Systems of equations
At the stationery store on the corner, Ximena paid 36 pesos for 2 pens and 3 erasers. Diego paid 24 pesos for 2 pens and 1 eraser. How much does each item cost? There are two unknowns (the price of a pen and the price of an eraser) and two clues. One equation isn't enough: you have to use both at the same time. That is a system of equations.
What a system is
To check a solution, substitute it into both equations, not just one. With $x = 9$ and $y = 6$: $2 \times 9 + 3 \times 6 = 36$ ✓ and $2 \times 9 + 6 = 24$ ✓.
Graphical meaning
Each equation of this kind is a line: all of its points are solutions of that equation. The solution of the system is the point where the two lines intersect. Move the sliders and watch what happens.
Two lines can be positioned in three ways. That gives three types of system:
In a role-playing video game, the experience points of two skills, Strength ($x$) and Magic ($y$), must satisfy the conditions of the following system of equations:…
In a role-playing video game, the experience points of two skills, Strength ($x$) and Magic ($y$), must satisfy the conditions of the following system of equations:
$$\begin{cases} x + y = 15 \\ 2x - y = 9 \end{cases}$$Check, by carrying out the necessary operations, which of these pairs of values is the correct solution to the system for the character to level up:
- a) $x = 10, y = 5$
- b) $x = 8, y = 7$
Translate into algebraic language and write down the corresponding system of equations for the following situation: "In a café, we were charged $100$ pesos for two soft…
Translate into algebraic language and write down the corresponding system of equations for the following situation:
"In a café, we were charged $100$ pesos for two soft drinks ($x$) and a bag of chips ($y$). If we buy one soft drink and one bag of chips, the price is $60$ pesos."
(It is not necessary to solve it, just set up the system).
Substitution method
You solve for one unknown in one equation and substitute it into the other. It works well when one unknown is already solved for or has coefficient $1$ or $-1$. The reason is that the other equation is left with a single unknown, and you already know how to solve that.
Solve the following system by the substitution method. Notice that in the second equation the unknown $y$ is already isolated, which makes the process easier:…
Solve the following system by the substitution method. Notice that in the second equation the unknown $y$ is already isolated, which makes the process easier:
$$\begin{cases} 3x + y = 10 \\ y = 2x \end{cases}$$Solve the following system of equations using the substitution method. Remember that it is advisable to rearrange for the unknown with the simplest coefficient: $$…
Solve the following system of equations using the substitution method. Remember that it is advisable to rearrange for the unknown with the simplest coefficient:
$$ \begin{cases} 3x - y = 7 \\ 5x + 2y = 19 \end{cases} $$Equating method
You solve for the same unknown in both equations and set the two expressions equal. It makes sense: if $x$ has the same value in both, those two expressions must be equal to each other.
Here are the two lines from the example (written as $y = x - 5$ and $y = (x - 2)/2$). Find the point $(8, 3)$:
Solve the following system using the equating method. To do this, equate the two expressions representing the value of $x$: $$\begin{cases} x = y + 5 \\ x = 2y + 2…
Solve the following system using the equating method. To do this, equate the two expressions representing the value of $x$:
$$\begin{cases} x = y + 5 \\ x = 2y + 2 \end{cases}$$Look at the following figure made up of two rectangles with different properties. Rectangle A has a perimetre of $22\text{ cm}$. Rectangle B is such that its base is…
Look at the following figure made up of two rectangles with different properties. Rectangle A has a perimetre of $22\text{ cm}$. Rectangle B is such that its base is twice that of rectangle A and its height is three times that of rectangle A, having a total perimetre of $52\text{ cm}$.
Calculate the dimensions $x$ (base) and $y$ (height) of rectangle A by setting up and solving a system using the equating method.
Addition and subtraction method
It is also called the reduction or elimination method. You multiply the equations by suitable numbers so that, when you add (or subtract) them, one unknown disappears. It works because adding two equalities gives another true equality.
If no unknown cancels on its own, you first multiply one or both equations.
Solve the following system using the elimination method. Notice that the unknown $y$ has opposite coefficients ($+1$ and $-1$), so you can add the two equations directly…
Solve the following system using the elimination method. Notice that the unknown $y$ has opposite coefficients ($+1$ and $-1$), so you can add the two equations directly to eliminate it:
$$\begin{cases} 4x + y = 13 \\ x - y = 2 \end{cases}$$Lucía has bought $2$ pens and $3$ rubbers at the stationer's for a total of $80$ pesos. Pedro has bought $2$ pens and $1$ rubber for $40$ pesos. If $x$ is the price of a…
Lucía has bought $2$ pens and $3$ rubbers at the stationer's for a total of $80$ pesos. Pedro has bought $2$ pens and $1$ rubber for $40$ pesos.
If $x$ is the price of a pen and $y$ that of a rubber, the system representing the situation is:
$$\begin{cases} 2x + 3y = 80 \\ 2x + y = 40 \end{cases}$$Find the price of each item using the elimination method (subtracting both equations).
In the school canteen, a group of friends paid $\$300$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $\$350$ for 3 soft drinks and 4 sandwiches.…
In the school canteen, a group of friends paid $\$300$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $\$350$ for 3 soft drinks and 4 sandwiches. Use the elimination method to find the price of a soft drink and that of a sandwich.
Systems with parentheses and fractions; special cases
Before choosing a method, put each equation in the form $ax + by = c$:
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Remove parentheses using the distributive property.
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If there are denominators, multiply the whole equation by the LCM of the denominators.
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Collect the terms with $x$ and with $y$ on the left side and the numbers on the right.
When there isn't just one solution
If an unknown disappears when you simplify and you are left with an equality with no letters, see whether it is false or true:
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False (for example $0 = 5$): the lines are parallel. The system is inconsistent: it has no solution.
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True (for example $0 = 0$): the two equations are the same line. The system is consistent dependent: it has infinitely many solutions.
For example, $x + y = 3$ and $x + y = 5$ cannot both hold at once (inconsistent). On the other hand, $x + y = 3$ and $2x + 2y = 6$ are the same line (infinitely many solutions).
Find the solution to the following system of equations using the method you consider most appropriate. Before solving it, check whether it is necessary to simplify the…
Find the solution to the following system of equations using the method you consider most appropriate. Before solving it, check whether it is necessary to simplify the expressions by expanding brackets:
$$ \begin{cases} 2(x + y) - 3x = 1 \\ x + 3(y - 1) = 2 \end{cases} $$Solve the following system of linear equations by first simplifying both expressions. To find the final solution, use the double elimination method (eliminating one…
Solve the following system of linear equations by first simplifying both expressions. To find the final solution, use the double elimination method (eliminating one unknown and then the other):
$$ \begin{cases} \dfrac{2(x-1)}{3} - \dfrac{y+2}{6} = -1 \\ \\ \dfrac{x+y}{2} + \dfrac{x-y}{3} = \dfrac{7}{6} \end{cases} $$Solve the following system using the method you consider most appropriate, after expanding brackets and collecting like terms: $$ \begin{cases} 3(x+2) - 5(y-1) = 2x + 5…
Solve the following system using the method you consider most appropriate, after expanding brackets and collecting like terms:
$$ \begin{cases} 3(x+2) - 5(y-1) = 2x + 5 \\ \\ 2(x+y) - 3(x-y) = 11 \end{cases} $$Once solved, state whether the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent.
Setting up word problems with systems
Always write down what each letter means (for example, $x$ = price of one pen) and what units you are working with.
On a small farm there is a total of $12$ animals, consisting of chickens ($x$) and rabbits ($y$). If we count all their legs, there is a total of $34$ legs. The system…
On a small farm there is a total of $12$ animals, consisting of chickens ($x$) and rabbits ($y$). If we count all their legs, there is a total of $34$ legs. The system of equations describing this situation is:
$$\begin{cases} x + y = 12 \\ 2x + 4y = 34 \end{cases}$$Solve the system using whichever method you prefer to find out how many chickens and how many rabbits there are on the farm.
In a strategy video game, to unlock an upgrade you need to collect blue gems and red gems. We know that three times the number of blue gems plus twice the number of red…
In a strategy video game, to unlock an upgrade you need to collect blue gems and red gems. We know that three times the number of blue gems plus twice the number of red gems gives a total of 34 energy. Furthermore, the difference between the number of blue and red gems is only 3.
- a) Set up a system of equations to find out how many gems of each type you need.
- b) Check whether the pair of values $x = 8, y = 5$ (where $x$ are the blue gems and $y$ are the red gems) is the solution to the system.
To organise a charity festival, two types of tickets were sold: children's tickets at $\$80$ and adult tickets at $\$140$. At the end of the event, a total of 240 people…
To organise a charity festival, two types of tickets were sold: children's tickets at $\$80$ and adult tickets at $\$140$. At the end of the event, a total of 240 people attended and the total raised was $\$27\,000$. How many children and how many adults attended the festival?
Common types of problems
These problems show up again and again. Each one has a typical way of being set up.
| Type | Unknowns | Typical equations |
|---|---|---|
| Ages | current age of each person | one relationship today and another "in $n$ years" (add $n$ to both ages) |
| Digits | $x$ = tens, $y$ = units | the number is $10x + y$; reversed it is $10y + x$ |
| Motion | speeds or time and distance | $d = v \cdot t$; moving toward each other the speeds add, in the same direction they subtract |
| Mixtures | liters of each solution | total liters, and amount of pure substance: liters $\times$ percent |
| Figures | base and height | perimeter $2x + 2y$ or area $x \cdot y$ |
A mother's age is currently three times that of her son. However, in 12 years' time, the mother's age will be only twice that of her son. How old are they both now? Use…
A mother's age is currently three times that of her son. However, in 12 years' time, the mother's age will be only twice that of her son. How old are they both now? Use a table to organise the age data before setting up the system.
Two cyclists, Ana and David, are a distance of $45\text{ km}$ apart. Ana sets off from point A at a constant speed of $12\text{ km/h}$ and David sets off at the same…
Two cyclists, Ana and David, are a distance of $45\text{ km}$ apart. Ana sets off from point A at a constant speed of $12\text{ km/h}$ and David sets off at the same time from point B to meet Ana at $18\text{ km/h}$.
If $x$ is the time they take to meet and $y$ is the distance covered by Ana:
- a) Set up a system of equations relating the time and distance for both.
- b) How long will it take them to cross paths and at what distance from point A will it happen?
Find the dimensions of a rectangle given that its perimetre is $40$ cm. If its base were tripled and its height halved, the perimetre would become $70$ cm. Once you have…
Find the dimensions of a rectangle given that its perimetre is $40$ cm. If its base were tripled and its height halved, the perimetre would become $70$ cm. Once you have found the base ($x$) and the height ($y$), calculate the area of the original rectangle.
When you've mastered these, try the harder problems on mixtures, digits, discounts, and cyclists meeting:
In a computer shop, a customer buys 3 mechanical keyboards and 2 wireless mice for a total of $135$ pesos. The next day, taking advantage of a sale, they buy 2 keyboards…
In a computer shop, a customer buys 3 mechanical keyboards and 2 wireless mice for a total of $135$ pesos. The next day, taking advantage of a sale, they buy 2 keyboards and 4 mice for $111$ pesos. However, the deal was that keyboards had a discount of $10\%$ and mice a discount of $20\%$ off their original price.
Calculate the original price of each item by setting up a system of equations.
A chemistry laboratory needs to prepare $10$ liters of a $25\%$ acid solution. To do this, it has two containers: one with an acid concentration of $15\%$ and another…
A chemistry laboratory needs to prepare $10$ liters of a $25\%$ acid solution. To do this, it has two containers: one with an acid concentration of $15\%$ and another with a concentration of $40\%$. How many liters must be mixed from each container to obtain the desired solution?
The sum of the two digits of a number is $12$. If we reverse the order of its digits, we obtain a new number that is $18$ units greater than the original. Set up a…
The sum of the two digits of a number is $12$. If we reverse the order of its digits, we obtain a new number that is $18$ units greater than the original. Set up a system of equations to find that number.
(Hint: A two-digit number is written as $10x + y$).
Two cyclists, Elena and Marco, are $36$ km apart. If they ride towards each other, they take $45$ minutes to meet. If they ride in the same direction (Elena chasing…
Two cyclists, Elena and Marco, are $36$ km apart. If they ride towards each other, they take $45$ minutes to meet. If they ride in the same direction (Elena chasing Marco), Elena takes $3$ hours to catch up with him.
Calculate the speed of each cyclist in km/h by setting up a system of equations.
A geometry review: lateral area of a frustum of a pyramid
One of the hard exercises mixes algebra with geometry: you have to paint the faces of a frustum of a pyramid. It doesn't involve a system of equations, but it does involve organized calculations. A frustum of a regular square pyramid is a pyramid whose top has been cut off with a plane parallel to the base. It has two square bases (the large one and the small one) and four identical lateral faces, each one a trapezoid.
The area of a trapezoid is $\dfrac{(B + b)}{2} \cdot h$. Here its height $h$ is not the height of the frustum, but the slant height $a$: the height of the slanted face. It is obtained with a right triangle (the Pythagorean theorem, $a^2 = c_1^2 + c_2^2$):
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one leg is the height of the frustum;
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the other leg is the difference of the halves of the sides: $\dfrac{B - b}{2}$;
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the hypotenuse is the slant height $a$.
Calculate the cost of painting the lateral surface of a regular quadrangular frustum of a pyramid whose bases have side lengths of $4\text{ m}$ and $2\text{ m}$ and…
Calculate the cost of painting the lateral surface of a regular quadrangular frustum of a pyramid whose bases have side lengths of $4\text{ m}$ and $2\text{ m}$ and whose height is $1.5\text{ m}$, knowing that the price is $85\text{ \$/m}^2$.
Back to the beginning: the stationery store system was $2x + 3y = 36$ and $2x + y = 24$. Solving it gave $x = 9$ and $y = 6$: each pen costs 9 pesos and each eraser 6 pesos.