Exercises: Systems of linear equations

Check if the ordered pair $(x, y) = (2, -1)$ 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 by setting the expressions equal to each other:

$y = 3x - 2$

$y = -x + 6$

Solve the following system of equations using the elimination method:

$5x + 3y = 21$

$2x - 3y = 0$

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

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

At a coffee shop, we were charged $\$7,50$ for $3$ coffees and $2$ croissants. If we buy $1$ coffee and $4$ croissants of the same type, the price is $\$6,50$. How much does each coffee and each croissant cost?

Solve the following system of equations using the algebraic method of your choice:

$4x - y = 7$

$2x + y = 11$

Solve the following system of linear equations, expanding the parentheses first:

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

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

Solve the following system by eliminating the denominators first:

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

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

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

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

A merchant mixes olive oil at $\$4$/gallon with sunflower oil at $\$2$/gallon to obtain $60$ gallons of a mixture that will be sold at $\$3,50$/gallon. Set up a system of equations to find out how many gallons of each type of oil were used.

There are $40$ coins in a piggy bank, some are $\$0,20$ and others are $\$0,50$. If there is a total of $\$12,50$, how many coins of each type does the piggy bank contain?

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

$x + y = 5$

$2x - y = 4$

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

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

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

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

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 store, a laptop and a tablet cost $\$1500$ together. During the sale week, a $15\,\%$ discount is applied to the laptop and a $10\,\%$ discount to the tablet, making the combined price $\$1305$. 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 cities, A and B, which are $78$ miles apart, and travel toward each other in a straight line. They take exactly $3$ hours to meet. If the speed of the first cyclist is $4$ mph faster than the second, at what speed is each one traveling?

A laboratory has two solutions of the same acid, one with a $10\,\%$ concentration and another with a $25\,\%$ concentration. The aim is to obtain $45$ gallons of a new solution with a $16\,\%$ concentration. How many gallons of each original solution must be mixed?

Look at the following graph 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 relationship must exist between the coefficients of both equations for this situation to occur.

A person invests a total of $\$25000$ 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 $\$1320$, how much money did they invest in each one?