Systems of equations
Marcos buys one controller and two games for 70 €. Lucía buys one controller and one game for 50 €. How much does each item cost? There are two things you don't know (the price of the controller and the price of a game) and two clues. One equation on its own isn't enough: you need a system.
What a system is
To check, substitute into both equations: $30 + 2 \cdot 20 = 70$ ✓ and $30 + 20 = 50$ ✓. If it satisfies only one, it is not a solution.
Graphical meaning
Each linear equation with two unknowns is a straight line (all of its solutions). The solution of the system is the point where the lines intersect. Move the lines:
There are three possibilities:
| Lines | Type of system | Solutions |
|---|---|---|
| They intersect | Consistent and independent | One |
| Parallel | Inconsistent | None |
| Coincident | Consistent and dependent | Infinitely many |
Look at the graphical representation of the two lines forming a system of equations. Without doing algebraic calculations, determine the solution to the system and…
Look at the graphical representation of the two lines forming a system of equations. Without doing algebraic calculations, determine the solution to the system and classify it as Consistent Independent (unique solution), Consistent Dependent (infinitely many solutions), or Inconsistent (no solution).
Copy and complete the following table of values for the two equations of the system in your notebook. Then, state the common solution by looking at the results: System:…
Copy and complete the following table of values for the two equations of the system in your notebook. Then, state the common solution by looking at the results:
System: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$
| $x$ | $y$ (Equation 1) | $y$ (Equation 2) |
|---|---|---|
| $1$ | ||
| $2$ | ||
| $3$ | ||
| $4$ |
Look at the following graphical representation of a system of two linear equations with two unknowns and answer the questions: a) State the coordinates of the…
Look at the following graphical representation of a system of two linear equations with two unknowns and answer the questions:
- a) State the coordinates of the intersection point $P$ of both lines.
- b) What does that point represent in relation to the system of equations?
- c) Classify the system according to the number of solutions it has.
- d) Is the point $P$ a solution of the following system?
$\begin{cases} 2x + y = 8 \\ x - 3y = -3 \end{cases}$
Briefly explain why it is or why it is not, checking both equalities.
Substitution method
You rearrange one equation to make one unknown the subject and substitute it into the other. It is the best choice when one unknown is "on its own" (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:
$\begin{cases} y = 3x \\ 2x + y = 10 \end{cases}$
Elimination by equating
You make the same unknown the subject in both equations and equate the expressions.
Solve the following system of equations using the equating method. Notice that the unknown $x$ is already isolated in both expressions:…
Solve the following system of equations using the equating method. Notice that the unknown $x$ is already isolated in both expressions:
$\begin{cases} x = 5 - y \\ x = 2y - 1 \end{cases}$
Elimination by addition
You multiply the equations by suitable numbers so that, when you add (or subtract) them, one unknown disappears.
Solve the following system using the elimination method. Notice that the coefficients of the unknown $y$ are equal in magnitude but opposite in sign, so you can add the…
Solve the following system using the elimination method. Notice that the coefficients of the unknown $y$ are equal in magnitude but opposite in sign, so you can add the equations directly:
$\begin{cases} 4x + y = 7 \\ 2x - y = 5 \end{cases}$
Setting up problems with systems
-
Choose two unknowns and write down what each one means ($x$ = price of the controller…).
-
Translate each piece of information in the problem into an equation.
-
Solve and check against the problem statement, not just in the equations.
Translate the following situation into algebraic language and set up a system of two equations in two unknowns, clearly stating what each variable represents: "In a…
Translate the following situation into algebraic language and set up a system of two equations in two unknowns, clearly stating what each variable represents:
"In a café, a group of friends paid $11 \text{ €}$ for $2$ soft drinks and $3$ bags of crisps. Another group paid $13 \text{ €}$ for $4$ soft drinks and $1$ bag of crisps."
In a video game shop, second-hand controllers and games are sold. Marcos bought one controller and one game for $50 \text{ €}$. If the price of the game is three times…
In a video game shop, second-hand controllers and games are sold. Marcos bought one controller and one 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 whichever method you prefer.
When organising an e-sports event, a total of 250 tickets were sold. There are two types of seats: "Gamer" tickets, which cost $15\ €$, and "Pro" tickets, which cost…
When organising an e-sports event, a total of 250 tickets were sold. There are two types of seats: "Gamer" tickets, which cost $15\ €$, and "Pro" tickets, which cost $25\ €$. If the total amount raised was $4\ 700\ €$, set up a system of equations and calculate how many tickets of each type were sold using the substitution method.
Back to the beginning: The controller costs 30 € and each game costs 20 € (Marcos's controller and two games add up to $30 + 2 \cdot 20 = 70$ €, and with Lucía's game they add up to $30 + 20 = 50$ €).