Question

Solve the following equations using step-by-step equational reasoning, and list each step.

a) Solve the following equation for x ∈ ℤ/43ℤ: x^{2} ≡
25 (mod 43)

b) Solve the following equation for x ∈ ℤ/19ℤ: x^{2} ≡ 5
(mod 19)

c) Solve the following equation for x ∈ ℤ/55ℤ: x^{2} ≡ 1
(mod 55)

d) Solve the following equation for x ∈ ℤ/41ℤ: 3 * x^{2}
≡ 7 (mod 41)

e) Solve the following equation for x ∈ ℤ/49ℤ: x^{2} ≡ 8
(mod 49)

f) Solve the following equation for x ∈ ℤ/21ℤ: (8 *
x^{2} ) + 4 ≡ 6 (mod 21)

Answer #1

Number Theory: Please show all work.
Solve each of the following equations for the unknown variable X
≡ 0, 1, 2, 3, 4, 5, 6 mod 7.
(i) 2X + 5 ≡ 6 mod 7.
(ii) 3X + 5 ≡ 6 mod 7.

Solve each of the following congruences or explain why it is not
solvable:
(i) x2 + 5x + 7 ≡ 0(mod 41).
(ii) x2 ≡ 11(mod 125)

find the following square roots.
A. √8( mod 41)
B. √7(mod 41)
C. √7 (mod 19)
D. √7 (mod 29)
E. √5 (mod 29)

For each of the following congruences if there is a solution,
express the solution in the form x
≡ some_number (mod
some_modulus), e.g. x ≡ 6 (mod 9). To standardize
answers, some_number should always be a value
in the range {0, 1, 2, ..., some_modulus -1}. For example
x ≡ 5 (mod 8) is OK but x ≡ 13 (mod 8) is not.
If there is no solution say "No solution". You don't have to
show work for any of the problems. Type...

Using Chinese Remainder Theorem solve for X:
x = 2 (mod 3)
x = 4 (mod 5)
x = 5 (mod 8)
I have the answer the professor gave me, but I can`t understand
what`s going on. So if you could please go over the answer and
explain, it would help a lot.
x = 2 (mod 3)
x = 3a + 2
3a + 2 = 4 (mod 5)
(2) 3a = 2 (2) (mod 5) -----> why number...

Problem #7-18 Using the following
equations, graph the constraints, and solve using the corner point
approach. NOTE: is a minimization
problem like Holiday Turkey example in book
X1 = number of undergraduate courses
X2 = number of graduate courses
Minimize cost = $2,500X1 +
$3,000X2
subject
to
X1 >= 30
X2 >= 20
X1 + X2 >= 60
X1, X2 >= 0

1)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express your answer in terms
of the parameters t and/or s.)
x1
+
2x2
+
8x3
=
6
x1
+
x2
+
4x3
=
3
(x1,
x2, x3)
=
2)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express...

Please show complete, step by step
working.
Solve the following equations for 0 ≤ x ≤ 2π, giving
your answers in terms of π where appropriate
i) cos
x = -√3/2
ii) 2 sin
2x = 1
iii) cot x =
1/√3
b) Solve the following for 0° ≤ q ≤
360°,
i) sin^2Q = 3/4
ii) 2 sin^2Q - sinQ = 0 {Hint: factor out sinθ)

2. Solve the following systems of equations.
y=x^2−4x+4and2y=x+4
Group of answer choices
(4,4)
(12,94)and(4,4)
(94,12)
3. The axis of symmetry for f(x)=2x^2−3x+4 is x = k. What is the
value of k?
4. The horizontal asymptote for f(x)=3^x+8 is y = k. What is the
value of k?
5. For which equation is the solution set {3, 4}?
Group of answer choices
x^2−9x=16
x^2+3x+4
x^2−7x+12
6. Find the smaller root of the equation
Group of answer choices
-2
2
-4...

Using the data below find the slope of the regression line.
x
y
xy
x2
y2
3
4
12
9
16
4
6
24
16
36
5
7
35
25
49
7
12
84
49
144
8
14
112
64
196
27
43
267
163
441

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