Correct olve the following system of equations. If there is a solution, write -3x-4y=20 -6x-8y=38

Answer: cos B-sin²A = cos²A-sin-B

Step-by-step explanation:

Answer: cos B-sin²A = cos²A-sin-B

Explanation:

Using the double angle identity, we can express the left side of the equation as:

cos(A + B).cos(A-B) = cos A.cos B - sin A.sin B

Now, using the Pythagorean identity, we can rewrite the right side of the equation as:

cos²A-sin-B = cos A.cos B - sin A.sin B

Hence, we can conclude that cos(A + B).cos(A-B) = cos B-sin²A = cos²A-sin-B

Answer: Read the step by step explanation for the answer.

Step-by-step explanation:

We can prove this trigonometric identity using the product-to-sum and Pythagorean identities as follows:

Starting with the left-hand side (LHS):

cos(A + B).cos(A - B)

Using the product-to-sum identity:

= [cos A cos B - sin A sin B][cos A cos B + sin A sin B]

= cos²A cos²B - sin²A sin²B

Using the Pythagorean identity:

= cos²A(1 - sin²B) - sin²A(1 - cos²B)

= cos²A - cos²A sin²B - sin²A + sin²A cos²B

= cos²A - sin²A + sin²A cos²B - cos²A sin²B

Using the Pythagorean identity:

= cos²A - sin²A + sin²A(1 - sin²B) - cos²A sin²B

= cos²A - sin²A + sin²A - sin²A sin²B - cos²A sin²B

= cos²A - sin²A + sin²A cos²B - sin²B cos²A

= cos²A - sin²A + cos²A sin²B - sin²B cos²A

Using the Pythagorean identity:

= cos²A - sin²A + cos²A - sin²B

= 2cos²A - sin²A - sin²B

= cos²A + cos²A - sin²A - sin²B

= cos²A + sin²A cos²B - sin²B

= RHS

Therefore, we have shown that (b) cos(A + B).cos(A-B) = cos B-sin²A = cos²A-sin-B.

The options which satisfy the given diagram are

m∠5 + m∠6 = 180°

m∠2 + m∠3 = m∠6

m∠2 + m∠3 + m∠5 = 180°

The sum of the interior angles of a triangle is 180°.

The exterior angle of a triangle is equal to the sum of its interior opposite angles.

Let the Triangle be ABC and the exterior angle be ∠D , then

∠A + ∠B + ∠C =  180°

∠A + ∠B = ∠D , ∠A and ∠B are interior opposite angles of the triangle

So, the Options that satisfied

m∠5 + m∠6 = 180° { linear pair}

m∠2 + m∠3 = m∠6 {exterior angle theorem}

m∠2 + m∠3 + m∠5 = 180° {Angle Sum Property}

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Answer: 5y²

Step-by-step explanation:

  Since 8y² and 3y² are similar terms, we can subtract the coefficients and keep the degree and variable the same.

8 - 3 = 5, so 8y² - 3y² = 5y²

The number of comic books is given as follows:

P/100 x 560.

In which P is the percentage equivalent to comic books on the circle graph.

The number comic books sold at the book fair is given by the proportions in the context of the problem.

The circle graph gives the percentage out of each type of book, hence the percentage of comic books is given as follows:

P/100.

Then the amount of comic books is obtained multiplying the percentage by the total amount of 560 as follows:

P/100 x 560.

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Step-by-step explanation:

red gumballs=6 blue gumballs=6

yellow gumballs=6 green gumballs=6

purple gumballs=6

total number of gumballs=30

probability of a student selecting a red gumballs

no of red gumballs 6 1

=------------------------ = ----- = -----

no of total gumballs 30 5

Using some rules for interior angles, we will see that:

x = 22.5

y = 22.5

Remember that the sum of the interior angles of any triangle is 180°.

Now, we can see that :

3x + k = 180°

Where k is the missing angle in the left triangle.

k = 180° - 3x

if we add the 3 interior angles, we will get:

2x + y + 180° - 3x = 180°

y - x = 0°

So x and y have the same value.

If now use the top triangle, we can write:

(90 - y) + 3x + 2x = 180

And we know that y = x

90 -x  + 3x + 2x = 180

90 + 4x = 180

4x = 180 - 90  =90

x = 90/4 = 22.5

Then:

x = 22.5

y = 22.5

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The statement  "If G is connected with 2 or more nodes, then all nodes must have a degree bigger or equal to 1." is true.

This is because a connected graph is defined as a graph in which there is a path between any two nodes. If a node has a degree of 0, that means it is not connected to any other nodes, and therefore the graph cannot be connected. So, in order for G to be connected, all nodes must have a degree of at least 1.

In conclusion, the answer is true.

A node is a point where several elements are connected.

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Answer:

60 pages

Step-by-step explanation:

64 + 56 = 120

180-120 = 60 pages

Using expression, the following provides one explanation for why there are two ways to add up the symbols on the tablet.

A statement, at least two variables or integers, and one or more arithmetic operations make up a mathematical expression. This mathematical procedure makes it possible to multiply, divide, add, or subtract quantities.

The hieroglyphic number system of ancient Egypt employed a decimal system with hieroglyphs to denote the powers of ten. It is challenging to give a precise response, though, without being able to see the precise symbols on the tablet.

It is feasible that there are two distinct ways to add the numbers represented by the symbols on the tablet, presuming that the tablet is employing a fundamental arithmetic operation like addition and that each symbol represents a different number.

Finding two different sets of integers that, when added together, yield the same answer is one way to approach this issue.

As an illustration, if the symbols stand for 2, 3, and 5, one technique to add them up would be:

2 + 3 + 5 = 10

They might also be added up as follows:

3 + 7 = 10

In this instance, the number 2 would be represented by the sign 2, and the number 5 by the symbol 3. This could be one reason for why the symbols on the tablet can be added in two different ways.

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By applying synthetic division concept, it can be concluded that f(x) = 4x³ + x² + x - 9 can be expressed as f(x) = (x + 1)(4x² -3x + 4) - 13

Synthetic division is a shorthand way of dividing polynomials where we can divide the coefficients of the polynomial by omitting variables and exponents. As a result, we get the coefficient of the quotient and the remainder.

Polynomial remainder theorem states that the value of p in argument b is equal to the remainder of the polynomial division p(x) / (x - b). Specifically, p(x) is divided by x - b with a remainder of zero if, and only if, b is a root of p.

We have the following polynomial:

f(x) = 4x³ + x² + x - 9

and we have k = -1

We will divide the polynomial f(x) by (x + 1). The steps are as follows:

1. Put the coefficients in a row and multiply the outside coefficient by the divisor: 4(-1)= -4

2. Add the inside coefficient to the product from the previous step: -4 + 1 = -3

3. Multiply the result from the previous step by the divisor: -3(-1) = 3

4. Add the next coefficient to the product from the previous step: 3 + 1 = 4

5. Multiply the result from the previous step by the divisor: 4(-1) = -4

6. Add the last coefficient to the product from the previous step: -4 - 9 = -13

Now we take the first coefficient and the result of the sum steps, we get the following numbers: 4, -3, 4, -13

Then we can write the expression as follows: 4x³ + x² + x - 9 = (x + 1)(4x² - 3x + 4) - 13

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Answer:

Let's assume that the original price of petrol is x per liter.

After the increase of R1 per liter, the new price of petrol is (x+1) per liter.

According to the problem, with the new price, you can buy 7 liters less for R504, so we can write:

7(x+1) = 504

Simplifying this equation, we get:

7x + 7 = 504

Subtracting 7 from both sides, we get:

7x = 497

Dividing both sides by 7, we get:

x = 71

Therefore, the original price of petrol was Rx71 per liter.

I - The collapse of the housing market due to low interest rates, easy credit, insufficient regulation, and toxic subprime mortgages: C. Problems in financial markets.

II - Consumers are very worried about the economy: B. Negative demand shift.

1 - The collapse of the housing market due to low interest rates, easy credit, insufficient regulation, and toxic subprime mortgages falls into the category of "Problems in financial markets" (C). This event can lead to a recession because it can cause a decrease in the wealth of households, leading to a decrease in consumer spending and a decrease in aggregate demand.

II - Consumers being very worried about the economy falls into the category of "Negative demand shift" (B). This event can lead to a recession because when consumers are worried about the economy, they tend to save more and spend less, leading to a decrease in aggregate demand. This decrease in demand can cause businesses to decrease production and lay off workers, leading to a decrease in income and further decrease in demand.

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The ratio of strawberries to bananas in the basket is 12:18 or 2:3. To find this ratio, you simply take the number of strawberries and place it over the number of bananas. In this case, there are 12 strawberries and 18 bananas, so the ratio is 12:18.

However, this ratio can be simplified by dividing both numbers by their greatest common factor, which is 6. So, the simplified ratio of strawberries to bananas is 2:3. This means that for every 2 strawberries, there are 3 bananas in the basket.

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Answer: 0.125 and 12.5%

Step-by-step explanation:

To convert any fraction to decimal form, we just need to divide its numerator by the denominator.

Here, the fraction is 1/8 which means we need to perform 1 ÷ 8.

This gives the answer as 0.125. So, 1/8 as a decimal is 0.125

To get its percentage form, you simply multiply the obtained decimal value by 100:

0.125 x 100 = 12.5

The true statement is all irrational and rational numbers have decimal expansions. (second option)

A rational number is a number that can be expressed as a fraction of two integers . Rational numbers have terminating decimals and they have decimal expansions. Examples of rational numbers are 1.5, 2 and 4.25

A irrational number is a number that cannot be expressed as a fraction of two integers . Irrational numbers have repeating decimals. Examples of irrational numbers are 22/7, 1/3.

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156 customers are expected to wear a black shoe.

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favorable outcomes/Total number of outcomes

As per the given data:

Customers with black shoes = 13 out of 30

Customers with brown shoes = 17 out of 30

Total customers in the mall = 360

Probability that a customer wears black shoes:

P(B) = 13/30 = 0.433

Number of customers out of 360 expected to wear black shoes:

= P(B) × 360

= 0.433 × 360

= 156 customers.

Hence, 156 customers are expected to wear a black shoe.

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The solution set for the inequality 6x + 1 ≤ 4x + 1 is x ≤ -3/2 can be written in interval notation as (-∞, 0].

To solve the inequality 6x + 1 ≤ 4x + 1, we need to isolate the variable on one side of the inequality. Here are the steps to do so:

1. Subtract 4x from both sides of the inequality: 6x + 1 - 4x ≤ 4x + 1 - 4x
2. Simplify: 2x + 1 ≤ 1
3. Subtract 1 from both sides of the inequality: 2x + 1 - 1 ≤ 1 - 1
4. Simplify: 2x ≤ 0
5. Divide both sides of the inequality by 2: 2x/2 ≤ 0/2
6. Simplify: x ≤ 0

The solution set in interval notation is (-∞, 0]. This means that any value of x less than or equal to 0 will satisfy the inequality.

"

Prper question:

Solve the inequality. Write the solution set in interval notation.

6x + 1 ≤ 4x + 1

"

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A package that weighs 5 pounds will cost $15 for both options.

The equation can be used to symbolize the price, y: Y is equal to 5 plus 2 times the weight of the package, x.

Another choice is for the buyer to pay a one-time price of $15, which is represented by the equation y = 15, in order to send the package.

We'll compare the equation as a whole.

In this case,

5 + 2x = 15

Compile similar terms,

2x = 15 - 5

2x = 10

x = 10 / 2

x = 5

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Answer:

Either x/9 or 0.75 per cookie

Step-by-step explanation:

Let the number of cookies in a box be x.

The cost of x=9$

Then, the cost of one cookie= x/9

alternatively:

Considering one box contains 12 cookies,

The cost of a box of cookies= 9$

Then, the cost of one cookie= 9/12=0.75

The solution of the equation 1/2x² + 3/8x - 2 using completing the square method are: 1.66 or -2.41.

To solve the equation 1/2x² + 3/8x -2 using completing the square method, follow these steps:

Step 1: Move the constant term to the right-hand side

1/2x² + 3/8x = 2

Step 2: Multiply both sides by 2 to clear the fraction

x² + 3/4x = 4

Step 3: Add the square of half of the coefficient of x to both sides of the equation

x² + 3/4x + (3/8)² = 4 + (3/8)²

The left-hand side is now a perfect square, specifically (x + 3/8)².

Step 4: Simplify the right-hand side of the equation

x² + 3/4x + 9/64 = 256/64 + 9/64

x² + 3/4x + 9/64 = 265/64

Step 5: Take the square root of both sides

x + 3/8 = ±√(265)/8

Step 6: Solve for x

x = -3/8 ± √(265)/8

x = 1.66 or - 2.41

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The cord length is k* theta radian, where theta radian is the central angle and k is the proportionality constant. This means that the rope length is proportional to the central angle in radians.

The complete cord length for a full central angle of 2pai is 2pai * r if the cord arc is a segment of a circle of radius r, where k=2pai is the proportionality constant.

What is arc?

In mathematics, an arc is a portion of the boundary of a circle or curve. It is sometimes referred to as an open curve.

The measurement around a circle that determines its edge is called the circumference, often known as the perimeter. As a result, the distance between any two locations along an arc's circumference is how it is defined.

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The reason that should be used to complete statement 2 "∠2 and ∠3 are supplementary" is linear pair postulate.

In Mathematics, the linear pair theorem is sometimes referred to as linear pair postulate and it states that the measure of two angles would add up to 180° provided that they both form a linear pair.

This ultimately implies that, the measure of the sum of two adjacent angles would be equal to 180° when two parallel lines are cut through by a transversal.

According to the linear pair theorem, we have the following supplementary angles:

∠2 + ∠3 = 180°

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The transformations that would occur to the parent function are a vertical translation, a horizontal translation to the right, and a vertical compression with a reflection over the x-axis.

The parent function of f(x)=-(1)/(3)(x-5)^(3)+3 is f(x)=x^(3). There are several transformations that would occur to the parent function in order to obtain the given function.

1. Vertical translation: The "+3" at the end of the given function indicates that the parent function is shifted up 3 units.

2. Horizontal translation: The "(x-5)" inside the parentheses indicates that the parent function is shifted to the right 5 units.

3. Vertical stretch/compression: The "-(1)/(3)" in front of the parentheses indicates that the parent function is vertically compressed by a factor of 1/3 and reflected over the x-axis.

Therefore, the transformations that would occur to the parent function of f(x)=-(1)/(3)(x-5)^(3)+3 are a vertical translation of 3 units up, a horizontal translation of 5 units to the right, and a vertical compression by a factor of 1/3 with a reflection over the x-axis.

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The final answer of Using Descartes's Rule of Signs, we can determine that there are either 2 or 0 positive real zeros and 0 negative real zeros of [tex]f(x)=8x^4-7x^3-4x^2-2x+7[/tex]

To determine the possible numbers of positive and negative real zeros of f(x)=8x^(4)-7x^(3)-4x^(2)-2x+7 using Descartes's Rule of Signs, we must first count the number of sign changes in the polynomial.

For positive real zeros, we look at the original polynomial: f(x)=8x^(4)-7x^(3)-4x^(2)-2x+7.

There are two sign changes: from 8x^(4) to -7x^(3) and from -2x to +7.

Therefore, there are either 2 or 0 positive real zeros.

For negative real zeros, we look at the polynomial with x replaced by -x: f(-x)=8(-x)^(4)-7(-x)^(3)-4(-x)^(2)-2(-x)+7.

Simplifying, we get [tex]f(-x)=8x^4+7x^3-4x^2+2x+7[/tex]. There are no sign changes in this polynomial, so there are 0 negative real zeros.

In conclusion, Using Descartes's Rule of Signs, we can determine that there are either 2 or 0 positive real zeros and 0 negative real zeros of [tex]f(x)=8x^4-7x^3-4x^2-2x+7[/tex]

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An equivalent expression for the total length of fencing is 11y+10 inches.

The whole distance encircling a shape is referred to as its perimeter. It is the length of any two-dimensional geometric shape's boundary or outline. Depending on the size, the perimeter of several figures can be the same.

From the given figure, length of rectangle is (3y+5) inches and breadth is y inches. Sides of equilateral triangle measures y inches.

a) We know that, the perimeter of a rectangle is 2 (Length+Breadth)

= 2(3y+5+y)

= 2(4y+5)

= 8y+10 inches

Perimeter of equilateral triangle is 3×Side

= 3×y

= 3y inches

So, total length = 8y+10+3y inches

b) Equivalent expression is

8y+10+3y

= 11y+10 inches

c) Factors of the expression is 1 and 11y+10.

Therefore, an equivalent expression for the total length of fencing is 11y+10 inches.

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The cost of one rose shrub costs $12 and one pot of ivy costs $24, respectively.

It takes two equations with two variables to solve a system of equations. The variables can then be solved for using algebraic techniques like substitution or elimination.

Let x be the cost of one rose bush and y be the cost of one pot of ivy.

From Mei's purchase, we have:

3x + y = 36

From Ming's purchase, we have:

9x + 8y = 168

Use the first equation to solve for y:

y = 36 - 3x

Substituting this into the second equation, we get:

9x + 8(36 - 3x) = 168

9x + 288 - 24x = 168

-15x = -120

x = 8

Substituting x = 8 into equation of y, we get:

y = 24

Hence, the cost of one rose bush is $8, and the cost of one pot of ivy is $24.

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On a roll of a number cube, the theoretical probability of the event of rolling a 8 is 0.

The Theoretical Probability is defined as the likelihood of an event occurring based on all possible outcomes in a given situation.

The theoretical probability is determined by dividing the number of ways the event can occur by the total number of possible outcomes.

The theoretical probability of rolling a specific number on a fair, six-sided number cube is 1/6, because there are 6 equally likely outcomes which are the numbers 1 to 6,

So, the theoretical probability of rolling an 8 on a number cube is 0 or 0%, because an 8 is not one of the possible outcomes when rolling a standard, six-sided number cube.

Therefore, the required probability is 0.

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One way to label the tick marks on the number lines is to use whole numbers for the longer tick marks and decimals for the shorter tick marks.

For example, on the first number line, you could label the longer tick marks as 0, 1, 2, 3, 4, and 5, and the shorter tick marks as 0.1, 0.2, 0.3, etc.

This fits with the structure of the base-ten system because each whole number is ten times greater than the corresponding decimal.

Another way to label the tick marks is to use fractions for the shorter tick marks. For example, on the second number line, you could label the longer tick marks as 0, 1, 2, 3, 4, and 5, and the shorter tick marks as 1/10, 2/10, 3/10, etc.

This also fits with the structure of the base-ten system because each fraction is one-tenth of the corresponding whole number.

Both of these labeling methods are accurate and fit with the structure of the base-ten system, so either one would be an appropriate way to label the tick marks on the number lines.

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