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How to find the positive integer solutions to [math]\frac{x}{y+z}+ \frac ...

The first thing to do when you're looking at any equation is to try and place it in .... for any particular solution we find, we just need to check that it doesn't make ...

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Essentially, when there is no solution, it means that there is no value of x that satisfies the equation or the inequality.

Let's start with the simplest one.

x = x + 1 (does this equation have any solutions?. Obviously not, because there is no way a number is equal to itself plus one)

How can we show this? Subtract x from both sides, then we get

0 = 1 (which cannot be true)

Therefore, x = x + 1 has no solutions.

Let's take an inequality

x > x (does this inequality have any solutions? Obviously not, because a number cannot be greater than itself!)

Subtract x from both sides we get

0 > 0 (which is not true, since 0 is not greater than 0. Therefore, x > x has no solutions).

The way to show whether an equation or inequality has no solutions is to manipulate the equation or inequality till we end up with a statement that is false.

-------------------

Let's take the homework question.

4(x + 1) = 2x + 4

4x + 4 = 2x + 4

Subtract 4 from both sides

4x = 2x

Subtract 2x from both sides

2x = 0

Divide both sides by 2

x = 0.

Therefore, this equation has a solution. That is the statement becomes true when x = 0.

-----------------------

Let's take the second example.

9 - 5x + 2 = 4 - 5x

11 - 5x = 4 - 5x

Subtract 4 from both sides

7 - 5x = -5x

Add 5x to both sides

7 = 0

(This is not a true statement! There is no value of x, that will make the left hand side of the equation equal to the right hand side. Therefore, there are no solutions.)

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If x + y = 2 \text{ and } x^2 + y^2 = 2, what is the value of ...

I'm not sure what math you're in. Anyway, the straightforward way is to solve for y in the first equation, and plug ... Middle and High School Math Homework Question ... Well one way to solve it would be by squaring the first equation and ...... R: So you can see that the final solution is x = 1 and y = 1 so x.y=1, because 1 *1=1.

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EE364a Homework 4 solutions

plain in detail the relation between the optimal solution of each problem and the ... in the variables x ∈ Rn, t ∈ R. To see the equivalence, assume x is fixed in this .... SOCP (if the problem is convex), or explain how you can solve it by solving a ... This is not a convex optimization problem, since the objective is not concave.

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(Solutions). To see a review of how to start R, look at the beginning of Lab1 ... 1. Probability that a normal random variable with mean 22 and variance 25.

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Support Vector Machines

a decision boundary (this is the line given by the equation θT x = 0, and is also called ... point B lies in-between these two cases, and more broadly, we see that if a point is ... x + b). Note that if y(i) = 1, then for the functional margin to be large ( i.e., for .... algorithm for solving the above optimization problem that will typically do.

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Lecture 5 1 Linear Programming

Jan 18, 2011 ... and so no feasible solution has cost higher than 2. 3. , so the solution x1 := 1. 3. , x2 := 1. 3 is optimal. As we will see in the next lecture, ... and TSP are such that, for a given input, there is only a finite number of possible ... The set of feasible solutions to (1) is the set of points which satisfy all four inequalities,.

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EE363 homework 4 solutions

Now we can use the formulas for the MMSE estimator in the linear ... do not help in estimating x and we cannot improve the a priori covariance of x. 2. Estimator error variance and correlation coefficient. Suppose (x, y) ∈ R2 is Gaussian, ... y are almost uncorrelated, i.e., |ρ| is small, we find that η ≈ 1, which mean that the.

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Triviality (mathematics) - Wikipedia

In mathematics, the adjective trivial is frequently used for objects that have a very simple structure. The noun triviality usually refers to a simple technical aspect of some proof or ... Trivial can also be used to describe solutions to an equation that have a very simple ... while a nontrivial solution is ... (See also Vacuous truth.).

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EE364a Homework 3 solutions

Solution. To show that W is quasiconcave we show that the sets {x | W(x) ≥ α} are convex for ... which holds for any differentiable convex function, applied to g(t) = t2/2. ... Solution. The feasible set is shown in the figure. x1 x2. (1,0). (2/5,1/5). (0, 1) ... right, we see that Qj = Qk. Thus the mapping from the index i to the index s is .

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Cubic Formula -- from Wolfram MathWorld

The cubic formula is the closed-form solution for a cubic equation, i.e., the ... However, Cardano was not the original discoverer of either of these results. ... To solve the general cubic (1), it is reasonable to begin by attempting to eliminate the a_2 ... The general cubic would therefore be directly factorable if it did not have an x ...

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If you take the reciprocal in an inequality, would it change the

Jul 4, 2016 ... I also remark that inverting a sum is not the same as inverting the ..... You can take reciprocals using only multiplication and division. In algebra, there are rules for what you can do to an equation or inequality to make sure it stays true. ... you can turn it into its reciprocal equation by taking the following steps:.

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asymptotics - Big Theta Proof on polynomial function - Computer ...

Sep 16, 2013 ... So your approach is valid, and it's possible that the solution from your class is ... is error-prone, though: it's easy for you to make a mistake along the way. It is ... Start from what you know is true, and then derive the implications of that, ending .... This style of proof will be better for you, and better for the reader.

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mathematics - 1 2 3 4 5 6 7 8 9 = 100 - Puzzling Stack Exchange

Apr 14, 2015 ... If you allow exponents, you can get away with just two: 1 .... I can confirm there is no better solution than Andrew Smith's if we use the operators +-*/. ... There are 215 possibilities total and the number of solutions for the number of ... I know that technically I do not have a shorter solution, but I applied the ...

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linear programming - How the dual LP solves the primal LP ...

If you use the simplex method or some variant of it, you are actually simultaneously solving the primal and dual. That is, from an optimal simplex tableau you can read off both an optimal solution ... From an optimal solution of either primal or dual, complementary slackness .... See my other answer for a ( hopefully) correct one.

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Simplex algorithm - Wikipedia

In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is .... It can also be shown that, if an extreme point is not a maximum point of the ...

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