Systems of linear equations
In a garage there are cars and motorbikes: 35 vehicles in total and 110 wheels, counting only the ones that touch the ground. A single equation with two unknowns cannot be solved, but the problem gives you two independent conditions. Two equations and two unknowns: that is a system, and with two clues there is a single solution.
What a system is and how to check a solution
To check a solution, substitute into both equations: $20+15=35$ ✓ and $4\cdot20+2\cdot15=80+30=110$ ✓. If it satisfies only one of the equations, 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:
$3x + 2y = 4$
$x - 4y = 6$
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 equations using whichever algebraic method you consider most appropriate:
$4x - y = 7$
$2x + y = 11$
Three 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:
$5x + 3y = 21$
$2x - 3y = 0$
Graphical meaning: two lines
Each equation of the system is a line in the plane. The solution of the system is the point where they intersect (if they do).
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: ratio of coefficients
Just as for a $2\times2$ system you can study the rank of the coefficient matrix and of the augmented matrix (you will see this formally in Bachillerato with the Rouché–Frobenius theorem), here you can anticipate the type of system by directly comparing the ratios of the coefficients of $\begin{cases} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2 \end{cases}$.
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):
$3x - my = 12$
$-6x + 8y = -24$
Problems with linear systems
A merchant mixes olive oil at $4$ €/litre with sunflower oil at $2$ €/litre to obtain $60$ litres of a mixture to sell at $3,50$ €/litre. Set up a system of equations to…
A merchant mixes olive oil at $4$ €/litre with sunflower oil at $2$ €/litre to obtain $60$ litres of a mixture to sell at $3,50$ €/litre. Set up a system of equations to find out how many litres of each type of oil were used.
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?
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…
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 was invested in each fund?