What is Linear equations: Definition and 239 Discussions
In mathematics, a linear equation is an equation that may be put in the form
a
1
x
1
+
⋯
+
a
n
x
n
+
b
=
0
,
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}+b=0,}
where
x
1
,
…
,
x
n
{\displaystyle x_{1},\ldots ,x_{n}}
are the variables (or unknowns), and
b
,
a
1
,
…
,
a
n
{\displaystyle b,a_{1},\ldots ,a_{n}}
are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients
a
1
,
…
,
a
n
{\displaystyle a_{1},\ldots ,a_{n}}
are required to not all be zero.
Alternatively a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken.
The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.
In the case of just one variable, there is exactly one solution (provided that
a
1
≠
0
{\displaystyle a_{1}\neq 0}
). Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown.
In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n.
Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.
This article considers the case of a single equation with coefficients from the field of real numbers, for which one studies the real solutions. All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field. For the case of several simultaneous linear equations, see system of linear equations.
Hi,
First of all, sorry if this is not the right place to post my question I was not sure where exactly to post this kind of question.
I'm wondering how can I find the value of a constant experimentally.
For instance, I have a equation ##l = AB^{4/3}##, with a set of data for ##I## and ##B##...
The point (1, 5) is on the curve: y=ax^2+bx+c. This point gives the linear equation: 5 = a + b + c. A second point on the curve, (2, 10) gives the linear equation 10=4a+2b+c. A student called Erika thinks that the point (2, 19) is also on the curve.
5 = a + b + c.
10=4a+2b+c
19=4a+2b+c
the...
(I don't know how to make augmented matrices in latex, so what I would do is to use an equal to sign)
$$
\begin{bmatrix}
5 &2&-6&2 \\
1&-1&1&-1 \\
\end{bmatrix}
= \begin{bmatrix}
-1\\
-2\\
\end{bmatrix}$$
## R_1 \to R_1 +2R_2##
$$
\begin{bmatrix}
7 &0&-4&0\\
1&-1&1&-1\\
\end{bmatrix}
=...
Given ## a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0 ##, if ##(x_1,y_1)## and ##(x_2,y_2)## are pairs of real numbers satisfying:
## ax_1 + by_1 = e, cx_1 + dy_1 =f ##
## ax_2 + by_2 = e, cx_2 + dy_2 = f ##
then ## (x_1,y_1) = (x_2,y_2). ##
Here is my attempt at a proof, I have gotten stuck...
I have the following equation to solve for the coefficients b:
\sum\limits_{m,n=0}^N b_m b_n^* (x_{mn}+y_{mn}) e^{ ik (ux_{mn} + vy_{mn}) } = 0
which must be satisfied for all u and v in the interval [-1,1]. Here k is a constant, b is a vector of length N with unit norm
||b||^2=1,
and x and y...
The solution from my book:
From $$\frac{3}{x+2}<\frac{13}{12}<\frac{3}{x} \tag1$$
It follows that ##13x<36<13(x+2)##
x<3, i.e. x = 1 or 2. By checking, x=1 is not the solution and x = 2 satisfies the equation.
However, how does the author deduce (1)?
I was reading about systems of linear equations. Even if we have the same number of unknowns and equations, we may still have infinitely many or no solutions. But if in addition to that the determinant of the matrix of coefficients does not vanish, then does it necessarily imply that we have a...
So this question is a math question having to do with me calculating the rate of population growth starting from a population of 100,000. I have already gotten the first 3 steps done (sex ratio, ratio of cycle time, and pregnancy ratio) after a week among those in the fertile timeframe...
a+b=0
so..
a= -b
-2a-2c = -1 = -2(-b)-2c = -1 = 2b-2c=-1
-a-3b+c= 1 = -(-b)-3b+c= 1 = b-3b+c=1 = -2b+c=1
i think this is right but i don't know where to go from here
Homework Statement
Sorry for all the posts lately. This should be the last one for a while.
Solve the following system of linear equations:
##\displaystyle 4x-y=10##
##\displaystyle x+y-3z=8##
##\displaystyle 3x-y+z=12##
Homework EquationsThe Attempt at a Solution
[/B]
I started with...
Homework Statement
Determine whether the following system of equations has a single point of intersection. If so, find the point of intersection.
4x+y-9z=0
x+2y+3z=0
2x-3y-5=0
Homework Equations
n1⋅(n2×n3)
The Attempt at a Solution
I have to pick a variable, use a pair of equations to...
I have a question about HHL algorithm https://arxiv.org/pdf/0811.3171.pdf for solving linear equations of the form:
A x = b
Where A, x and b are matrices
Take for example
4x1 + 2x2 =14
5x1 + 3x2 = 19
HHL apply the momentum operator eiAτto/T on the state, do a Fourier Transform on |b> and...
Dear Everybody,
I have a question about an example:
"Solve the system:
$x'(t)=3x(t)-4y(t)+1$
$y'(t)=4x(t)-7y(t)+10t$
We write the system using the operator notation:
$(D-3)[x]+4y=1$
$-4x+(D+7)[y]=10t$
We can eliminate x from this system by adding 4 times the first equation to $(D-3)$...
Homework Statement
Determine whether of not each system has a nonzero solution.
x + 3y - 2z = 0
2x - 3y + z = 0
3x - 2y + 2z = 0
Homework EquationsThe Attempt at a Solution
Using a ti-84 I put the matrix in rref and I get a nonzero solution for x and y and zero for z but when I find the...
So, my linear algebra book, if you can call it that, says the following:
$Ax=b$ is a system of linear equations with $m$ equations and $n$ variables. ${v}_{1}, {v}_{2}, ..., {v}_{n}$ are the vectors in the columns of $A$. The following are equivalent:
(1) The system $Ax=b$ is possible for...
Homework Statement
Solve the system of equations: { (1/2)x-y=3 and x=6+2y
Homework Equations
NA
The Attempt at a Solution
The solution is 3=3, which is an identity, which means that there is an infinite amount of solutions to the system. Here's where my question lies (asked my teacher but...
In this paper on the history of functional analysis, the author mentions the following example of an infinite system of linear equations in an infinite number of variables ##c_i = A_{ij} x_j##:
\begin{align*}
\begin{array}{ccccccccc}
1 & = & x_1 & + & x_2 & + & x_3 & + & \dots \\
1 & = & &...
Hello,
its been a while since I have taken linear algebra and I am having trouble understanding what a target vector is. I need to solve a system of linear equations in matrix form using back substitution and with different target vectors. I don't have a problem with back substitution, but I...
Hi All,
In my work I would like to solve a thousands of moderate size systems of linear equations, parallelly. The uniquum in my problem is that, I am not interested in the whole solutions of the systems, but only several elements of the solution vectors are interesting.
To be more strict:
Let...
The term simultaneous in simultaneous linear equations does not make sense to me? Would you explain the what simultaneous mean there?
Example: "We have all solved simultaneous linear equations - for example,
2x + y = 4
x - 2y = -3 "
Source: Linear Algebra by Fraleigh/Beauregard.
Thank you.
Homework Statement
3.For which values of ##\lambda## does the following system of equations also have non trivial solutions
Homework EquationsThe Attempt at a Solution
What I tried doing first is to put all variables on the same side and got
##
v+y-\lambda*x=0\\
x+z-\lambda*y=0\\...
Homework Statement
What is the next radius outwards of this Apollonian gasket?
R = radius of outer circle = 5
r1 = radius of largest inner circle = 3
r2 = radius of second largest inner circle = 1
a = unknown radius
Homework Equations
C = 2πr
A = πr2
d = 2r
The Attempt at a Solution
Make a...
I understand that for many iterative methods, convergence rates can be shown to depend on the condition number of the coefficient matrix A in the linear equation
$$Ax=y.$$
Therefore, if a preconditioner satisfies
$$P \approx A,$$
then by solving the transformed linear equation
$$(AP^{-1})...
Homework Statement
This is an ordinary differential equation using the differential operator.
Given the system:
d^2x/dt - x + d^2y/dt^2 + y = 0
and
dx/dt + 2x + dy/dt + 2y = 0
find x and y equation
Answer: x = 5ce^(-2t)
y = -3ce^(-2t)
Homework EquationsThe Attempt at a Solution
We change...
I want to understand which of these is computationally expensive (in the sense of computational time) which is more accurate. Also I want to understand which of these two problems (computations time + accuracy) of iterative methods are addressed by multi-grid methods?
Hi. I have read many times that the Schrodinger equation is a linear equation and so if Ψ1 and Ψ2 are both solutions to the equation then so is Ψ1 + Ψ2. Is this use of the word linear the same as generally used for differential equations ? As the Schrodinger equation is also an eigenvalue...
Hello!
1. Homework Statement
Please, take a look at the problem and at my solution. Have I done it correctly and is my logic correct?
How much of a 10 liter 30% acid solution must be replaced with pure acid to obtain 10 liters
of a 50% solution?
Homework Equations
3. The Attempt at a...
Hello!
1. Homework Statement
Here is the problem I can't solve, and will be grateful for your help on this - please, guid me to the understanding of how to solve such problems. The difficulty lies in replacing some of the solution with water. It would be much easier to solve if we add water...
Is there a difference between the following statement
"The number of dimes is 5 times more than twice the number of nickels" and "The number of dimes is 5 more than twice number of nickels"?
The 5 times more... and the 5 more than... confuses me. Please clarify this for me. Thanks!
Homework Statement
i am doing homework about linear equations, i am just asking if i am ok, whith my selection
the question is: from the next equations, which ones are line equations, in x1,x2 and x3?
Homework Equations
[/B]The Attempt at a Solution
the next are line; a,c,f
the next arent...
Homework Statement
Suppose y(t) is a complex-valued solution of y'' +py' + qy=0 where p and q are real numbers. Show that if y(t)=yre(t) + iyim(t), where yre(t) and yim(t) are real valued, then both yre(t) and yim(t) are solutions of the second-order equation.
Homework Equations
We can use the...
Homework Statement
I got following Question from a book:
I am doing it using a different method but my answer is wrong. Can somebody please guide me, what is the problem with it??
The question is:In 1999, Diana read 10 English books and 7 French books. In 2000, she read twice as many French...
Homework Statement
Homework EquationsThe Attempt at a Solution
No clue really.
I went ahead and tried to simplify this by turnining it into an echelon matrix.
But I am sort of stuck now, since I can't divide by -k because I don't know whether or not it is equal to 0?
Homework Statement
Find the general solution:
http://puu.sh/ngck4/95470827b1.png
Homework Equations
Method: Gaussian Elimination by row operations.
The Attempt at a Solution
http://puu.sh/ngcml/7722bef842.jpg
I am getting the wrong answer( w = -27/5). The solutions provided to me says the...
Homework Statement
find the general solution of the given system of equations:
http://puu.sh/ncKaS/57a333f5b9.png
Homework Equations
Row Echelon Operations
The Attempt at a Solution
http://puu.sh/ncKcm/3e2b2bd5ab.jpg
The correct answer given is x = 1, y = 1, z = 2, w = −3
I have done...
Hi,
While solving a system of linear equations, there are three possible cases - unique / infinite / no solutions - to the system.
One geometric interpretation is when one looks at a set of planes intersecting at one / many / no points respectively, for each of the above cases.
While going...
I've been taught that for any system of linear equations, it has a corresponding matrix.
Why do people sometimes use systems of linear equations to describe something and other times matrices?
Is it all just a way of writing things down faster or are there things you could do to matrices that...
Hey guys, I am a little bit stuck on a recent math question and i was wondering if i could get some help about the best way to go about doing it
i have a matrice which is
1 2 -1 / -3
0 1 (-k-3) /-5
0 0 (k^2-2k) /(5k+11)
and i need to find...
Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values.
3x1+6x2 = −6
3x1+9x2−6x3 = −12...
Homework Statement
Problem:
Consider a system of linear equations in echelon form with r equations and n unknowns.
Prove the following.:
(i) If r = n, then the system has a unique solution.
(ii) If r < n, then we can arbitrarily assign values to the n - r free variables and solve uniquely...
Hi
I am working on a programming assignment that requires me to implement the successive over-relaxation algorithm. We are given the wikipedia page for this: http://en.wikipedia.org/wiki/Successive_over-relaxation.
I have read through the wikipedia page for this numerous times but am still...
(Wave)
i have a test on friday that I am studying for so i was working through some problems in my textbook. i came across this question and I am stuck on what to do. can anyone help me out?
thanks
I got three equations:
l-cm-bn=0
-cl+m-an=0
-bl-am+n=0
In my textbook, its written "eliminating l, m, n we get:"
$$
\begin{vmatrix}
1& -c& -b\\
-c& 1& -a\\
-b& -a& 1\\
\end{vmatrix}=0
$$
but if I take l, m, n as variables and since ##l=\frac{\Delta_1}{\Delta}## (Cramer's rule) and...
Homework Statement
How do I solve the following system?
x+y+z = 100
120x + 50y + 25z = 4000
where x, y, and z must all be at least one and are all integers. There will be more than one solution right
Homework Equations
none.
The Attempt at a Solution
I solved for z in terms of the other two...