Systems of 2×2 linear equations
At an auto repair shop in Puebla there are cars and motorcycles: 35 vehicles in total and 110 tires, not counting spares. A single equation with two unknowns can't be solved, but the problem gives you two conditions. Two equations and two unknowns: that is a system. If the two conditions are truly independent (they don't say the same thing and don't contradict each other), the system has a single solution; later you will see what happens when they are not.
What a system is and how to check a solution
To check a solution, substitute into both equations: $20+15=35$ ✓ and $4\times20+2\times15=80+30=110$ ✓. If it satisfies only one, it is not a solution of the system.
Check whether the pair of values $(x, y) = (2, -1)$ is a solution to the following system of linear equations: $3x + 2y = 4$ $x - 4y = 6$
Check whether the pair of values $(x, y) = (2, -1)$ is a solution to the following system of linear equations:
$3x + 2y = 4$
$x - 4y = 6$
Algebraic methods
Solve the following system of equations using the substitution method: $x = 2y + 5$ $3x - 4y = 11$
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 equating method:
$y = 3x - 2$
$y = -x + 6$
Solve the following system of equations using the elimination method: $5x + 3y = 21$ $2x - 3y = 0$
Solve the following system of equations using the elimination method:
$5x + 3y = 21$
$2x - 3y = 0$
Systems with parentheses and fractions
Before applying a method, it helps to put each equation in the form $ax+by=c$: remove parentheses using the distributive property and clear denominators by multiplying the whole equation by the LCM of the denominators.
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 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: $\frac{x}{2} + \frac{y}{3} = 4$ $\frac{x}{3} - \frac{y}{2} = \frac{1}{2}$
Solve the following system, first clearing the denominators:
$\frac{x}{2} + \frac{y}{3} = 4$
$\frac{x}{3} - \frac{y}{2} = \frac{1}{2}$
Solve the following system of linear equations, first simplifying the expressions with fractions and brackets: $\frac{3(x - 1)}{2} + \frac{y + 2}{4} = \frac{11}{4}$…
Solve the following system of linear equations, first simplifying the expressions with fractions and brackets:
$\frac{3(x - 1)}{2} + \frac{y + 2}{4} = \frac{11}{4}$
$\frac{x + y}{5} - \frac{x - y}{2} = \frac{3}{2}$
Graphical meaning: two lines
Each equation of the system is a line in the plane. The solution is the point where they intersect. This is the graphing method: solve for $y$ in each equation, make a table with two or three points, draw the lines, and read off the intersection point. Always check in both equations, because reading a graph is approximate.
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…
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).
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…
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.
Classifying without solving
Two linear equations don't always give a unique solution: it depends on whether the conditions are independent. You can tell what type a system $\begin{cases} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2 \end{cases}$ is by comparing the ratios of its coefficients. If the lines have the same slope, they are parallel or the same line.
Determine the value of the parameter $k$ so that the following system of equations is inconsistent (that is, the lines representing it are parallel): $kx + 3y = 5$…
Determine the value of the parameter $k$ so that the following system of equations is inconsistent (that is, the lines representing it are parallel):
$kx + 3y = 5$
$4x - 2y = 7$
Determine the value of the parameter $m$ so that the following system has infinitely many solutions (is a consistent dependent system): $3x - my = 12$ $-6x + 8y = -24$
Determine the value of the parameter $m$ so that the following system has infinitely many solutions (is a consistent dependent system):
$3x - my = 12$
$-6x + 8y = -24$
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…
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.
Problems with linear systems
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.
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…
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?
$5$ 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…
$5$ 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?
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…
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?
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?
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 $110$…
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?
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…
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.
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…
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…
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?
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…
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?
Back to the start: with $x$ cars and $y$ motorcycles, $x+y=35$ and $4x+2y=110$ are two independent conditions (the lines intersect) and give $x=20$ cars and $y=15$ motorcycles.