Exercises: Systems of linear equations

Check whether the pair of values $(x, y) = (2, -1)$ is a solution to the following system of linear equations:

$3x + 2y = 4$

$x - 4y = 6$

Solve the following system of equations using the substitution method:

$x = 2y + 5$

$3x - 4y = 11$

Solve the following system of equations using the equating method:

$y = 3x - 2$

$y = -x + 6$

Solve the following system of equations using the elimination method:

$5x + 3y = 21$

$2x - 3y = 0$

Look at the graphical representation of two lines that form a system of linear equations. State the solution to the system and classify it according to its number of solutions (consistent independent, consistent dependent, or inconsistent).

The sum of two numbers is $54$ and their difference is $12$. Set up a system of linear equations and find the value of both numbers.

In a café, $3$ coffees and $2$ croissants cost $150$ pesos. If we buy $1$ coffee and $4$ croissants of the same type, the price is $130$ pesos. How much does each coffee and each croissant cost?

Solve the following system of equations using whichever algebraic method you consider most appropriate:

$4x - y = 7$

$2x + y = 11$

Solve the following system of linear equations, first expanding to remove the brackets:

$2(x - 3) + 3(y + 1) = 4$

$x - 4(y + 2) = -10$

Solve the following system, first clearing the denominators:

$\frac{x}{2} + \frac{y}{3} = 4$

$\frac{x}{3} - \frac{y}{2} = \frac{1}{2}$

$5$ years ago, a mother was three times as old as her son. In $10$ years, the mother's age will be double the son's age. Set up a system of linear equations to calculate their current ages.

The perimeter of a rectangular plot is $120$ meters. If the length of the plot is $10$ meters longer than the width, what are the dimensions of the plot?

A merchant mixes olive oil at $80$ pesos/liter with sunflower oil at $40$ pesos/liter to obtain $60$ liters of a mixture to sell at $70$ pesos/liter. Set up a system of equations to find out how many liters of each type of oil were used.

In a piggy bank there are $40$ coins, some of $4$ pesos and others of $10$ pesos. If the total amount is $250$ pesos, how many coins of each type are in the piggy bank?

Plot the graph of the following system of linear equations and state its solution from the graph:

$x + y = 5$

$2x - y = 4$

You can do it on paper and send a photo with your answer.

In a vehicle repair shop there are $35$ vehicles in total, including cars and motorbikes. If we count all their wheels (excluding spare tyres), there is a total of $110$ wheels. How many cars and how many motorbikes are in the shop?

Solve the following system of linear equations, first simplifying the expressions with fractions and brackets:

$\frac{3(x - 1)}{2} + \frac{y + 2}{4} = \frac{11}{4}$

$\frac{x + y}{5} - \frac{x - y}{2} = \frac{3}{2}$

Determine the value of the parameter $k$ so that the following system of equations is inconsistent (that is, the lines representing it are parallel):

$kx + 3y = 5$

$4x - 2y = 7$

In an electronics shop, a laptop and a tablet cost a total of $30\,000$ pesos. During sale week, a discount of $15\,\%$ is applied to the laptop and $10\,\%$ to the tablet, bringing the combined price to $26\,100$ pesos. Set up a system of equations to find the original price of each item.

Determine the value of the parameter $m$ so that the following system has infinitely many solutions (is a consistent dependent system):

$3x - my = 12$

$-6x + 8y = -24$

Two cyclists leave at the same time from two towns, A and B, $78$ km apart, travelling towards each other in a straight line. They take exactly $3$ hours to meet. If the speed of the first cyclist is $4$ km/h faster than that of the second, at what speed is each cyclist travelling?

In a laboratory, two solutions of the same acid are available, one at a concentration of $10\,\%$ and the other at $25\,\%$. We wish to obtain $45$ liters of a new solution at a concentration of $16\,\%$. How many liters of each original solution should be mixed?

Look at the following graphical representation of a system formed by two lines, $r_1$ and $r_2$, whose equations are of the form $ax+by=c$ and $a'x+b'y=c'$. From the graph, classify the system according to its number of solutions and explain what algebraic relation must exist between the coefficients of both equations for this situation to occur.

A person invests a total of $500\,000$ pesos in two different investment funds. The first fund offers an annual interest rate of $4\,\%$, while the second offers $6\,\%$. If after one year the total profit obtained from both funds amounts to $26\,400$ pesos, how much money was invested in each fund?