how many solutions does the system of equations 4x+2y=6 and y=-2x+6 have?

1). W is closed under addition.  

  W is not closed under scalar multiplication

2. W is not closed under addition

   W is closed under scalar multiplication

1. Let V be the vector space of all 2x2 matrices with real entries. Consider the subset W of V consisting of matrices of the form:

| a b |

| 0 c |

where a, b, and c are real numbers. Note that W is non-empty, since the matrix |0 0| is in W.

|0 0|

Now, let A and B be any two matrices in W, so that A has the form:

| a1 b1 |

| 0 c1 |

and B has the form:

| a2 b2 |

| 0 c2 |

Then the sum of A and B has the form:

| a1+a2 b1+b2 |

| 0 c1+c2 |

which is clearly also in W. Therefore, W is closed under addition.

However, W is not closed under scalar multiplication. Let A be any matrix in W, so that A has the form:

| a b |

| 0 c |

where a, b, and c are real numbers. Then, if we multiply A by a non-real scalar k = xi (where x is a real number and i is the imaginary unit), we get:

kA = xi * | a b | = | xia xib |

| 0 c | | 0 xc |

which is not in W, since the entry in the (1,2) position is non-zero. Therefore, W is not closed under scalar multiplication.

2. Let V be the vector space of all polynomials with real coefficients. Consider the subset W of V consisting of all polynomials of degree at most 2.

Note that W is non-empty, since the polynomial p(x) = 0 is in W.

Now, let c be any real scalar, and let p(x) be any polynomial in W.

Then cp(x) is a polynomial of degree at most 2, since multiplying a polynomial of degree at most 2 by a scalar does not change its degree. Therefore, W is closed under scalar multiplication.

However, W is not closed under addition. Let p(x) and q(x) be any two polynomials in W of degree at most 2. Then the sum p(x) + q(x) may have degree greater than 2, and hence may not be in W. For example, if p(x) = x^2 + 2x + 1 and q(x) = -x^2 + x - 1, then p(x) + q(x) = 3x, which is not in W. Therefore, W is not closed under addition.

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a) Tοtal cοst C(x) = 200x + 1500

   Revenue cοst R(x) = 300x

The definitiοn οf an equatiοn in algebra is a mathematical statement that demοnstrates the equality οf twο mathematical expressiοns. Fοr instance, the equatiοn 3x + 5 = 14 cοnsists οf the twο equatiοns 3x + 5 and 14, which are separated by the 'equal' sign.

Let x be the number οf scοοters.

It cοsts yοu $1500 fοr tοοls and equipment tο get started, and the materials cοst $200 fοr each scοοter.

Here fixed cοst = $1500 and variable cοst = $200

We knοw that cοst functiοn = variable cοst per unit + fixed cοst

Cοst functiοn :

C(x) = 200x+1500

Yοur scοοter sell fοr $300

Revenue functiοn :

R(x) = 300x

(b) The sοlutiοn means

When yοu start a business assembling initial cοst is $1500 fοr tοοls and equipment.

The material cοst will increase $200 fοr each number οf scοοter increases.

The revenue will increase $300 fοr each number οf scοοter increases.

c) Let us take number οf scοοters x=50

Then cοst C(50)=200*50+1500 = 10000+1500=$11500

Revenue Cοst R(50)=300*50=$15000

Then prοfit = Revenue- cοst = 15000-11500 = $3500.

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Answer:

744416

Step-by-step explanation:

Just add everything together

Answer:

To find the area of the trapezoid-shaped playground, we need to use the formula:

Area = (1/2) × (base1 + base2) × height

In the given diagram, the length of the top base is 24 feet, the length of the bottom base is 48 feet, and the height is 52 feet.

So, the area of the playground is:

Area = (1/2) × (24 + 48) × 52

= (1/2) × 72 × 52

= 1,872 square feet

Therefore, the area of the playground is 1,872 square feet.

The closest option to this answer is option B, which is 1,768 square feet. However, the correct answer is actually 1,872 square feet.

The distance between the points C=(-7,6) and E=(-2,2) is 6.4 units.

To calculate the distance between two points, we use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, the points are C=(-7,6) and E=(-2,2), so we can plug in the values into the formula:

d = √((-2 - (-7))^2 + (2 - 6)^2)

d = √((5)^2 + (-4)^2)

d = √(25 + 16)

d = √(41)

d = 6.4

Therefore, the distance between the points C=(-7,6) and E=(-2,2) is 6.4 units.

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After 5 hours of draining, the depth of the water in Mara's swimming pool will be 11/4 feet

The depth of the water variations via- 3/ 4 foot each hour, this means that the depth decreases by employing 3/4 foot every hour.

Still, also after one hour of draining, the intensity of the water might be

If the primary depth of the water was five bases.

5-(3/4) = 41/4 ft

After hours, the depth might be

-(3/4) = 31/2 fr

After three hours

-(3/4) = 23/4 ft

After 4 hours

-(3/4) = 2 ft

After five hours

2-(3/4) = 11/4 bases

Thus, after 5 hours of draining, the depth of the water in Mara's swimming pool will be 11/4 feet

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Answer:

Zoe will save 30% of the original cost of the sweater, which is $40. To calculate the amount saved, we must multiply 30% by the original cost of the sweater, which gives us 0.30 x $40 = $12. Therefore, Zoe will save $12 on the sweater with the 30% discount

Step-by-step explanation:

Answer:

She will save $12.00

Step-by-step explanation:

.3 x 40 = 12

30% as a decimal is .3.

The cost of the sweater on sale would be

.7 x 40 = 28

If we take 30% off, we are leaving 70% on.

28 + 12 = 40 the original cost.

Helping in the name of Jesus.

Phil spent 18.26% of his pocket money on eating out for one meal.

The following is the formula for calculating the percentage difference between two values:

100% of ((new value - old value) old value)

In order to describe the change as a percentage, this formula calculates the difference between the new and old values, divides that difference by the old value, and multiplies the result by 100.

Given that, Phil spent $21 of his $115 pocket money.

Thus,

$21 ÷ $115 × 100% = 18.26%

Hence, Phil spent 18.26% of his pocket money on eating out for one meal.

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Given that Jordan cut strips of border for a design for a triangular sign to put on a bulletin board. Two of the strips were 15 inches long and the third was 30 inches long. We need to determine if the design can be made.

To determine whether the design can be made or not, we will check whether the sum of the lengths of any two sides of the triangle is greater than the length of the third side or not. Let a, b and c be the three sides of the triangle such that c is the longest side. According to the Triangle Inequality Theorem, For a triangle to be formed, the sum of the lengths of any two sides of the triangle should be greater than the length of the third side.Thus, a + b > cIf the above condition is satisfied, then the design can be made. If not, then the design cannot be made.
Let's check for the given design:
a + b > ca + b = 15 + 15 = 30 (Since two of the strips were 15 inches long)
Therefore, 30 > 30 (Since the third strip was 30 inches long)The given design satisfies the Triangle Inequality Theorem. Hence, the design can be made.

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A figure which is a dilation of figure E include the following: A. figure F.

In Geometry, dilation can be defined as a type of transformation which typically changes the size of a geometric shape, but not its shape. This ultimately implies that, the size of the geometric shape would be increased (enlarged) or decreased (reduced) based on the scale factor applied.

In Geometry, a scale factor is the ratio of two corresponding length of sides or diameter in two similar geometric figures such as equilateral triangles, quadrilaterals, and other types of polygons.

Based on the image (see attachment), we can logically deduce that only figure E and figure F share the same (common) line of symmetry.

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The value of Function f(x) =  5x²+ 3x + 1 at x=2 is 27.

A function is an expression, rule, or law in mathematics that describes a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are common in mathematics and are required for the formulation of physical relationships in the sciences.

Given:

We have the Function as

f(x) = 5x²+ 3x + 1

Now, we have to find value of function at x= 2 so

f(x) = 5x²+ 3x + 1

f(2) = 5(2)²+ 3(2) + 1

f(2) = 5(4)+ 6 + 1

f(2) = 20 + 7

f(2)= 27

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The answer of the given question based on the Irons is trying to lay out the bases for a game of kickball such that the infield is a square the answer is  first base and third base should be about 35.4 feet apart.

The Pythagorean theorem is  fundamental concept in geometry that relates to three sides of right triangle. It states that in right triangle, the square of  length of  hypotenuse (the side opposite the right angle) is equal to  sum of the squares of  lengths of other two sides.

We can use the Pythagorean theorem to find this distance.

Let's call the distance between home base and first base "a". We know that the distance between home base and third base is also "a", because the infield is a square. We also know that the distance between home base and second base (which is the hypotenuse of the right triangle formed by home base, first base, and second base) is 25 feet.

Using the Pythagorean theorem, we can solve for "a":

a² + 25² = a² + a²

2a² = 25²

a² = (25²)/2

a = sqrt((25²)/2) ≈ 17.7 feet

So the distance between first base and third base should be approximately 35.4 feet (2a). Rounding to the nearest tenth of a foot, this is 35.4 feet. Therefore, first base and third base should be about 35.4 feet apart.

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The time can be obtained from 1/0.042 * ln (p(t)/1000)

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is a variable. The base a is usually a number greater than 1, although it can be any positive number.

Exponential functions have a distinctive characteristic that sets them apart from other types of functions: the value of the function increases or decreases exponentially as the value of x increases or decreases. In other words, the rate of change of the function is proportional to its current value, which results in a rapid growth or decay.

We have that;

p(t) = 1000e^0.042t

p(t)/1000 = e^0.042t

ln (p(t)/1000) = 0.042t

t = 1/0.042 * ln (p(t)/1000)

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Answer:

t = 56 degree

Step-by-step explanation:

A triangle is 180 degrees.

The angle on the right is a vertical angle to the 34-degree angle, meaning their angles are equal.

We know two angles; one is 90 degrees, and the other is 34 degrees. To find the angle t, we take

180 - 90 - 34 = 56 degree

So, t = 56 degree

Therefore , the solution of the given problem of unitary method comes out to be a 2-liter soda costs $4.98 and a platter of nachos costs $11.99.

The measurements taken from this femtosecond section must be multiplied by two in order to complete the task using the unitary variable technique. In essence, the characterised by a group and the hue groups are both removed from the unit approach when a desired object is present. For example, 40 pens with a expression price will indeed cost Inr ($1.01). It's possible that one country will have total influence over the approach taken to accomplish this. Almost every living creature has a distinctive quality.

Here,

Let's use "N" for the price of a platter of nachos and "S" for the price of a 2-liter soda.

The first aspect of the issue reveals that:

=> 5N + 2S = 67.87

The second component of the issue reveals the following to us:

=>2N + S = 27.94

=> S = 27.94 - 2N

When we use this expression in place of S in the first equation, we obtain:

=> 5N + 2(27.94 - 2N) = 67.87

When we simplify and account for N, we obtain:

=> 5N + 55.88 - 4N = 67.87\sN = 11.99

We can determine S by substituting this number for N in the second equation:

=> 2(11.99) + S = 27.94\sS = 4.98

As a result, a 2-liter soda costs $4.98 and a platter of nachos costs $11.99.

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The measure of the angle that is created when we move the hour hand to 5 and the minute hand to 1 is 144°.

To find the measure of the angle that is created when we move the hour hand to 5 and the minute hand to 1, we need to calculate the difference between the two positions on the clock.

Each hour on the clock represents (360°/12) = 30°, and each minute represents (360°/60) = 6°.

Therefore, the hour hand at 5 represents 150° (5 x 30°) and the minute hand at 1 represents 6° (1 x 6°). The difference between the two positions is 144° (150° - 6°).

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Since you sailed 0.5 miles 60º north of east to reach the marina, the distance you sailed eastward is 0.5cos(60º) = 0.25 miles and the distance you sailed northward is 0.5sin(60º) = 0.433 miles.

To return to your cabin, you sailed directly south for 0.433 miles, forming the adjacent side of a right triangle, and then sailed directly west for 0.25 miles, forming the opposite side of the same right triangle.

Therefore, the distance you sailed on your return trip is the hypotenuse of the right triangle, which can be found using the Pythagorean theorem:

distance = sqrt((0.433)^2 + (0.25)^2) ≈ 0.51 miles

Rounded to the nearest hundredth, the distance you sailed on your return trip is 0.51 miles, which is closest to option (a) 0.68 miles.

The sum of the polynomials is 6x^(2)-3x-6.

To add the polynomials using a vertical format, we will first write each polynomial in a column, lining up like terms vertically. Then, we will add the coefficients of each like term together to find the sum.

Here is the solution in a vertical format:

So, the sum of the polynomials is 6x^(2)-3x-6.

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x = 18 ...... y = 20/3

x = 30 ..... y = ?

____________

[tex]y = \dfrac{30 \cdot 20}{3 \cdot 18} = \dfrac{100}{9}[/tex]

The center of the circular track with the equation x² - 18x + y² - 22x = -177 is (20, 0). See below for other solutions

The equation of a circle that passes through the origin is represented as

x² + y² = r²

Where

r = radius

For circle 13, we have

Point = (0, 6)

So, the radius is

0² + 6² = r²

r = 6

This gives

x² + y² = 6²

For the point (√11, 5), we have

(√11)² + 5² = 6²

36 = 36 --- true

The point (√11, 5) is on the circle

For circle 14, we have

Point = (-7, 0)

So, the radius is

-7² + 0² = r²

r = 7

This gives

x² + y² = 7²

For the point (√14, 6), we have

(√14)² + 6² = 7²

50 = 49 --- falsee

The point (√14, 6) is not on the circle

Andy's error is that he did not square 12 in (√23)² + (11)² ≠ 12

The correct solution is

(√23)² + (11)² = 12²

144 = 144

The point (√23, 11) is on the circle

The equation of a circle is represented as

(x - a)² + (y - b)² = r²

For circle 16, we have

Center = (-1, 5)

Radius, r = 4

So, we have

Equation: (x + 1)² + (y - 5)² = 4²

For circle 17, we have

Center = (2, 0)

Point = (-2, 3)

So, we have

Equation: (x - 2)² + (y - 0)² = (-2 - 2)² + (3 - 0)²

Equation: (x - 2)² + y² = 25

Given that

x² - 18x + y² - 22x = -177

This gives

x² - 40x + y² = -177

Factorize

(x - 20)² + y² = -177 + 400

Evaluate

(x - 20)² + y² = 223

From the above, we have

Center = (20, 0)

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The company will need 58,800 cubic inches of wax to make 420 candles.

To calculate the amount of wax required to make 420 candles,

The volume of a rectangular prism is given by the formula V = l x w x h, where l is the length, w is the width, and h is the height.

So, we have

V = 5 in x 4 in x 7 in = 140 cubic inches

To find the total amount of wax required to make 420 candles,

We have

Total amount of wax = 140 cubic inches/candle x 420 candles

= 58,800 cubic inches

Hence, the company needs 58,800 cubic inches

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Reflect point A over the line to get point A', -1,2 is A'.

What is a graph?

A graph is a structure that fundamentally consists of a set of items where some pairs of the objects are "connected" in some way. This definition comes from discrete mathematics, more especially graph theory. The items are represented by mathematical abstractions called vertices, and each pair of connected vertices is known as an edge.

A graph is frequently depicted in a diagram by a collection of dots or circles for the vertices and lines or curves for the edges. Among the topics covered by discrete mathematics are graphs.

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In a case whereby Malcolm is finding the solutions of this quadratic equation by factoring 3x²-24x=-45, the correctly factored equation that can be used to find the solutions is 3(x-3)(x-5)=0

The given equation is  3x²-24x=-45

The equation can be re written as;

3x²-24x=-45

3x²-24x+45 = 0

Then we can divide by 3 and have

x²-8x + 12= 0

thene if we factorize we have

x= 5, x= 3

Therefore, option B is correct.  which is 3(x-3)(x-5)=0

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To solve this problem, we can use a system of equations.

Let's call the number of Php 5 coins x and the number of Php 10 coins y. The first equation will represent the total number of coins: x + y = 120. The second equation will represent the total amount of money: 5x + 10y = 950. Now we can use the elimination method to solve for one of the variables. Let's multiply the first equation by -10 and then add the two equations together:

-10x - 10y = -1200

5x + 10y = 950

-----------------

-5x = -250

Divide both sides by -5 to get x:

x = 50. Now we can plug this value back into the first equation to solve for y:

50 + y = 120

y = 70

So there are 50 Php 5 coins and 70 Php 10 coins.

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"f " is continuous at x = 0.

a) To determine if f is continuous from the left at x = 0, we need to evaluate the limit of f(x) as x approaches 0 from the left. This means we need to use the first piece of the function, x + sin x, since this is the piece that applies when x < 0:

lim(x->0-) f(x) = lim(x->0-) (x + sin x) = 0 + sin 0 = 0

Since the limit exists and is equal to the value of the function at x = 0 (which is 2), f is continuous from the left at x = 0.

b) To determine if f is continuous from the right at x = 0, we need to evaluate the limit of f(x) as x approaches 0 from the right. This means we need to use the third piece of the function, 2/(1+x^2), since this is the piece that applies when x > 0:

lim(x->0+) f(x) = lim(x->0+) (2/(1+x^2)) = 2/(1+0^2) = 2

Since the limit exists and is equal to the value of the function at x = 0 (which is 2), f is continuous from the right at x = 0.

c) To determine if f is continuous at x = 0, we need to make sure that the limits from the left and right both exist and are equal to each other and to the value of the function at x = 0. We already found that the limits from the left and right are both equal to 2, which is also the value of the function at x = 0. Therefore, f is continuous at x = 0.

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Using synthetic division, the value of h(-4) is - 24.

1. Set up the synthetic division grid with the divisor (-4) in the top left corner and the coefficients of the polynomial in the top row.
-4 | 2   0   -25   4

2. Bring down the first coefficient to the bottom row.
-4 | 2   0   -25   4
   |                      
     2

3. Multiply the divisor (-4) by the first number in the bottom row (2) and put the result (-8) in the second column of the top row.
-4 | 2   0   -25   4
   |     -8            
     2

4. Add the numbers in the second column (0 and -8) and put the result (-8) in the second column of the bottom row.
-4 | 2   0   -25   4
   |     -8            
     2  -8

5. Repeat steps 3 and 4 for the remaining columns.
-4 | 2   0   -25   4
   |     -8     32  -28    
     2  -8      7   -24

6. The last number in the bottom row (-24) is the remainder and the value of h(-4). The other numbers in the bottom row (2, -8, 7) are the coefficients of the quotient polynomial.

Therefore, h(-4) = 24.

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\(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\)

Let \(n \in \mathbb{N}\) with \(n>2\). We consider the set \( S = \{a \in \mathbb{Z}_{n} \ | \ a^{2} = [1] \in \mathbb{Z}_{n}\} \). We have to prove that \( S \neq \emptyset \).

We prove by contradiction. Suppose \( S = \emptyset \). This implies that for all \( a \in \mathbb{Z}_{n}, \ a^{2} \neq [1] \in \mathbb{Z}_{n}\). Thus, \( [1] \) is not a square in \(\mathbb{Z}_{n}\). But since \(n >2\), \([1]\) has at least two square roots in \(\mathbb{Z}_{n}\) which implies that \( S \neq \emptyset \).

Therefore, \(S \neq \emptyset\) and thus there exists \(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\).

This proves that there exists an \(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\).

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