What is the solution to the system of linear equations
A. (-3,0)
B. (-3,3)
C. (0,2)
D. (3,1)

Answer: The answer is (B) This is the graph of a function, but it is not one-to-one.

Step-by-step explanation:  We are given a graph and four options out of which we are to select the statement that best describes the given graph.

Since the graph of a linear function is a straight line, so the first option is not correct.

The given graph is of a function because for a particular value of y, there is a fixed value of 'x'.

Also, since there are two different values of x having same value for y, so the function is not one-to-one. For example, If y = f(x) is the function, then f(1.5) = f(-1) = 3.

Therefore, the graph represents a function, but it is not one-to-one.

Thus, the correct option is (B).

The numbers, their sum is 7, and their product is 15 are,

(7 + √11 i)/2 and (7 - √11 i)/2.

Use the concept of complex numbers defined as:

A real number and an imaginary number are effectively combined to create a complex number. The complex number is written as a+ib, where a and ib are real and imaginary numbers, respectively.

Let the numbers x and y such that,

x+y = 7  .....(i)

xy = 15  ......(ii)

Since we know that,

(x - y)² = x² + y² - 2xy

Add 2xy on both sides,

(x - y)² + 2xy = (x² + y² + 2xy) - 2xy

(x - y)² + 2xy = (x + y)² -2xy

put the values from equations (i) and (ii),

(x - y)² + 30 = 49 - 30

Simplify it,

(x - y)²  = -11

Take square root on both sides,

x - y = √11 i   .....(iii)                 ( i = √-1)

Add equations (i) and (iii),

2x = 7 + √11 i

x = (7 + √11 i)/2

Substitute this value of x in equation (iii),

y =  (7 - √11 i)/2

Hence,

The required numbers are (7 + √11 i)/2 and (7 - √11 i)/2.

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The complete question is:

Find the complex or real numbers such that,

Their sum is 7, and their product is 15.

The amount of money that is presently in my investment account is $11,820.

The formula that can be used to determine the amount in my account presently is:

P = FV / (1 + r)^n

25,000 / (1.0425)^18 = $11,820

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Final answer:

To find the current amount needed to reach $25,000 in 18 years at a 4.25% interest rate, the formula P = A / (1 + r)^n is used. Plugging in the values gives P ≈ $11,820, indicating that approximately $11,820 is currently in the account. Thus, the correct answer is (b) $11,820.

Explanation:

The question asks how much money needs to be present in an account now to reach a future value of $25,000 in 18 years at an annual compound interest rate of 4.25%. To find the present value, we use the formula for compound interest: P = A / (1 + r)^n, where P is the present value, A is the amount of money accumulated after n years, inclusive of interest, r is the annual interest rate (decimal), and n is the number of years the money is invested for.

In this case, A = $25,000, r = 0.0425 (4.25%), and n = 18 years. Plugging these values into the formula gives: P = $25,000 / (1 + 0.0425)^18. Calculating this gives us P ≈ $11,820, which means the amount of money currently in the account to reach $25,000 in 18 years at a 4.25% annual interest rate compounded annually is approximately $11,820. Therefore, the correct answer is (b) $11,820.

Using the Quadratic formula

x^4 = (x^2)^2

hence

(x^2)^2 + 4x +(-1) = 0

ax^2=1(x^2)^2

bx = 4(x)

c=(-1)

By substituting the value of one number in terms of the other from the difference equation into the sum of squares equation, we find the two numbers to be 10 and -3.

Let's solve the problem step-by-step. Given that two numbers differ by 13 and the sum of their squares is 125, we can set up the following equations where x and y are the two numbers:

Substitute the value of y from equation (1) into equation (2), giving us:

x^2 + (x - 13)^2 = 125

Solving this, we find that x = 10 and hence y = -3, which means the two numbers are 10 and -3.

The center of the circle that can be circumscribed about a triangle with vertices A(2, 6), B(2, 0), and C(10, 0) is calculated by finding the intersection of the perpendicular bisectors of the sides of the triangle. In this case, the center of the circle is at (6,3).

For the triangle with vertices A(2, 6), B(2, 0), and C(10, 0), the circle that you can circumscribe about it is known as the circumcircle. The center of this circle is the circumcenter which is obtained by the intersection of the perpendicular bisectors of the sides of the triangle.

Since points A and B lie along the same vertical line, the perpendicular bisector of line AB is a horizontal line halfway between them at y=3. The bisector of line BC is a vertical line halfway between points B and C at x=6.

The intersection of these two bisectors (x=6, y=3) is the circumcenter. So, the center of the circumcircle is at (6,3).

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An expression is a sum of cubes if it can be written in form of a³ + b³, where a and b can be any constant/variables.

So, this means the given expression 125x³ + 144 can only be sum of cubes if the both the terms i.e. 125x³ and 144 are perfect cubes.

125x³ = (5x)³ . This means 125x³ is a perfect cube. However, 144 is not a perfect cube.

Hence we can conclude that the expression 125x³ + 144 is not a sum of cubes. So the answer to this question if FALSE

The last operation performed when evaluating (8-2x)^2+4 for x=3 is the addition operation.

Equations are known to make up of variables and constants. Given the expression;

(8-2x)^2+4

If x = 3, substitute into the expression to have:

(8-2x)^2+4

= (8-2(3))^2+4
=  (8 - 6)^2 + 4

Evaluate the expression in parenthesis

= 2^2 + 4
= 4 + 4

Add the resulting expression

(8-2x)^2+4  = 8

Hence the last operation performed when evaluating (8-2x)^2+4 for x=3 is addition operation

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Before a theorem is proven, it is known as a proposition. This term indicates that the statement is proposed to be true and is pending proof.

A theorem called before it is proven is known as a proposition. A proposition is a statement that is proposed to be true and is awaiting proof or disproof. In contrast, a postulate is a statement that is assumed to be true without proof for the purposes of reasoning in mathematics or science, whereas a contradiction refers to a statement that is always false. A tautology is a statement that is true by necessity or by virtue of its logical form.

Final answer:

A theorem before it is proven is known as a proposition. Propositions are statements considered in mathematics and philosophy for investigation and proof. Other terms like postulate, contradiction, and tautology have different distinct meanings in logic.

Explanation:

Before a theorem is proven, it is known as a proposition. A proposition is a statement that proposes an idea, which can be either true or false, but has not yet been proven to be true. In mathematics and philosophy, this term is used to describe concepts that are considered and investigated for proof. Once a proposition is proven conclusively, it becomes a theorem. Other options like postulate, contradiction, and tautology have different meanings within mathematical logic and reasoning.

For a triangle with an obtuse angle B and the longest side opposite it (side c), the correct statement is a^2 + c^2 > b^2.

The correct answer is option B.

In a triangle with sides a, b, and c, where angle B (opposite side b) is obtuse, and side c is the longest side (opposite the largest angle), we can apply the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles:

c^2 = a^2 + b^2 - 2ab * cos(B)

Given that angle B is obtuse, cos(B) will be negative. Therefore, the term -2ab * cos(B) will subtract from the sum of a^2 + b^2. The Law of Cosines now becomes:

c^2 = a^2 + b^2 + 2ab * cos(B)

Considering the given options:

A. a^2 + c^2 < b^2 - This is not necessarily true in this context.

B. a^2 + c^2 > b^2 - This is true based on the Law of Cosines.

C. b^2 + c^2 < a^2 - This is not necessarily true in this context.

D. a^2 + b^2 < c^2 - This is not necessarily true in this context.

Therefore, from the given options the correct one is option B.

The solutions are as follows:

11) 110 - alternate exterior angles

12) 84 - alternate interior angles

13) 80 - supplementary angles

14) 111 - alternate interior angles

15) 125 - alternate interior angles

16) 47 -  alternate exterior angles

17) 53- corresponding angles

18) 113-alternate interior angles

Here, we have,

from the given figure, we get,

11) the required angle is alternate exterior angle of 110,

so, the value is 110.

12) the required angle is alternate interior angle of  84

so, the value is 84

13) the required angle is supplementary angle of 100

so, the value is 80

14) the required angle is alternate interior angle of 111

so, the value is 111

15) the required angle is alternate interior angle of 125

so, the value is 125

16) the required angle is alternate exterior angle of 47,

so, the value is 47.

17) the required angle is corresponding angle of 53

so, the value is 53.

18) the required angle is alternate interior angle of  113

so, the value is 113.

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