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.
Problem statement : Let me copy and paste the problem to the right as it appears in the text.
Solution attempt (mine) : There are mainly three cases to consider.
(1) ##\boldsymbol{x\ge 3\; :}## Using the relevant equations given above, the problem statement reduces to $$x-3+x-2 = 1\Rightarrow...
Write a linear equation.
A school district purchases a
high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment
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A pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of
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Solution:
I am looking for W(S).
S = monthly sales
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A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it will be in use.
Let t =...
A pharmaceutical salesperson receives a monthly salary of $5000 plus a commission of
7% of sales. Write a linear equation for the salesperson’s monthly wage W in terms of monthly sales S.
I will let W(S) = monthly wage W in terms of monthly sales S.
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i get preparation for my university entering exam and i studying linear equation and they define it linear equation are those which graph line now my question is that why one variable linear equation called linear for example y=3 that actually give us just line why we draw line for this...
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Where did I make a mistake?
I don't know the terms so I'm sorry if the informations at summary above is unclear. But I add a detailed photo of my calculations below. I use Gauss' Elimination laws.
https://www.physicsforums.com/attachments/9471
ok lots of options to solve this but I would start by $R3-R1\to R3$
if I remember correctly if get a diagonal of ones and the rest zeros in A we will have B from Ax=B
$\tiny{USMA = United \,States\, Military\, Academy}$
[solved] Basis for set of solutions for linear equation
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