Systems of Linear Equations
At the concession stand, Jayden pays $\$7$ for 2 slices of pizza and 1 lemonade. Aaliyah pays $\$11$ for 3 slices and 2 lemonades. How much is one slice? How much is one lemonade? There are two unknowns (the slice and the lemonade) and two clues. One equation is not enough, so you need to use both at the same time. That is a system of equations.
What Is a System?
To check a solution, plug it into both equations. Try $(0.5, 1)$ in the system above: $2(0.5) + 3(1) = 4$ ✓ and $2(0.5) + 1 = 2$ ✓. So $(0.5, 1)$ is the solution.
Order matters. In the pair $(x, y)$, the first number is always $x$.
What the solution looks like on a graph
Each linear equation with two unknowns is a line: every point on the line is a solution of that one equation. A solution of the system must lie on both lines, so it is the point where the lines intersect. Move the lines and watch the intersection point.
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 performing the necessary operations, which of these pairs of values is the correct solution to the system so that the character can level up:
- a) $x = 10, y = 5$
- b) $x = 8, y = 7$
The Substitution Method
Isolate one variable in one equation, then substitute that expression into the other equation. The method works best when a variable is already isolated or has a coefficient of $1$ or $-1$.
Sometimes you have to isolate a variable first. Choose the one with coefficient $1$ or $-1$ so you can avoid fractions.
Solve the following system using the substitution method. Notice that in the second equation the unknown $y$ is already isolated, which makes the process easier:…
Solve the following system using 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 isolate the variable with the simplest coefficient: $$…
Solve the following system of equations using the substitution method. Remember that it is advisable to isolate the variable with the simplest coefficient:
$$ \begin{cases} 3x - y = 7 \\ 5x + 2y = 19 \end{cases} $$Setting Expressions Equal
If the same variable is isolated in both equations, then both expressions equal the same number. So you can set them equal to each other. This is just substitution: you replace one expression with the other, and you get one equation with one unknown. Some problems call this the equating method.
Here is what it looks like on a graph. For $y = 2x + 1$ and $y = -x + 7$, setting the right sides equal gives $2x + 1 = -x + 7$, so $x = 2$ and $y = 5$. The lines cross at $(2, 5)$.
Solve the following system using the equating method. To do this, equate the two expressions that represent the value of $x$: $$\begin{cases} x = y + 5 \\ x = 2y + 2…
Solve the following system using the equating method. To do this, equate the two expressions that represent the value of $x$:
$$\begin{cases} x = y + 5 \\ x = 2y + 2 \end{cases}$$The Elimination Method
Add or subtract the two equations so that one variable disappears. If the coefficients of a variable are not opposites, first multiply one or both equations by a number to make them opposites (or equal, if you plan to subtract).
When no variable cancels right away, multiply to create opposites.
Solve the following system using the elimination method. Observe that the unknown $y$ has opposite coefficients ($+1$ and $-1$), so you can add the two equations…
Solve the following system using the elimination method. Observe that the unknown $y$ has opposite coefficients ($+1$ and $-1$), so you can add the two equations directly to eliminate it:
$$\begin{cases} 4x + y = 13 \\ x - y = 2 \end{cases}$$Lucy bought $2$ pens and $3$ erasers at the stationery store for a total of $\$4$. Michael bought $2$ pens and $1$ eraser for $\$2$. If $x$ is the price of a pen and $y$…
Lucy bought $2$ pens and $3$ erasers at the stationery store for a total of $\$4$. Michael bought $2$ pens and $1$ eraser for $\$2$.
If $x$ is the price of a pen and $y$ the price of an eraser, the system that represents the situation is:
$$\begin{cases} 2x + 3y = 4 \\ 2x + y = 2 \end{cases}$$Find the price of each item using the elimination method (subtracting both equations).
In the school cafeteria, a group of friends paid $15\text{ \$}$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $17,50\text{ \$}$ for 3 soft drinks…
In the school cafeteria, a group of friends paid $15\text{ \$}$ for 4 soft drinks and 2 sandwiches. The next day, another group paid $17,50\text{ \$}$ for 3 soft drinks and 4 sandwiches. Use the elimination method to find the price of a soft drink and a sandwich.
How Many Solutions?
Two lines in a plane can cross, be parallel, or be the same line. That gives three kinds of systems.
You can tell which case you have from the algebra. If all the variables cancel, look at what is left:
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A false statement like $0 = 2$: no solution (inconsistent system, parallel lines).
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A true statement like $0 = 0$: infinitely many solutions (consistent dependent system, same line).
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A single value for each variable: one solution (consistent independent system).
Clean Up First
Many systems do not come in the form $Ax + By = C$. Before you choose a method, simplify each equation.
Find the solution to the following system of equations using the method you consider most appropriate. Before solving it, observe if 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, observe if it is necessary to simplify the expressions by removing parentheses:
$$ \begin{cases} 2(x + y) - 3x = 1 \\ x + 3(y - 1) = 2 \end{cases} $$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 removing parentheses and combining like terms: $$ \begin{cases} 3(x+2) - 5(y-1) = 2x + 5…
Solve the following system using the method you consider most appropriate, after removing parentheses and combining like terms:
$$ \begin{cases} 3(x+2) - 5(y-1) = 2x + 5 \\ \\ 2(x+y) - 3(x-y) = 11 \end{cases} $$Once solved, indicate whether the system is consistent independent, consistent dependent or inconsistent.
Writing a System from a Story
Word problems are where systems really shine. Follow these steps:
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Define two unknowns and write what each one means (for example, $x$ = number of blue gems).
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Turn each fact in the story into its own equation.
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Solve the system with the method you like best.
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Check in the story, not just in your equations, and answer the question with units.
Translate into algebraic language and write the system of equations corresponding to the following situation: "In a cafeteria, for two soft drinks ($x$) and a bag of…
Translate into algebraic language and write the system of equations corresponding to the following situation:
"In a cafeteria, for two soft drinks ($x$) and a bag of chips ($y$) we were charged $\$5$. If we buy one soft drink and one bag of chips, the price is $\$3$."
(It is not necessary to solve it, only to set up the system).
In a strategy video game, to unlock an upgrade you need to collect blue gems and red gems. We know that triple the blue gems plus double the red gems adds up to a total…
In a strategy video game, to unlock an upgrade you need to collect blue gems and red gems. We know that triple the blue gems plus double the red gems adds up to 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 if the pair of values $x = 8, y = 5$ (where $x$ are the blue ones and $y$ are the red ones) is the solution to the system.
Counts and Prices
In this type of problem, one equation counts the items and the other adds up the money. For example, if $s$ student tickets cost $\$5$ and $a$ adult tickets cost $\$8$:
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Number of tickets: $s + a$.
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Money collected: $5s + 8a$.
The graph shows the same story. The horizontal axis is student tickets and the vertical axis is adult tickets. The lines meet at $(70, 50)$.
On a small farm there is a total of $12$ animals, including chickens ($x$) and rabbits ($y$). If we count all their legs, there is a total of $34$ legs. The system of…
On a small farm there is a total of $12$ animals, including chickens ($x$) and rabbits ($y$). If we count all their legs, there is a total of $34$ legs. The system of equations that describes this situation is:
$$\begin{cases} x + y = 12 \\ 2x + 4y = 34 \end{cases}$$Solve the system using the method of your choice to find out how many chickens and how many rabbits there are on the farm.
To organize a charity festival, two types of tickets were sold: children's at $4\text{ \$}$ and adults' at $7\text{ \$}$. At the end of the event, it is known that a…
To organize a charity festival, two types of tickets were sold: children's at $4\text{ \$}$ and adults' at $7\text{ \$}$. At the end of the event, it is known that a total of 240 people attended and that the total amount collected was $1350\text{ \$}$. How many children and how many adults attended the festival?
In a computer store, a user buys 3 mechanical keyboards and 2 wireless mice for a total of $\$135…
In a computer store, a user buys 3 mechanical keyboards and 2 wireless mice for a total of $\$135$. The next day, taking advantage of an offer, they buy 2 keyboards and 4 mice for $\$111$. However, the offer consisted of a $10\%$ discount on the keyboards and a $20\%$ discount on the mice from their original price.
Calculate the original price of each item by setting up a system of equations.
Ages and Digits
Ages. Make a table with rows for each person and columns for "now" and "later." Everyone gets older by the same number of years.
Digits. A two-digit number with tens digit $x$ and ones digit $y$ is worth $10x + y$. For example, $37 = 10(3) + 7$. Reversing the digits gives $10y + x$.
A mother's age is currently triple that of her son. However, in 12 years, the mother's age will only be double that of her son. How old are they both currently? Use a…
A mother's age is currently triple that of her son. However, in 12 years, the mother's age will only be double that of her son. How old are they both currently? Use a table to organize 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 $18$ greater than the original. Set up a system of…
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 $18$ greater than the original. Set up a system of equations to find this number.
(Hint: A two-digit number is written as $10x + y$).
Geometry Problems
When a problem gives facts about a shape, use the formulas: the perimeter of a rectangle is $P = 2x + 2y$ and the area is $A = xy$. A change to the shape (such as "the base is doubled") changes the expression in the equation.
Observe the following figure composed of two rectangles with different properties. Rectangle A has a perimeter of $22\text{ cm}$. Rectangle B is such that its base is…
Observe the following figure composed of two rectangles with different properties. Rectangle A has a perimeter of $22\text{ cm}$. Rectangle B is such that its base is double that of rectangle A and its height is triple that of rectangle A, adding up to a total perimeter 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.
Find the dimensions of a rectangle knowing that its perimeter is $40$ in. If its base were tripled and its height halved, the perimeter would become $70$ in. Once the…
Find the dimensions of a rectangle knowing that its perimeter is $40$ in. If its base were tripled and its height halved, the perimeter would become $70$ in. Once the base ($x$) and height ($y$) are found, calculate the area of the original rectangle.
Mixtures and Motion
Mixtures. Two things add up: the total amount and the amount of the pure ingredient. The pure amount is the percent (as a decimal) times the amount of the mixture.
Motion. Use $d = rt$ (distance = rate × time). Two travelers give two equations, and the situation tells you how to combine the speeds:
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Moving toward each other: the gap closes at the sum of the speeds.
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Moving in the same direction: the gap closes at the difference of the speeds.
When the unknowns are the time and a distance, write one equation for each traveler's distance.
Two cyclists, Ashley and David, are at a distance of $45\text{ mi}$. Ashley leaves from point A at a constant speed of $12\text{ mph}$ and David leaves at the same time…
Two cyclists, Ashley and David, are at a distance of $45\text{ mi}$. Ashley leaves from point A at a constant speed of $12\text{ mph}$ and David leaves at the same time from point B to meet Ashley at $18\text{ mph}$.
If $x$ is the time they take to meet and $y$ is the distance traveled by Ashley:
- a) Set up a system of equations relating the time and distance of both.
- b) How long will it take for them to cross paths and at what distance from point A will it occur?
A chemical laboratory needs to prepare $10$ gallons of an acid solution at $25\%$. To do this, it has two containers: one with an acid concentration of $15\%$ and…
A chemical laboratory needs to prepare $10$ gallons of an acid solution at $25\%$. To do this, it has two containers: one with an acid concentration of $15\%$ and another with a concentration of $40\%$. How many gallons from each container must be mixed to obtain the desired solution?
Two cyclists, Olivia and Marcus, are at a distance of $36$ miles. If they travel toward each other, they take $45$ minutes to meet. If they travel in the same direction…
Two cyclists, Olivia and Marcus, are at a distance of $36$ miles. If they travel toward each other, they take $45$ minutes to meet. If they travel in the same direction (Olivia chasing Marcus), Olivia takes $3$ hours to catch up with him.
Calculate the speed of each cyclist in mph by setting up a system of equations.
Back to the start: Let $p$ = price of a slice and $l$ = price of a lemonade. Then $2p + l = 7$ and $3p + 2l = 11$. Multiply the first by $2$: $4p + 2l = 14$. Subtract the second: $p = 3$. Then $l = 7 - 6 = 1$. One slice costs $\$3$ and one lemonade costs $\$1$.