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TRENTO, A.A. 2019/20

MATHEMATICS FOR DATA SCIENCE/BIOSTATISTICS EXERCISE SHEET # 1

Important! In solving the exercises

• explain what you are doing,

• explain why you are doing what you are doing, and

• spell out all intermediate steps.

Exercise 1.1. Discuss the solutions x of the equation ax = b, where a, b are real numbers.

(Suggerimento: You should say under which conditions on a, b the equation has solutions, and then how many; and say under which conditions on a, b the equation has no solutions.)

Exercise 1.2. Discuss the solutions x, y of the equation

(1) ax + by = c.

In detail,

(1) Discuss the case a = b = 0.

(2) Show how to find the solutions when a ̸= 0.

(3) Show how to find the solutions when b ̸= 0.

(4) Show that if a, b are both non-zero, and if x

0

, y

0

is a solution of (1), then the solutions of (1) are exactly of the pairs x, y of the form

{

x = −bt + x

0

y = at + y

0

where t is an arbitrary real number. (Suggerimento: We are going to discuss this in the lectures, but try and do it yourself.)

Exercise 1.3. Find the solution(s) of {

2x + 3y = 5 7x − 4y = 8 with the elimination methods.

Then try the substitution mathod(s):

(1) first solve the first equation, writing x in terms of y. Substitute this value of x in the second equation to get an equation in y alone;

(2) then solve the second equation, writing y in terms of x. Substitute this value of y in the second equation to get an equation in x alone.

Verify that all methods give the same results.

(2)

2 EXERCISE SHEET # 1

Exercise 1.4. Show that the system {

2x + 3y = 5 6x + 9y = 15 has infinite solutions, and spell them out.

Exercise 1.5. Show that the system {

2x + 3y = 5

6x + 9y = −1

has no solutions.

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