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Mohamed J.
Experienced Tutor. Math Enrichment at Rochester Math and Science Academy.
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Linear Algebra
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Question:

Solve the following system of equations $$x+y+z=27$$ $$x+y-2z=9$$ $$2x+3y+z=59$$

Mohamed J.

This is equivalent to the matrix system: $$\begin{bmatrix} 1 &1 &1 \\ 1&1 &-2 \\ 2 &3 &1 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 27\\ 9\\59 \end{bmatrix}$$ Using row operations, we get the reduced row echelon matrix $$\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0 & 0 &1 \end{bmatrix} \begin{bmatrix} 10\\ 11\\ 6 \end{bmatrix}$$ So the solution to the system is: $$x=10, y=11 and z=12 Number Theory TutorMe Question: Prove that the product of two consecutive integers is even? Mohamed J. Answer: Let the two integers be represented by a,a+1 Since the integers are consecutive, one of them must be even. Suppose that the first integer is even (and the second will be odd). We can re-write the integers as$$ 2k, 2k+1 a*(a+1) =2k*(2k+1)2k*(2k+1)=2[k*(k+1)]$$Since,$$k*(k+1)$$is an integer, a*(a+1) is even. Therefore, the product of two consecutive integers is even. Statistics TutorMe Question: Cumulative SAT scores are approximated well by a normal model,$$ N(\mu=1500, \sigma=300)$$a) What is the probability that a random student will score greater than 1600 on their SAT? b) Given that a student scores in the 90th percentile, what is their SAT score? Mohamed J. Answer: a) First calculate the z-score that corresponds to a 1500 score. Recall that$$ Z=\frac{x-\mu}{ \sigma} Z=\frac{1600-1500}{300} $$=0.33 Using the standard normal table, the probability of getting a Z-score of 0.33 is 0.63 or 63%. b) Since the student scored in the 90th percentile, their Z-score is 0.90 (90% of students score more than them). Using the standard normal table, Z=1.29 So,$$ Z=\frac{x-1500}{300} $$=1.29 Solving the above equation for x, we get:$$ x=1887 Therefore, the students scored 1887 on their SAT.

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