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Another was to write it without using "division" is that you want the number that results in 1 (or 1+0i) when you multiply by your given number 12* (1/12) = 1 (-9 - 17i) * (x + iy) = (1 + 0i) Multiply the two binomials using FOIL and separate real and imaginary parts to get two equations in...

OK - that makes more sense for the statement of the question. You started with three equations, BUT you discovered that one was a linear combination of the others, so you only have two equations that are linearly independent. Yes, I am saying y (or x) could be the free variable. With only two...

Since the 3rd equation you wrote is a tautology, what you have is a system of two equations in three unknowns. There is no "obvious" choice for the free variable - any one of them will do. The other two variables can be expressed as functions of the one you chose to be "free." If you want it to...

Perhaps another way to interpret what you said about rotation is that the domino may be either vertical or horizontal. How many ways for each orientation? Have you considered a single row or column? You don't have to MOVE the domino, just count how many ways it can be PLACED on the board.

We need to see your work to see where you are stuck. Basically, what you have to understand is that a line intersecting two parallel lines does so at the same angle. You should be able to find two equations in two unknowns.

The centered dot is one way to indicate multiplication. You have 6 objects to arrange. 6 choices for what is first, 5 choices for 2nd, . . . 6 * 5 * 4 * 3 * 3 * 1 = 6! is the numerator of the result BUT not all of those permutations are unique, because some of the objects are identical. The two...

Yes. The domain refers to values of x for which the function is defined, and that is "all real numbers." The range is the set of y that occur for all x in the domain. The range is exactly what you stated.

Perhaps at a 45° angle.

If either b or b' is 0, you can't write the expression in the form where they are in denominators. The respective line would be x=constant.

Since the numeral "4" appears in the problem, the base must be 5 or greater. Carrying out the multiplication on the left side, x^3 + 7x^2 + 13x + 4 = 2x^3 + x + 4 and combining like terms x^3 - 7x^2 - 12x = 0 Eliminating (by dividing out) the trivial root x=0, x^2 - 7x - 12 = 0 That is NOT...

OOPS - I fear JeffM is right! Whereas the probability that Alex will be on team A is exactly 1/2, we must use CONDITIONAL probability for which team you will be on. Since one of the positions on team A is already taken, your probability of being one of the (x-1) players on the same team, out of...

Look at the equation for coefficients \vec{a}: ......\vec{a} = \alpha^{-1}\ \vec\beta,....where....\displaystyle\beta_i = \sum y\ x_i Assume the independent variables are known perfectly, and that all of the measurements y_k have the same Variance, V_y. Suppose that you have weighted all of...

We need to see your work to know how to help.. The choices for each of you are "random" - perhaps you could think of flipping a coin? What is the chance that two flips are the same, either HH or TT? Is that easier than you thought? EDIT: this was an oversimplification, leading to the wrong...

So far so good - except it would have been clearer if you used parentheses to show that lnx^2 meant ln(x^2). Exponentiate both sides of your final step. That is, the inverse operation of "ln". THEN you will have a quadratic in x. You will have to use the quadratic formula, because one of the...

I would interpret the question as being that you are asked to FIND a 10-digit number with no repeating digits, such that all prime factors are 13 or less. Unfortunately, I don't have many clues to offer. A good guess for the rightmost (units) digit is 0, because there are likely to be factors of...

The linear model for y(x1,x2,...x32) is \displaystyle \hat y = \vec a \cdot \vec x = \sum_i\ a_i\ x_i A measurement consists of a value of \vec x_k and the corresponding y_k, which deviates from the model value by an amount \epsilon_k = \hat y_k - y_k = \sum_i\ a_i\ x_{ik} - y_k Let S be...

I will try to find time [currently on vacation] to work this through, so I can make a coherent reply. Two points: 1) All of your 1000 data must be statistically independent - no correlation (or covariance) between measurements. 2) In principle you have to know the statistical uncertainties...

The problem is probably related to Covariances. Variance is a matrix, not a vector. The elements of the covariance matrix are indeed related to the correlation coefficients, as you suspected. The diagonal elements of the covariance are the Variances of the coefficients C_i, and the off-diagonal...

I interpolated in the table! Instead of rounding z to 0.092, I left it as z=0.0917 and interpolated between the table entries for x=0.91 and x=0.92. I am probably a little more fussy than necessary, but to quote the probability precise to 4 digits I think the interpolation would be needed. On...

There are two minor errors. The first is the "continuity correction." Since the normal distribution is continuous, any number less than 725.5 will round to be an integer less than 726. You need to use z(725.5), rather than 726. The second error is a matter of semantics: the ratio (x-mu)/sigma...