Systems of Equations
At the school store, Jayden buys 1 hoodie and 2 T-shirts for $\$70$. Aaliyah buys 1 hoodie and 1 T-shirt for $\$50$. How much does each item cost? You don't know two things (the price of a hoodie and the price of a T-shirt), and you have two clues. One equation is not enough. You need a system.
What Is a System?
To check a solution, plug it into both equations: $30 + 2 \cdot 20 = 70$ ✓ and $30 + 20 = 50$ ✓. If it works in only one equation, it is not a solution of the system.
Each equation is a line
Each linear equation with two unknowns has many solutions $(x, y)$, and together they make a line. A solution of the system must be on both lines, so it is the point where the lines cross. Move the first line and watch where they meet.
A table of values
You can also find the solution by building a table. For $x + y = 7$ and $x - y = 3$, solve each for $y$ and compare:
| $x$ | $y$ from $x + y = 7$ | $y$ from $x - y = 3$ |
|---|---|---|
| 3 | 4 | 0 |
| 4 | 3 | 1 |
| 5 | 2 | 2 |
| 6 | 1 | 3 |
At $x = 5$ both equations give $y = 2$. The common point is $(5, 2)$. That is the solution.
Copy and complete the following table of values for the two equations of the system in your notebook. Then, indicate which is the common solution by observing the…
Copy and complete the following table of values for the two equations of the system in your notebook. Then, indicate which is the common solution by observing the results:
System: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$
| $x$ | $y$ (Equation 1) | $y$ (Equation 2) |
|---|---|---|
| $1$ | ||
| $2$ | ||
| $3$ | ||
| $4$ |
Two friends, Lucy and Marcus, are accumulating points in their favorite video game to unlock a new reward. The stealth missions are represented by the unknown $x$ and…
Two friends, Lucy and Marcus, are accumulating points in their favorite video game to unlock a new reward. The stealth missions are represented by the unknown $x$ and the assault missions by the unknown $y$. The system of equations that describes the objectives they must complete today is the following:
$$\begin{cases} x + y = 5 \\ 2x + 3y = 12 \end{cases}$$Check numerically if the pair of values $(x = 3, y = 2)$ is the solution to the system. Looking at the following graph, indicate whether the friends have managed to find the exact point where both conditions are met (total missions and accumulated points).
Solving by Graphing
Two lines can be placed in three ways. That gives three kinds of systems:
| Lines | Number of solutions | Type of system |
|---|---|---|
| Cross | One | Consistent and independent |
| Parallel | None | Inconsistent |
| Same line | Infinitely many | Consistent and dependent |
Consistent means the system has at least one solution. Inconsistent means it has none. Some problems use the older labels compatible determined, incompatible, and compatible indeterminate for these same three cases, in the same order as the table.
Getting an equation from a graph
To find a line's equation, use two points you can read, such as the intercepts. Then use the slope $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ and the y-intercept $b$ in $y = mx + b$.
Observe the graphical representation of the two lines that form a system of equations. Determine, without performing algebraic calculations, what the solution to the…
Observe the graphical representation of the two lines that form a system of equations. Determine, without performing algebraic calculations, what the solution to the system is and classify it as Compatible Determined, Compatible Indeterminate or Incompatible.
Observe the following graphical representation of a system of two linear equations with two unknowns and answer the questions raised: a) State the coordinates of the…
Observe the following graphical representation of a system of two linear equations with two unknowns and answer the questions raised:
- a) State the coordinates of the point of intersection $P$ of both lines.
- b) What does this point represent in relation to the system of equations?
- c) Classify the system according to the number of solutions it has.
$\begin{cases} 2x + y = 8 \\ x - 3y = -3 \end{cases}$
Briefly explain why it is or why it is not, verifying both equalities.
Observe the following graphical representation of two linear functions that form a system of equations. a) Identify the equations of both lines from their points of…
Observe the following graphical representation of two linear functions that form a system of equations.
- a) Identify the equations of both lines from their points of intersection and their gradient.
- b) Determine the solution of the system graphically.
- c) Classify the system according to its number of solutions (compatible determined, compatible indeterminate, or incompatible).
Substitution
Graphing is slow and not always exact. Substitution is an exact algebra method: solve one equation for one unknown, then substitute that expression into the other equation. It works best when an unknown has coefficient $1$ or $-1$.
Solve the following system using the substitution method, taking advantage of the fact that one of the unknowns is already isolated in the first equation:…
Solve the following system using the substitution method, taking advantage of the fact that one of the unknowns is already isolated in the first equation:
$\begin{cases} y = 3x \\ 2x + y = 10 \end{cases}$
Setting Expressions Equal
Sometimes both equations are already solved for the same variable, like $y = \ldots$ and $y = \ldots$. Both expressions equal $y$, so they equal each other. Set them equal and solve. This is just substitution where one expression replaces $y$. Some problems call it the equating method.
If a system is not already solved for the same variable, solve both equations for it first (or just use substitution).
Solve the following system of equations using the equating method. Note that the unknown $x$ is already isolated in both expressions:…
Solve the following system of equations using the equating method. Note that the unknown $x$ is already isolated in both expressions:
$\begin{cases} x = 5 - y \\ x = 2y - 1 \end{cases}$
Elimination
With elimination, you add (or subtract) the two equations so one variable cancels. If the coefficients of one variable are opposites, just add. If they are not, multiply one or both equations by numbers to make them opposites first.
When the coefficients are equal (like $5x + 2y = 19$ and $3x + 2y = 13$), subtract instead: $2x = 6$, so $x = 3$, and then $y = 2$.
Solve the following system using the elimination method. Observe that the coefficients of the unknown $y$ are equal but of opposite sign, so you can add the equations…
Solve the following system using the elimination method. Observe that the coefficients of the unknown $y$ are equal but of opposite sign, so you can add the equations directly:
$\begin{cases} 4x + y = 7 \\ 2x - y = 5 \end{cases}$
No Solution or Infinitely Many
Sometimes elimination or substitution produces a statement with no variables left. It tells you which case you are in.
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A false statement (like $0 = 4$) means no solution. The lines are parallel.
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A true statement (like $0 = 0$) means infinitely many solutions. Both equations describe the same line.
If the second equation had been $2x + 2y = 6$, you'd get $0 = 0$. It is the same line, so every point on it is a solution.
Slide the number $k$ and watch the second line move onto the first:
Observe the graphical representation of the lines $r$ and $s$ that form a system of equations. a) Identify the equations of both lines from their points of intersection…
Observe the graphical representation of the lines $r$ and $s$ that form a system of equations.
- a) Identify the equations of both lines from their points of intersection with the axes.
- b) Determine, analytically and using the graph, whether the system is compatible determined, indeterminate or incompatible.
- c) Find the point of intersection $(x, y)$.
Word Problems with Systems
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Choose two unknowns and write what each one means, with units.
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Turn each fact into an equation.
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Solve, then check in the story, not just in the equations.
When there are two kinds of items, one equation usually counts them ("how many") and the other counts value ("how much money").
Translate the following situation into algebraic language and set up a system of two equations with two unknowns, clearly indicating what each variable represents: "In a…
Translate the following situation into algebraic language and set up a system of two equations with two unknowns, clearly indicating what each variable represents:
"In a coffee shop, a group of friends has paid $11 \text{ \$}$ for $2$ soft drinks and $3$ bags of chips. Another group has paid $13 \text{ \$}$ for $4$ soft drinks and $1$ bag of chips".
In a video game store, used controllers and games are sold. Marcus has bought a controller and a game for $50 \text{ \$}…
In a video game store, used controllers and games are sold. Marcus has bought a controller and a game for $50 \text{ \$}$. If the price of the game is three times that of the controller, how much does each item cost?
Set up a system of equations and solve it using the method of your choice.
In organizing an esports event, a total of 250 tickets have been sold. There are two types of tickets: "Gamer", which cost $15\ \$$, and "Pro", which cost $25\ \$$. If…
In organizing an esports event, a total of 250 tickets have been sold. There are two types of tickets: "Gamer", which cost $15\ \$$, and "Pro", which cost $25\ \$$. If the total revenue was $4\ 700\ \$$, set up a system of equations and calculate how many tickets of each type have been sold using the substitution method.
Rectangles and Ages
Rectangles. Perimeter gives $2l + 2w = P$. A sentence like "the length is 4 more than the width" gives $l = w + 4$.
Ages. "In $n$ years" means add $n$ to each person's current age. Write each new age in parentheses.
A new rectangular play area is being designed in the school playground. The architect has established that the total perimeter must be $56$ feet. Furthermore, so that…
A new rectangular play area is being designed in the school playground. The architect has established that the total perimeter must be $56$ feet. Furthermore, so that there is enough space for the basketball hoops, the length of the area must be $4$ feet less than three times the width.
Calculate the dimensions of the play area by setting up a system of equations and solving it using the equating method.
Two well-known streamers* are comparing their years of experience on the platform. The sum of their current ages is $44$ years. In $10$ years' time, the age of the older…
Two well-known streamers are comparing their years of experience on the platform. The sum of their current ages is $44$ years. In $10$ years' time, the age of the older one will be exactly double the age that the younger one is currently.
Set up a system of linear equations and find out the current age of each one.
A courier company uses drones and vans for its deliveries. We know that double the number of drones exceeds triple the number of vans by 4 units. Furthermore, if they…
A courier company uses drones and vans for its deliveries. We know that double the number of drones exceeds triple the number of vans by 4 units. Furthermore, if they bought 2 more drones and lost one van, they would have three times as many drones as vans. Set up a system of linear equations and find how many vehicles of each type the company has.
Digit Problems
A two-digit number is not the sum of its digits. In $47$, the $4$ is worth $40$. If $t$ is the tens digit and $u$ is the ones digit, the number is $10t + u$. Reversing the digits gives $10u + t$.
A two-digit number has a curious property: the tens digit is double the units digit. If we subtract $27$ from this number, we obtain the number resulting from reversing…
A two-digit number has a curious property: the tens digit is double the units digit. If we subtract $27$ from this number, we obtain the number resulting from reversing the order of its digits.
Translate the statement into a system of equations (remember to express the number in polynomial form: $10x + y$) and find out what number it is.
Find a two-digit number knowing that the sum of its digits is $12$ and that, if the order of these digits is reversed, the resulting number is $18$ units greater than…
Find a two-digit number knowing that the sum of its digits is $12$ and that, if the order of these digits is reversed, the resulting number is $18$ units greater than the original number.
Note: Remember that a two-digit number $xy$ is written algebraically as $10x + y$.
Mixtures
Mixture problems always have two equations:
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an amount equation: the parts add up to the whole mixture;
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a value equation: the price (or the amount of salt, juice, acid…) in the parts adds up to the amount in the whole mixture.
For percents, change each percent to a decimal first. In the diagram above, the value equation is $0.10a + 0.40b = 2$. Multiply it by $10$ to get $a + 4b = 20$, then subtract $a + b = 8$ to get $3b = 12$. So $b = 4$ and $a = 4$.
Some problems give amounts in other units, such as kilograms or liters. The setup is exactly the same: keep the units consistent, and answer in the unit the question asks for.
In a bulk food store, we want to prepare a special mixture of $20\ kg$ of nuts that works out at a price of $8,50\ \$/kg$. To do this, we will use cashews that cost $10\…
In a bulk food store, we want to prepare a special mixture of $20\ kg$ of nuts that works out at a price of $8,50\ \$/kg$. To do this, we will use cashews that cost $10\ \$/kg$ and peanuts that cost $6\ \$/kg$.
Apply the elimination method to find out how many pounds of cashews and how many of peanuts we must mix.
In a chemistry laboratory, they want to mix two saline solutions. The first has a salt concentration of $15\%$ and the second of $40\%$. If exactly $20$ gallons of a…
In a chemistry laboratory, they want to mix two saline solutions. The first has a salt concentration of $15\%$ and the second of $40\%$. If exactly $20$ gallons of a mixture with a concentration of $30\%$ are needed, how many gallons of each solution must be used? Set up and solve the system of equations.
Motion Problems
Use $d = r \cdot t$ (distance = rate × time). Two ideas solve most problems:
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Chase: when one traveler catches up to the other, both have gone the same distance, but the one who started later traveled for less time.
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Loop or track: going in opposite directions, the speeds add. Going in the same direction, the speeds subtract.
Two cyclists, Paula and Sam, ride around a circular track of $400$ feet. If they ride in opposite directions, they pass each other every $20$ seconds. If they ride in…
Two cyclists, Paula and Sam, ride around a circular track of $400$ feet. If they ride in opposite directions, they pass each other every $20$ seconds. If they ride in the same direction, Paula (who is faster) overtakes Sam every $100$ seconds.
Knowing that the distance travelled is equal to speed multiplied by time ($e = v \cdot t$), set up a system of equations to calculate the speed of each cyclist in feet per second.
Two cyclists, Ashley and Brian, are taking part in a long-distance race. Brian leaves the starting line at a constant speed of $24\text{ km/h}$. Half an hour later,…
Two cyclists, Ashley and Brian, are taking part in a long-distance race. Brian leaves the starting line at a constant speed of $24\text{ km/h}$. Half an hour later, Ashley sets off in pursuit at a speed of $32\text{ km/h}$.
- a) How long will it take Ashley to catch up with Brian?
- b) At what distance from the starting line will they meet?
Hint: Use the system of equations based on the formula $e = v \cdot t$.
Harder Equations You May Meet
Some problems use the same ideas with trickier equations. Here are three tools.
Equations with fractions
Multiply every term by the LCD (least common denominator) to clear the fractions. Put parentheses around each numerator.
How many solutions does a quadratic have?
For $ax^2 + bx + c = 0$, the discriminant is $D = b^2 - 4ac$.
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$D > 0$: two real solutions.
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$D = 0$: one solution (a repeated root), $x = -\dfrac{b}{2a}$.
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$D < 0$: no real solutions.
A sum and a product
If you know two numbers' sum and product, look for factor pairs. For a rectangle with perimeter $28$ ft and area $40$ ft², the sum of length and width is $l + w = 14$ and the product is $lw = 40$.
If no nice pair exists, substitute $w = 14 - l$ into $lw = 40$ and solve the quadratic $l^2 - 14l + 40 = 0$.
Solve the following linear equation, paying special attention to the simplification of the fractions and the negative sign preceding the parentheses: $$\frac{2(x -…
Solve the following linear equation, paying special attention to the simplification of the fractions and the negative sign preceding the parentheses:
$$\frac{2(x - 3)}{5} - \frac{x - 1}{2} = \frac{3x - 4}{10} - \frac{5(1 - x)}{4}$$Study the discriminant of the following quadratic equation and determine for what value of the parameter $k$ the equation has a single real solution (repeated root).…
Study the discriminant of the following quadratic equation and determine for what value of the parameter $k$ the equation has a single real solution (repeated root). Once $k$ is found, solve the equation:
$$x^2 - 12x + k = 0$$Find the dimensions of a rectangular wooden frame knowing that its perimeter is $60\text{ cm}$ and its area is $216\text{ cm}^2$. To solve it, you must set up a system…
Find the dimensions of a rectangular wooden frame knowing that its perimeter is $60\text{ cm}$ and its area is $216\text{ cm}^2$. To solve it, you must set up a system where one of the equations is quadratic or solve it using a quadratic equation by applying the relationship between the sum and the product of the roots.
Back to the start: a hoodie costs $\$30$ and a T-shirt costs $\$20$. Jayden pays $30 + 2 \cdot 20 = 70$ dollars and Aaliyah pays $30 + 20 = 50$ dollars.