The drama club sold 220 tickets for the spring play and collected $\$1{,}250$. Student tickets cost $\$5$ and adult tickets cost $\$8$. How many of each type did they sell? Solving a system is usually the easy part. The hard part is turning sentences into equations, and deciding how many unknowns you need: two, or three when the problem mixes three quantities.
Setting up: as many equations as unknowns
Back to the start: the drama club sold 170 student tickets and 50 adult tickets. Each equation is a line, and the solution is the point where the lines cross.
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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 there are hens and cows. If the animals' heads are counted, there is a total of $45$; if the legs are counted, there are $140$. How many animals of each type…
On a farm there are hens and cows. If the animals' heads are counted, there is a total of $45$; if the legs are counted, there are $140$. How many animals of each type are there on the farm?
A coastal hotel has double and single rooms. In total there are $50$ rooms and $87$ beds. Assuming all rooms are fully occupied, how many rooms of each type does the…
A coastal hotel has double and single rooms. In total there are $50$ rooms and $87$ beds. Assuming all rooms are fully occupied, how many rooms of each type does the hotel have?
Price and mixture problems all follow one pattern: (amount) × (price or rate) = (total value). One equation adds the amounts. Another adds the values.
Percent changes work the same way. If a quantity $x$ increases by 10%, the new amount is $1.10x$. If it decreases by 20%, the new amount is $0.80x$.
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In a hardware store, 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 store, 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 $\$4$/gallon and another at $\$5,50$/gallon, to obtain a barrel of $300$ gallons of mixture that will be sold at…
In a winery, two types of wine are mixed, one at $\$4$/gallon and another at $\$5,50$/gallon, to obtain a barrel of $300$ gallons of mixture that will be sold at $\$4,60$/gallon. How many gallons of each type of wine have been used to make the mixture?
A family allocates a part of their monthly income to pay the mortgage and another part to food. Last month, the sum of these two expenses was $\$1200…
A family allocates a part of their monthly income to pay the mortgage and another part to food. Last month, the sum of these two expenses was $\$1200$. This month, the mortgage payment has increased by $5\%$ and the food expense has decreased by $10\%$, with the total expense for these two items being $\$1170$. Set up a system of equations to calculate how much they spent on each item last month.
Ages. When time passes, everyone gets older by the same amount. "In 12 years" adds 12 to every person's age.
Digits. A two-digit number with tens digit $t$ and units digit $u$ is worth $10t + u$, not $t + u$. Reversing the digits gives $10u + t$.
Geometry. Perimeter gives a linear equation, like $2x + 2y = P$. Area ($xy = A$) and diagonals ($x^2 + y^2 = d^2$) are not linear. You will handle those in the nonlinear section below.
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A father's current age is three times that of his son. In $10$ years, the father's age will be exactly double that of the son. Determine the current ages of both.
A father's current age is three times that of his son. In $10$ years, the father's age will be exactly double that of the son. Determine the current ages of both.
The perimeter of a rectangular plot measures $130$ feet. It is known that the length of the plot is $15$ feet more than the width. Set up a system of equations and…
The perimeter of a rectangular plot measures $130$ feet. It is known that the length of the plot is $15$ feet more than the width. Set up a system of equations and determine the dimensions of the plot.
The sum of the three digits of a number is $12$. The tens digit is the mean of the other two. If the order of the digits is reversed, the resulting number is $198$ units…
The sum of the three digits of a number is $12$. The tens digit is the 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 out what the number is.
A cyclist makes a round trip between two towns separated by a distance $d$. On the outward journey, with a tailwind, they travel at a constant speed of $30$ mph. On the…
A cyclist makes a round trip between two towns separated by a distance $d$. On the outward journey, with a tailwind, they travel at a constant speed of $30$ mph. On the return journey, with a headwind, they travel at $20$ mph. If the total time taken for the complete round trip is $2,5$ hours, set up a system of equations and find the distance between the two towns and the time taken for each journey.
Two faucets fill a tank in $6$ hours if they are opened at the same time. If the first faucet takes $5$ hours less than the second to fill the tank separately, set up a…
Two faucets fill a tank in $6$ hours if they are opened at the same time. If the first faucet takes $5$ hours less than the second to fill the tank separately, set up a system of equations (which will include algebraic fractions) to determine how long each faucet would take to fill it individually.
Two trains depart simultaneously from two stations A and B separated by $450$ miles, traveling on parallel tracks and in opposite directions. They pass each other after…
Two trains depart simultaneously from two stations A and B separated by $450$ miles, traveling on parallel tracks and in opposite directions. They pass each other after $3$ hours. If the train leaving A had departed $1$ hour before the train from B, they would have passed each other $306$ miles from station A. Set up a system of equations to calculate the constant speeds of both trains.
When a problem has three unknown quantities, you need a $3 \times 3$ system. You solve it by elimination: reduce it to a $2 \times 2$ system, solve that, then substitute back.
Some word problems give you a relationship that makes things easier, like "the number of A equals the sum of B and C" or "A gets double what B gets." Use it to substitute right away and reduce the system.
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A sports shop sells soccer balls, basketballs, and volleyballs. The price of a soccer ball is $\$25$, a basketball is $\$30$, and a volleyball is $\$20…
A sports shop sells soccer balls, basketballs, and volleyballs. The price of a soccer ball 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 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 have been purchased.
A company distributes its annual profits among its three partners A, B, and C. Partner A receives double that of partner B, and partner C receives $\$4000…
A company distributes its annual profits among its three partners A, B, and C. Partner A receives double that of 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.
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 out how many medals of each type the team has won.
Three unknowns with percents, mixtures, and weights
Systems with three unknowns often hide decimals and fractions. Two habits make them much easier:
Equations must compare the same kind of quantity. For a mixture, compare the amount of pure substance (liters, mL, ounces), not percents.
Clear decimals and fractions first. Multiply by 10 or 100 to remove decimals, or by the LCD to remove fractions. For example, take $\dfrac{x}{20} + \dfrac{y}{10} + \dfrac{z}{8} = 10$ and multiply both sides by 40: you get $2x + 4y + 5z = 400$.
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In a chemistry laboratory, there are three solutions of the same acid with concentrations of $10\%$, $20\%$, and $40\%$. We want to obtain $100$ mL of a new solution at…
In a chemistry laboratory, there are three solutions of the same acid with concentrations of $10\%$, $20\%$, and $40\%$. We 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 must be mixed?
A company manufactures three models of computers: basic, medium, and advanced. The total daily production is $400$ units. The manufacturing cost is $\$300…
A company manufactures three models of computers: basic, medium, and advanced. The total daily production is $400$ units. The manufacturing cost is $\$300$ for the basic, $\$500$ for the medium, and $\$800$ for the advanced, with a total daily cost of $\$195000$. If the company decides to increase the production of the basic model by $20\%$ and reduce that of the advanced model by $10\%$, the 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 $120$ ounces. It is known that the weight of the gold and silver together…
An alloy is composed of gold, silver, and copper. The total weight of a piece of this alloy is $120$ ounces. It is known that the weight of the gold and silver together exceeds the weight of the copper by $40$ ounces. When the piece is submerged in water, it loses $10$ ounces of its apparent weight due to hydrostatic upthrust. If it is known 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 ounces of each metal the piece contains.
Area, diagonals, and the Pythagorean Theorem give equations that are not linear. The strategy is the same as before, but the substitution leads to a quadratic equation.
The graph shows why there are two solutions: the hyperbola $y = \dfrac{48}{x}$ and the line $y = 14 - x$ cross at $(6, 8)$ and $(8, 6)$.
Boxes from cardboard. Cutting a square of side $c$ from each corner and folding up the sides gives a box of height $c$. Each side of the sheet loses $2c$, because you cut from both ends.
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The dimensions of a rectangular playing field are such that its area is $6000$ sq ft and the length of its diagonal is $130$ ft. Set up a non-linear system of equations…
The dimensions of a rectangular playing field are such that its area is $6000$ sq ft and the length of its diagonal is $130$ ft. 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 triangle knowing that its perimeter is $60$ inches and that the height drawn to the hypotenuse measures $12$ inches. Set up…
Determine the lengths of the sides of a right triangle knowing that its perimeter is $60$ inches and that the height drawn to the hypotenuse measures $12$ inches. Set up a non-linear system of equations with three unknowns (legs and hypotenuse) to solve it.
A box without a lid is to be made from a rectangular sheet of cardboard by cutting squares of side $5$ inches from each corner and folding up the edges. The length of…
A box without a lid is to be made from a rectangular sheet of cardboard by cutting squares of side $5$ inches from each corner and folding up the edges. The length of the original sheet is double the width. If the volume of the resulting box must be $1500$ cubic inches, set up the necessary equation or system and determine the dimensions of the original cardboard sheet.
Before you solve a problem completely, ask whether it has a solution, and whether it has only one.
Nonlinear problems. After you substitute, look at the quadratic. Its discriminant $b^2 - 4ac$ tells you how many real solutions exist: two if it is positive, one if it is zero, and none if it is negative. A fence along a river or a wall replaces one side of the rectangle, so the perimeter equation has only three sides in it.
Systems with a parameter. Some models contain an unknown constant $k$, such as the traffic split on three routes. To discuss the system, solve it with a letter in it, find which values of $k$ cause trouble, and test those values separately.
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A farmer wants to fence a rectangular field that borders a straight river, so no fence is needed on the river side. They have exactly $400$ feet of fencing. Knowing that…
A farmer wants to fence a rectangular field that borders a straight river, so no fence is needed on the river side. They have exactly $400$ feet of fencing. Knowing that the enclosed area must be $20000$ sq ft, 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.
A system of linear equations dependent on a parameter $k$ models the distribution of vehicles on three alternative routes $x$, $y$, $z$:…
A system of linear equations dependent 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 means physically for the system to be inconsistent.