x+2y=8

Ilnaus5
2022-09-20
Answered

x+2y=8

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Usually linear regression involves two variables (x,y), i.e. an independent variable x and a dependent variable y, and they are related by the following expression

$y={a}_{0}+{a}_{1}x$

where$a}_{0$ and $a}_{1$ are parameters that define the linear model. In linear regression we have one equation of this form for each couple of observed variables $({x}_{i},{y}_{i})$ , thus we have a linear system and its solution gives us $a}_{0$ and $a}_{1$ .

Let's consider that we have two set of independent-dependent variables, namely (x,y) and (w,z). The first two variables (x,y) are related by the previous equation, while the second two variables (w,z) are related by the following

$z={b}_{0}+{b}_{1}w$

where$b}_{0$ and $b}_{1$ are parameters that define the linear relation between z and w. Also in this case a set of observation $({w}_{j},{z}_{j})$ leads to a linear system and its solution gives us $b}_{0$ and $b}_{1$ .

In general, if$a}_{0},{a}_{1$ and $b}_{1$ are independent, then we can solve the two linear systems separately. But now, let's suppose that $a}_{0$ and $b}_{0$ are independent, while $a}_{1}={b}_{1$ . In this case, the two linear systems should be solved simultaneously.

I've solved this problem just definying one linear system of equation involving both the two sets of equations, but I would like to know if this problem has a specific name and how to correctly approach it. In particular, I want to know how to assessing the fit quality (for example, with an equivalent of the$R}^{2$ ).

where

Let's consider that we have two set of independent-dependent variables, namely (x,y) and (w,z). The first two variables (x,y) are related by the previous equation, while the second two variables (w,z) are related by the following

where

In general, if

I've solved this problem just definying one linear system of equation involving both the two sets of equations, but I would like to know if this problem has a specific name and how to correctly approach it. In particular, I want to know how to assessing the fit quality (for example, with an equivalent of the

asked 2022-06-03

I am having difficulty with true/false statements and their justifications regarding systems of linear equations.

(a) A linear system of three equations in five unknowns is always consistent (i.e. it has at least one solution)

(b) A linear system of five equations in three unknowns cannot be consistent

(c) If a linear system in echelon form is triangular then the system has the unique solution

(d) If a linear system of n equations in n unknowns has two equations that are multiples of one another, then the system is inconsistent.

So far, for (a) I have said False, as it will always be consistent if it is homogeneous, but not if it is non-homogeneous.

For (b) I have said false, but am having difficulty justifying this assertion

c) I know to be true.

(d) I believe may be false as having equations that are multiples could result in free variables and hence infinite solutions?

I am rather unsure on what I have done so far.

Any assistance is greatly appreciated.

(a) A linear system of three equations in five unknowns is always consistent (i.e. it has at least one solution)

(b) A linear system of five equations in three unknowns cannot be consistent

(c) If a linear system in echelon form is triangular then the system has the unique solution

(d) If a linear system of n equations in n unknowns has two equations that are multiples of one another, then the system is inconsistent.

So far, for (a) I have said False, as it will always be consistent if it is homogeneous, but not if it is non-homogeneous.

For (b) I have said false, but am having difficulty justifying this assertion

c) I know to be true.

(d) I believe may be false as having equations that are multiples could result in free variables and hence infinite solutions?

I am rather unsure on what I have done so far.

Any assistance is greatly appreciated.