Systems of two and three unknowns. Classification
In 1st Bachillerato you already learned to solve and discuss systems of two and three unknowns using substitution, elimination or Gauss. This year you are going to take a step further: instead of looking equation by equation, you will look at matrices. The same information —coefficients, unknowns, constant terms— is packed into tables of numbers, and on those tables we define the rank, the tool that in the next sections will let you classify any system without solving it, however large it is.
Review: classifying a system of two unknowns
Classify the following systems of equations with two unknowns according to the number of solutions they have (Consistent Independent System, Consistent Dependent System…
Classify the following systems of equations with two unknowns according to the number of solutions they have (Consistent Independent System, Consistent Dependent System or Inconsistent System) after attempting to solve them using your preferred method:
- a) $\begin{cases} 2x + 3y = 8 \\ 4x + 6y = 15 \end{cases}$
- b) $\begin{cases} x - y = 2 \\ 2x + y = 7 \end{cases}$
- c) $\begin{cases} 3x + y = 5 \\ 6x + 2y = 10 \end{cases}$
Three unknowns: the solution triple
With three unknowns, each equation $ax+by+cz=d$ represents a plane in space, and the solution of a system of three equations is a triple $(x,y,z)$: the point, line or plane where the three planes meet at the same time.
Check whether the ordered triple $(x, y, z) = (1, 2, -1)$ is a solution to the following system of equations:…
Check whether the ordered triple $(x, y, z) = (1, 2, -1)$ is a solution to the following system of equations:
$\begin{cases} 3x - y + z = 0 \\ x + y + z = 2 \\ 2x + 3y - z = 9 \end{cases}$
Solve the following system of equations with three unknowns which is already in row-echelon form:…
Solve the following system of equations with three unknowns which is already in row-echelon form:
$\begin{cases} 2x + y - z = 5 \\ \quad \quad 3y + 2z = 7 \\ \quad \quad \quad \quad 4z = 8 \end{cases}$
Matrix form: $A \cdot X = B$
Express the following system of linear equations in its matrix form $A \cdot X = B$, clearly identifying the coefficient matrix $A$, the matrix of unknowns $X$ and the…
Express the following system of linear equations in its matrix form $A \cdot X = B$, clearly identifying the coefficient matrix $A$, the matrix of unknowns $X$ and the matrix of constant terms $B$:
$\begin{cases} x - 2y + 3z = 0 \\ 2x + y - z = 4 \\ 3x - y + 2z = -1 \end{cases}$
Homogeneous systems: they always have the trivial solution
Without performing complex calculations, give reasons why the following homogeneous system is Consistent Dependent by analysing the relationship between its equations:…
Without performing complex calculations, give reasons why the following homogeneous system is Consistent Dependent by analysing the relationship between its equations:
$\begin{cases} x + y + z = 0 \\ 2x + 2y + 2z = 0 \\ 3x - y + 2z = 0 \end{cases}$
In a gaming peripherals shop, three friends buy different products. Marcos buys 2 mice and 1 keyboard for $110$ €. Julia buys 1 mouse and 2 keyboards for $130$ €. Set up…
In a gaming peripherals shop, three friends buy different products.
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Marcos buys 2 mice and 1 keyboard for $110$ €.
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Julia buys 1 mouse and 2 keyboards for $130$ €.
Set up a system of linear equations to determine the price of each item and classify it according to the number of solutions you expect to obtain.