Exercises: Systems of equations

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.

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".

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:

$\begin{cases} y = 3x \\ 2x + y = 10 \end{cases}$

Solve the following system of equations using the equating method. Note that the unknown $x$ is already isolated in both expressions:

$\begin{cases} x = 5 - y \\ x = 2y - 1 \end{cases}$

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:

$\begin{cases} 4x + y = 7 \\ 2x - y = 5 \end{cases}$

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)
$1$
$2$
$3$
$4$

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.

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).

Observe the following graphical representation of a system of two linear equations with two unknowns and answer the questions raised:

1. a) State the coordinates of the point of intersection $P$ of both lines.
2. b) What does this point represent in relation to the system of equations?
3. 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.

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.

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 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.

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.

Observe the following graphical representation of two linear functions that form a system of equations.

1. a) Identify the equations of both lines from their points of intersection and their gradient.
2. b) Determine the solution of the system graphically.
3. c) Classify the system according to its number of solutions (compatible determined, compatible indeterminate, or incompatible).

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.

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.

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). Once $k$ is found, solve the equation:

$$x^2 - 12x + k = 0$$

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.

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.

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.

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 $32\text{ km/h}$.

1. a) How long will it take Ashley to catch up with Brian?
2. b) At what distance from the starting line will they meet?

Hint: Use the system of equations based on the formula $e = v \cdot t$.

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 $10x + y$.

Observe the graphical representation of the lines $r$ and $s$ that form a system of equations.

1. a) Identify the equations of both lines from their points of intersection with the axes.
2. b) Determine, analytically and using the graph, whether the system is compatible determined, indeterminate or incompatible.
3. c) Find the point of intersection $(x, y)$.