac{1}{20}$ of its weight" is $\tfrac{x}{20}$. The total lost adds up what each ingredient loses.
Density and weight loss in water
Some alloy problems come from physics. Density is mass divided by volume: $\rho = \dfrac{m}{V}$. When you submerge an object in water, the water pushes it upward with a force equal to the weight of the water it displaces (buoyant force). That is why the object "weighs less" in water: its apparent weight drops by exactly the weight of that volume of water.
Since water weighs $1$ g per cm³, an object of volume $V$ cm³ loses $V$ grams. And since $V = \dfrac{m}{\rho}$, a denser metal loses a smaller fraction of its weight. So "copper loses $\tfrac{1}{8}$ of its weight in water" means that $m$ grams of copper lose $\tfrac{m}{8}$ grams. The total loss of an alloy is the sum of the losses of each metal.
With three metals you do the same thing: one equation for the total weight, another for the loss in water, and another with the relationship the problem gives between the weights.
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.
A company manufactures three computer models: basic, intermediate and advanced. The total daily production is $400$ units. The manufacturing cost is $6\,000$ pesos for…
A company manufactures three computer models: basic, intermediate and advanced. The total daily production is $400$ units. The manufacturing cost is $6\,000$ pesos for the basic model, $10\,000$ pesos for the intermediate model and $16\,000$ pesos for the advanced model, with a total daily cost of $3\,900\,000$ 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.
Problems with a parameter
Sometimes one piece of data is a letter $k$. Then you are not asked for a solution, but for which values of $k$ give one solution, none, or infinitely many. The idea is to eliminate unknowns and see which final equation is left.
Move $k$ on the graph: for $k = 2$ the two lines are parallel and never cross.
In a $3 \times 3$ system the procedure is the same: eliminate $x$ and get two equations in $y$ and $z$. Eliminate $y$ and you will reach something like $(k - a)z = b$. If $k \neq a$ there is exactly one solution. If $k = a$ and $b \neq 0$ it is inconsistent. If $k = a$ and $b = 0$ there are infinitely many solutions.
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.