What is Linear equation: Definition and 133 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.
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
during the 10 years it...
Write a linear equation for the application.
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.
Solution:
I am looking for W(S).
S = monthly sales
Let...
You are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during
the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 16 represent 2016.)1...
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.
$5000 plus a commission of 7% of sales =...
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...
I use 2x -4 as the LCD and turn 8/(x - 2) - (13/2) = 3 into 16 - 13x - 4 = 3, I then get 12 - 13x = 3 which leads me to 13x = -9 so x = -9/13 which is the wrong answer.
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
Hi,
I have this problem I was working through, but I'm not sure that I've approached it from the right way. The problem consists of 3 parts, which build off of each other. I'm pretty confident about the first two parts, but no so much...
Homework Statement
I need some help with a question on my assignment. It asks to set up a matrix from the linear equations, y=25x+70 and y=35x+40.
Homework Equations
How do I set this matrix up?
The Attempt at a Solution
I think that I have to rewrite it as 25x-y=-70 and 35x-y=-40. But then I...
Homework Statement
convert:
Y − 200 = −4 (X − 15)
to
X = −0.25 ⋅ Y + 65.
with a given Δy/Δx = -36/4
Homework Equations
point slope, slope intercept
The Attempt at a Solution
I understand point slope, slope intercept, and standard form, I understand how to convert one to the other, but I...
Imagine that you own a grove of orange trees, and suppose that from past experience you know that when 100 trees are planted, each tree will yield about 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases...
As per my understanding, a linear equation with two variables form a line segment (ax=by+c or ax+by=c) and linear equation with three variables form a plane (ax=by+cz+d or ax+by+cz=d). Am I right? And if I am right, does an equation with four variables form a cube?
Imagine that you own a grove of orange trees, and suppose that from past experiences you know that when 100 trees are planted, each tree will yield about 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases...
I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one...
The problem
I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ##
$$
\begin{cases}
x = 1 + s - t \\
y = 2 - s \\
z = -1 + 2s
\end{cases}
$$
The attempt
## s, t ## are the parameters for the two directional vectors which "support" the plane.
$$...
I am encountering a difficult(atleast to me) mathematical problem? I need to calculate the impact of a formulae on two sets of data for two different years
Consider for example I have a simple formulae A * B * C * D = F where A,B,C,D are 4 parameters that affect the value of F. Now I have a 2...
Hello,
i have been studying for finals and i am stuck on a question on my study guide. the question is to make a scatter plot of a set of data, find the equation of the best fit line, and approximate the value of y for x = 5.
the data is like this: x: 0 2 4 6 7
and the y is like : y: 2 7 14...
any particular solution plus the general solution to the homogeneous equation.
I'm having difficuilty understanding this proof from my lecture notes
Theorem
: Let T : V → W be a linear transformation. Let w ∈ W and suppose T(u0) = w
T(v) = 0. where v ∈ V (the kernel )
to prove:
T(u) = w...
Am I forgetting some critical basic knowledge?
It says: In the plane (identified by R^2) a linear equation ax+by=c is a straight line. If b=0 then this straight line is parallel with the y-axis; in the other case it is a straight line with slope -a/b. A (2 by 2)-system linear equation.
Why is...
Homework Statement
I am given the equation (m1 – m2)g = (m1 + m2 + I/R2)a and the experiment is to validate this equation.
Homework Equations
The Attempt at a Solution
After following the lab guide, it tells you to plot the weight difference (m1– m2)g against acceleration and determine...
Hi,
I wanted to know if the endpoints of an nth dimension linear equation will be guaranteed to contain a min and max over that interval.
For 1D ( like a line), if I find f(x) over an interval [x0, xn], I'm guaranteed that the two end points will be either an max or min.
So I was wondering if...
Homework Statement
Hello!
Please, take a look at the screeshot with a problem description.
Homework Equations
I am trying to solve this, but I seem to have a wrong understanding of the problem.
The Attempt at a Solution
It is said that a company charges 2.50 for the first 1/5 of a mile...
In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following:
$$x^2, y^2, xy$$
What surprised me was the fact that xy was listed as a nonlinear term.
In...
How the system of equations
##y=-2x+1##
##y=x+1##
##y=2x+1##
are linearly dependent. In wiki its written for the above system of equations "one equation linearly dependent on the others"
The problem
Adam is saving 1, 5 and 10 dollar bills. Adam has 165 bills. The amount of one dollar bills is twice as high as 10-dollar bills. The total value of his savings is 735 dollars. How many 5-dollar bills does Adam have?
This problem was translated. Sorry for grammatical errors.
The...
Homework Statement
Mass 2 collides with mass 1 as shown in the image, mass 1 is attached to the stick and it is initially stationary. Consider that the stick is massless and can rotate around the point O. The entire system is on a frictionless table.
Which magnitudes are conserved in the system...
I know that linear equations have variables, which have a power no greater than one.
So, for example, 5x + 2 = 15 is linear, because the x is to the first power only.
But what about this equation:
x/x+2 = 80
This has an x in the denominator. Could we consider this linear still, because...
If I have a similar question \frac{y}{5}+\frac{7}{20}=\frac{5-y}{4} should I go about the same process as the http://mathhelpboards.com/pre-algebra-algebra-2/solve-following-equation-12605.html? Try and cancel out the denominators of 5, 20 and 4? Which would be 20? So times the equation by 20?
Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots.
\begin{equation}
\frac{\gamma*q*P_Q}{k_p*(1-q)*P_C} = \frac{I*\alpha}{k_f+k_d+\frac{k_n*\lambda_b*\gamma*q*P_Q}{\lambda_b* \gamma...
Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots.
\begin{equation}
\frac{\gamma*q*P_Q}{k_p*(1-q)*P_C} =...
each of two brothers can wash a car in 1hr; however, their sister can wash a car in 45min. If the three work together, how long will it take to wash the car?
my solution,
$\frac{1}{60}+\frac{1}{60}+\frac{1}{45}=\frac{1}{x}$
multiply by 36,480x I get
$810x+1215x-36480=1350x$
solving for x I...
please I need assistance with this problem
help me get started
Roughing It Milo and Bernard are planning a three-day canoe trip
on the Roaring Fork River. Their friend Vince will drop them off at
the Highway 14 bridge. From there they will paddle upstream for
12 hours on the first day...
Olympic Track. To host the Summer Olympics, a city plans to
build an eight-lane track. The track will consist of parallel 100-m
straightaways with semicircular turns on either end as shown in
the figure. The distance around the outside edge of the oval track
is 514 m. If the track is built...
1. Average Speed. Junior drove his rig on Interstate 10 from San
Antonio to El Paso. At the halfway point he noticed that he had
been averaging 80 mph, while his company requires his average
speed to be 60 mph. What must be his speed for the last half of
the trip so that he will average 60...
1. Metallurgy. How much pure gold should be melted with
15 grams of 14-karat gold to produce 18-karat gold?
2. Road Construction. A new machine that deposits cement
for a road requires 12 hours to complete a one-half mile section of road. An older machine requires 16 hours to pave the same...
I need help with these problems.
1. Speed of Sound in Air. Two seconds after firing a rifle at a
target, the shooter hears the impact of the bullet. Sound travels
at 1100 feet per second and the bullet at 1865 feet per second.
Determine the distance to the target (to the nearest foot).
2...
If 3 m e n do a job in 12 days and two of the me n are three time s as fast as the third,
how long will it take one of the faster men to do the job ?
this is how I solved it
let x = required time for one of the faster men to do the job alone
3x = time taken by third to finish a job...
Forty men were engaged to finish a work in 90 days.
Afer 60 days, some men stopped working.
The remaining men finished the job at a same rate in 40 more days.
How many men stopped?
this is how far I can get to
let x = number of men remained
40-x = number of men stopped
Please help. thanks!
here's the other problem.
1. An army of soldiers is marching down a road at 5 mi/hr. A messenger on horseback rides from the front to the rear and returns immediately, the total time taken being 10 minutes. Assuming that the messenger rides at the rate of 10mi/hr...
1. A clay contains 45% silica and 10% water. Determine the percentage of silica in the clay on a dry (water-free) basis. All percentages are by weight.
my answer is 100% silica. because if the clay is water free the clay is 100% silica. is this correct?
2. A coal contains 2.4%...
please help me with this problem
please use single variable only.
The length of a rectangular swimming pool is twice its width. The pool is surrounded by a cement walk 4ft wide. If the area of the walk is 748 ft^2, determine the dimensions of the pool.
let x = width of...