What is Solving equations: Definition and 75 Discussions
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.
A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.
An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions.
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers always gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1).
The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y.
However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... to denote the known variables, which are often called parameters. This is typically the case when considering polynomial equations, such as quadratic equations. However, for some problems, all variables may assume either role.
Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a given interval. When the task is to find the solution that is the best under some criterion, this is an optimization problem. Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the best solution.
Hello :
i was looking into youtube the other day and watching some lectures , but start to wonder if there is a playlist for exercise solving on youtube for example for undergraduate math and physics ? do anyone knows of such channels or playlists ?
Best Regards
Hagop
Problem statement : Given the equation ##\sin\theta+2\cos\theta=1##, find the general solution for the angle ##\theta##.
Attempt : For the general case where we have ##a\sin\theta+b\cos\theta=c##, the line of approach is to take ##a=r\cos\alpha## and ##b=r\sin\alpha## wherein we will have...
Let me explain what I mean.
When you try to solve an algebraic equation, or even sometimes a differential equation it is useful to change variables so the new form of the equation is easier to solve, or takes the form of an equation that has already been solved.
Another technique is making...
Summary: I need to translate points on coordinate axes as part of a calculation process
Hello everyone,
I've created the diagram below to try and explain what I am trying to do as part of an existing software app that's used to generate profiles and programs to drive a CNC machine to grind...
Equations:
Question:
Using the equation for t' I got the answer -0.00000499756s (Wrong anyway)
As we know the values:
v = 0.67c m/s
x = 3900 m
t = 0.000005 s
Also how can time be negative? Does this mean that event 2 happened before the first event when t = 0?
My book states the following :
##{\frac{(y_1)-(y_2)}{(x_1)-(x_2)}}*\frac{(3+1)}{(0-0)} = -1## ...(1)
##\implies y_1 - y_2 = 0##
(A) Is this a valid deduction?
Context :
The Problem :If ##A(0,-1)## and ##B(0,3)## are two opposite vertices of a square, then find the other two vertices...
Homework Statement
Hi
I am having trouble solving for y for the following equation:
e^(y) = y^(2) - 2
Homework Equations
ln(x)^(y) = y*ln(x)
ln(e^(y)) = y
ln(2) = 1
The Attempt at a Solution
My attempt:
e^(y) = y^(2) -2
ln(e^(y)) = ln(y^(2)) - ln(2)
y = 2*ln(y) - 1
y - 2*ln(y) + 1 = 0...
Task
Use the Bisection, Newton, and Secant methods to solve (to at least 8
signicant figures) the equation
sin(x) = 0.98 cos(2x2)
over the interval [0, 2.5], in radiant units. For the Newton method, try with
several different initial guesses including x0 = 1. With the Secant method,
use the same...
Homework Statement
I am trying to solve a Eigen vector matrix:
##\begin{bmatrix}9.2196& 6.488\\4.233& 2.9787\end{bmatrix}\cdot\begin{bmatrix}x\\y\end{bmatrix}-\lambda\begin{bmatrix}x\\y \end{bmatrix}=0##
I have found ##\lambda_1 = 0## and ##\lambda_2 = 12.1983##
However, I can't solve the...
Homework Statement
its a nuclear physics problem :
a deuterium atom with energy 4Mev collides with a boron atom B5/10 in an elastic collision
producing an H1/1 and B5/11 ,the Q of the reaction = 9.23 Mev .
find the E,E' , the energy of the boron and the hydrogen
the unknowns :
E,E', theta of...
I read that in solving an equation, the reault is only a possibility for a solution, and not necessarily a solution. For example, in solving x+3 = 5, x=2 is not a solution until you explicitly plug it back into show that it actually. Why is the statement x=2 not enough to show that 2 is actually...
I am solving the equation ##\sqrt{x + 3} + 4 = \sqrt{8x + 1}##. I understand that , generally, to solve it, we have to eliminate the radicals by isolating a radical expression to one side and then squaring both sides of the equation.
I end up obtaining two solutions: ##x = 6## and ##x =...
Hi!
I have a problem: I need to solve an equation, Ax=y, where A is a known matrix, y is a known column vector and x is an unknown column vector. Unfortunately, A is singular so I cannot do the simple solution of inverse(A)*y=x. Does anybody know of any way that I can obtain the coefficients...
Homework Statement
A billiard ball moving at 2 m/s collides (0 degrees) with another of the same weight and at rest, in a perfectly elastic collision. Demonstrate with equations why the balls trade velocities.
Homework EquationsThe Attempt at a Solution
In the provided explanation, I can...
Is there any program that could find approximate solutions to any equations system? For example to N-body problem (equations)? Could it also find a function from differential equations like N-body problem ones?
If so ,then why are N-body simulators needed?
So let's say I have 2 matrices A and B. I need to solve 2 eqns involving specific elements of each matrix.
e.g. A(1)+B(2)=4; A(1)-B(2)=2.
Is there any way to do this? My efforts with Fsolve and solve have failed.
Here's what I've done so far:
function F=myfun(A,B)...
Solve for real solutions of the following system:
$2z+z^2x=x$
$2y+y^2z=z$
$2x+x^2y=y$
I know, this isn't a very hard problem, thus, we should compete in terms of how many minutes (or hours) that we used to come up with a good solution. I said good but not smart because to me:
$\text{all...
Homework Statement
Show that (1+i) is a root of the equation z4=-4 and find the other roots in the form a+bi where (a) and (b) are real.Homework Equations
Using De Moivre's Theorem
zn=[rn,nθ]
Modulus(absolute value of z) = 4
Argument = ?
The Attempt at a Solution
r4=4 → r = (4)^(1/5)...
Solving equations with "mod"
Homework Statement
I have a homework problem. I'm in computer science, however one of my programs requires me to solve this equation. I'm afraid it's beyond my expertise.
(a * x) mod b = 1
This needs to be rearranged in the form of x = ...
Due to the existence...
Are there general methods for solving equations of the form
a+bx = mod(c+dx, m),
where, in the notation I have made up here, mod is the modulo function which resets the argument to zero when it reaches m. I hope it's clear what I mean here.
Problem:
You make a primitive yo-yo by wrapping a massless string around a solid cylinder of mass M and radius R. You hold the freee end of the string stationary and release the cylinder from rest. The string unwinds but does not slip of stretch as the cylinder descends and rotates. Find the...
$y=2x+1$
$y=2x^2+3x+5$
$ a+3b=1$
$2a - b=1$
Multiply through by 3
$6a - 3b = 3$
Subtract equation (3) from (1)
$6a - 3b = 3$
$ a + 3b = 1$
$7a - 0b = 4$
a =$\frac{4}{7}$
Just learning latex on here and trying to understand how I can keep everything in line, which seems difficult with...
Homework Statement
Solve the equation in the interval [0,2∏)
3 sinx-6 = sinx-5
Homework Equations
The Attempt at a Solution
I originally tried to set it equal to zero and then divide by the 3 and then factor out the sinx but I didn't think that made any sense? HELP!
I am trying to solve an equation that involves substitution.
I was given k= 7 in the equation n=k+c
i was also given n=2^c-1
I figured that 7+c could be substituted into the equation as n, so I got
7+c=2^c-1 and therefore c=2^c-8.
The problem is that the answer I want is c=4...but...
Homework Statement
http://desmond.imageshack.us/Himg191/scaled.php?server=191&filename=daumequation13325015229.png&res=medium
where A,B,C & D are constants.
2. The attempt at a solution
Actually i am an electrical engineering student and these 2 equations are transistor equations that we...
Homework Statement
Solve each of the following equations. This is part of a practice mock test I have for an exam so I have all the answers but they aren't lining up somewhere. I got a) right but b) and c) wrong
Homework Equations
a)4x+6=18
b)5x+7=7x-9
c)1/x+2 = 7 (giving answer...
Homework Statement
Can someone please help or guide me with these problems.
Thanks in advance.
less than or equal to = </=
1 (x+3)(x-3)>0
2 x^2-2x-15</= 0
3 sin2x=sinx, 0</=x</=2pi
4 logx+log(x-3)=1
The Attempt at a Solution
OK so for number 1 I do not know where to...
SOLVED
Homework Statement
Hello,
I was wondering if someone could please tell what functions are required to solve this problem. I was trying to solve P*v = 1*8.314*T for T were P=[2494 1247 831 623 499 416]; and v=1:6; I was trying to come up with a series of commands that would compute...
Need to find optimal solution by solving 2 equations
X= VR*(R2+R1)/R2 - VN*R1/R2
Y= VR*(R2+R1)/R2 - VP*R1/R2
VP, VN are fixed. VP=5 & VN=0
VR can change. R1,R2 should be same.
How do I find a combo of VR, R1, R2 so that -
X = 0.46 & Y = 0.214 (this part is easy)
The same...
Homework Statement
I am having trouble with question 4(c)... part i and ii...
Homework Equations
The Attempt at a Solution
would it be okay for her to write iff between each line? I do not see why not but I cannot find the proper reasoning.
Hi everyone,
Iam just wondering on why my codes (with Mathematica) are not valid for the case of m = 5 and n = 13. However, they work perfectly for other cases for example, when m = 1 and n = 1. Note that m = 1,3,5,7,..99 and n = 1,3,5,7,...99.
Please let me know about this matter...
The question is x2/3 - x1/3 - 2 = 0
So the first thing I did was: x2/3 - x1/3 = 2
Then, I put both sides to the power of three, so I got:
x2 - x1 = 8
From there I factorized: x (x - 1) =8
And got the answers: x = 8 or x = 9, the book however, says the correct answers are -1 and 8.
Any...
I really struggle with these and would just like someones input on how to best solve these, I can never seem to cancel everything out except the variable I am looking for. I have:
E-i1-Ri2=0
Ri2-Ri3=0
i1+i2+i3
I am looking to solve for i1
All R's are equal to 0.73 x10^6 ohms so I just...
Homework Statement
See figure attached for problem statement.
Homework Equations
The Attempt at a Solution
Here's as far as I got,
The part that confuses me is the range we should solve this equation. It says,
\text{from } t=0 \text{ to } 1d
I put the ^{-1} in...
Homework Statement
I can't find a step by step explanation for solving these types of equations
eg.
99 = [2x+1]/3
Homework Equations
eg.
99 = [2x+1]/3
or
48 = 4[2x/3]
How do you handle the multipliers iand constants inside the brackets?
thx
[b]3. The Attempt at a...
Homework Statement
Hey there..
I would like to make a path for a spacecraft from Earth to moon, and back again. I found this pdf that explains how, but i can't figure it out.Homework Equations
http://courses.ncssm.edu/math/NCSSM%...r%203/Moon.pdf
"Part II: The slingshot around the Moon...
Homework Statement
http://gyazo.com/f4c02f2a57b24a39dfee575837e0a807.png
Homework Equations
Ax = A cos (?)
Ay = A sin(?)
The Attempt at a Solution
I tried putting the magnitude of x cos (?) but honestly have no idea what to do. Thanks
Just came across this for a program I'm writing. I need to be able to solve an equation of the form ln(x) =f(x) where f isn't a logarithm. specifically, it's ln(x)=-x^2+ a*x+b.
Is this solvable for x or do I need a numeric approximation?
Homework Statement
Find all solutions to the equation below such that -180° \leq x \leq 90°
2sin2x + sinx = 0Homework Equations
The Attempt at a Solution
2sin2x + sinx = 0
sinx[2sinx + 1] = 0
sin x = 0 sinx= -1/2
x = {0° + 360°n
{-180° + 360°n
n\epsilonI
Hi,
I have been trying to solve some mechanics problems and I can come up with the right equation, but I don't know how to solve it...
For example:
490.5cos(theta) - 3600cos(theta)sin(theta) + 1800sin(theta) = 0
i need to solve for theta here
its one equation and one unknown...
the idea is let us suppose i must solve
f(x)= 0 (1)
let us suppose that f(x) have SEVERAL (perhaps infinite ) inverses, that is there is a finite or infinite solutions to the equation
f(x)= y by g(y)= x with f^{-1}(x)=g(x)
then solution to equation (1) would be g(0)=x...
Can you please set up the equations for me to help me solve these questions.:
1) How many collisions per second occur on a container wall with an area of
1.00 cm^2 for a collection of Ar particles at 1.10 atm and 290.1 K?
I think I should use this equation to answer this question:
ZW =...