Systems of equations
At the school stationery store, 2 pencils and 3 erasers cost $\text{RD}\$130$; 2 pencils and 1 eraser cost $\text{RD}\$90$. How much does each item cost? There are two unknowns (the pencil and the eraser) and two clues. A single equation is not enough: you have to solve them 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 $x = 35$ and $y = 20$: $2 \times 35 + 3 \times 20 = 130$ ✓ and $2 \times 35 + 20 = 90$ ✓.
Graphical meaning
Each equation of this type is a line: its points are the solutions of that equation. The solution of the system is the point where the two lines intersect. Move the lines and observe:
Depending on how many solutions it has, the system is classified as follows:
| Lines | Solutions | Name |
|---|---|---|
| They intersect | one | consistent and independent |
| They coincide | infinitely many | consistent and dependent |
| Parallel | none | inconsistent |
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 $300$ 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 $300$ 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 $180$ 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 very well when one unknown is already isolated or has coefficient $1$ or $-1$.
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} $$Equating method
You solve for the same unknown in both equations and set the two expressions equal. It is a disguised substitution: if $x$ equals one thing and also equals another, those two things are equal.
On the graph, with $x$ on the horizontal axis, the lines are $y = x - 5$ and $y = (x - 2)/2$:
Solve the following system using the equating method. To do this, equate the two expressions representing the value of $x$: $$\begin{cases} x = y + 5 \\ x = 2y + 2…
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.
Elimination method
You multiply the equations by suitable numbers so that, when you add (or subtract) them, one unknown disappears.
When neither unknown has equal or opposite coefficients, you multiply 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 $240$ pesos. Pedro has bought $2$ pens and $1$ rubber for $120$ pesos. If $x$ is the price of…
Lucía has bought $2$ pens and $3$ rubbers at the stationer's for a total of $240$ pesos. Pedro has bought $2$ pens and $1$ rubber for $120$ pesos.
If $x$ is the price of a pen and $y$ that of a rubber, the system representing the situation is:
$$\begin{cases} 2x + 3y = 240 \\ 2x + y = 120 \end{cases}$$Find the price of each item using the elimination method (subtracting both equations).
In the school canteen, a group of friends paid $\text{RD\$}\ 900$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $\text{RD\$}\ 1\,050$ for 3 soft…
In the school canteen, a group of friends paid $\text{RD\$}\ 900$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $\text{RD\$}\ 1\,050$ for 3 soft drinks and 4 sandwiches. Use the elimination method to find the price of a soft drink and that of a sandwich.
Setting up problems with systems
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Choose two unknowns and write what each one means.
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Translate each piece of data from the problem into a different equation.
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Solve using the method that suits you best.
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Check in the original problem, not just in the equations.
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 $\text{RD\$}\ 240$ and adult tickets at $\text{RD\$}\ 420$. At the end of the…
To organise a charity festival, two types of tickets were sold: children's tickets at $\text{RD\$}\ 240$ and adult tickets at $\text{RD\$}\ 420$. At the end of the event, a total of 240 people attended and the total raised was $\text{RD\$}\ 81\,000$. How many children and how many adults attended the festival?
Ages, digits, and shapes
Ages. A table organizes the data. Digits: a two-digit number $xy$ has value $10x + y$; with the digits reversed it has value $10y + x$. Rectangles: the perimeter is $2x + 2y$ and the area is $x \cdot y$.
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.
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 $10x + y$).
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.
Motion: meeting and catching up
With constant speed, $\text{distance} = \text{speed} \times \text{time}$. There are two typical situations with two moving bodies on the same route:
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They approach each other (opposite directions): the speeds add. $d = (v_1 + v_2)\,t$.
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One chases the other (same direction): the speeds subtract. $d = (v_1 - v_2)\,t$.
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?
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.
Discounts and mixtures
Percents. A $10\%$ discount leaves the price at $0.9$ of the original; a $20\%$ discount, at $0.8$. So an item with original price $x$ costs $0.9x$ with a $10\%$ discount.
Mixtures. Two equations are set up: one for the amounts and one for the amount of pure substance (percent $\times$ amount).
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?
Systems with parentheses and fractions
Sometimes the system doesn't come ready to use. Before choosing a method, bring it to the form $ax + by = c$:
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Remove parentheses (distributive property).
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Remove denominators: multiply each equation by the LCM of its denominators.
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Gather the terms with $x$ and $y$ on the left and the numbers on the right.
When simplifying, the system may turn out to be special. See what happens when you eliminate:
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If you end up with something false like $0 = 5$, the system is inconsistent (parallel lines). Example: $x + y = 3$ and $2x + 2y = 8$.
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If you end up with something always true like $0 = 0$, it is consistent and dependent (the same line). Example: $x + y = 3$ and $2x + 2y = 6$.
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If you get values for $x$ and $y$, it is consistent and independent.
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 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} $$Once solved, state whether the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent.
Solve the following system of linear equations by first simplifying both expressions. To find the final solution, use the double elimination method (eliminating one…
Solve the following system of linear equations by first simplifying both expressions. To find the final solution, use the double elimination method (eliminating one unknown and then the other):
$$ \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} $$An auxiliary geometry calculation: lateral surface area of a frustum of a pyramid
A frustum of a regular square pyramid has two square bases (with sides $B$ and $b$) and four lateral faces that are identical trapezoids. The height of each trapezoid is the slant height $a$ of the frustum, and it is found with the Pythagorean theorem: if $h$ is the height of the frustum,
$$a = \sqrt{h^2 + \left(\frac{B - b}{2}\right)^2}$$Lateral surface area $= 4 \times \dfrac{B + b}{2} \times a$.
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 $85\text{ \text{RD\$}\ /m}^2$.
Back to the beginning: the stationery store system was $\begin{cases} 2x + 3y = 130 \\ 2x + y = 90 \end{cases}$. By subtracting we got $y = 20$ and $x = 35$: each pencil costs 35 pesos and each eraser 20 pesos.