The cake store is having a 30 % off sale on all of its cakes. if the cake you want regularly costs $10 how much would you save with the discount?

Answer:

Option (2) is correct.

The given system of equation is Consistent and independent.

Step-by-step explanation:

Now consider the given graph,

Since, they intersect at a unique point. Hence, they are consistent.

Also the two given equations are independent of each other .Thus, independent also.

Thus, the given system of equation is Consistent and independent.

Thus, option (2) is correct.

 3.1     Pull out like factors :

   5x2 - 25x - 70  =   5 • (x2 - 5x - 14) 

 3.2     Factoring  x2 - 5x - 14 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -5x  its coefficient is  -5 .
The last term, "the constant", is  -14 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -14 = -14 

Step-2 : Find two factors of  -14  whose sum equals the coefficient of the middle term, which is   -5 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -7  and  2 
                     x2 - 7x + 2x - 14

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-7)
              Add up the last 2 terms, pulling out common factors :
                    2 • (x-7)
Step-5 : Add up the four terms of step 4 :
                    (x+2)  •  (x-7)
             Which is the desired factorization

 4.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

 4.2      Solve :    5   =  0

This equation has no solution.
A a non-zero constant never equals zero.

 4.3      Solve  :    x+2 = 0 

 Subtract  2  from both sides of the equation : 
                      x = -2 

 4.4      Solve  :    x-7 = 0 

 Add  7  to both sides of the equation : 
                      x = 7 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

 5.1      Find the Vertex of   y = x2-5x-14

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 1 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   2.5000  

 Plugging into the parabola formula   2.5000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 2.50 * 2.50 - 5.0 * 2.50 - 14.0 
or   y = -20.250

Root plot for :  y = x2-5x-14
Axis of Symmetry (dashed)  {x}={ 2.50} 
Vertex at  {x,y} = { 2.50,-20.25}  
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-2.00, 0.00} 
Root 2 at  {x,y} = { 7.00, 0.00} 

 5.2     Solving   x2-5x-14 = 0 by Completing The Square .

 Add  14  to both side of the equation : 
   x2-5x = 14

Now the clever bit: Take the coefficient of  x , which is  5 , divide by two, giving  5/2 , and finally square it giving  25/4 

Add  25/4  to both sides of the equation :
  On the right hand side we have :
   14  +  25/4    or,  (14/1)+(25/4) 
  The common denominator of the two fractions is  4   Adding  (56/4)+(25/4)  gives  81/4 
  So adding to both sides we finally get :
   x2-5x+(25/4) = 81/4

Adding  25/4  has completed the left hand side into a perfect square :
   x2-5x+(25/4)  =
   (x-(5/2)) • (x-(5/2))  =
  (x-(5/2))2 
Things which are equal to the same thing are also equal to one another. Since
   x2-5x+(25/4) = 81/4 and
   x2-5x+(25/4) = (x-(5/2))2 
then, according to the law of transitivity,
   (x-(5/2))2 = 81/4

We'll refer to this Equation as  Eq. #5.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(5/2))2   is
   (x-(5/2))2/2 =
  (x-(5/2))1 =
   x-(5/2)

Now, applying the Square Root Principle to  Eq. #5.2.1  we get:
   x-(5/2) = √ 81/4 

Add  5/2  to both sides to obtain:
   x = 5/2 + √ 81/4 

Since a square root has two values, one positive and the other negative
   x2 - 5x - 14 = 0
   has two solutions:
  x = 5/2 + √ 81/4 
   or
  x = 5/2 - √ 81/4 

Note that  √ 81/4 can be written as
  √ 81  / √ 4   which is 9 / 2 

 5.3     Solving    x2-5x-14 = 0 by the Quadratic Formula .

 

Answer:

the value of p=-1

Step-by-step explanation:

General equation of a parabola is

[tex]y^2= 4p(x)[/tex] where vertex is at (0,0)

now we compare the given equation with general form

[tex]y^2=-4x[/tex]

We have -4 in the place of 4p

[tex]4p=-4[/tex]

Divide both sides by 4

so p = -1

the value of p = -1

Answer:

Option D -[tex]x+y\geq 120[/tex] , [tex]1.35x + 1.2y \leq 175[/tex]

Step-by-step explanation:

Given : You want at least three bratwursts or hamburgers for each of your 40 guests. Bratwursts cost $1.35 each and hamburgers $1.2 each. Your budget is $175.

To find : Which system of inequalities represents this situation?

Solution :

Let x be the number of bratwursts and y the number of hamburgers.

You want at least three bratwursts or hamburgers for each of your 40 guests.

i.e, you want at least [tex]3\times(40)=120[/tex]  bratwursts or hamburgers.  

We can write the equation as,

Number of bratwursts + number of hamburgers at least 120

[tex]x+y\geq 120[/tex]

Bratwursts cost $1.35 each and hamburgers cost $1.2 each. Your budget is $175.

We can write equations as ,

Cost of bratwursts + cost of hamburgers at most 175

[tex]1.35x + 1.2y \leq 175[/tex]

Therefore, The system of linear equation form is [tex]x+y\geq 120[/tex], [tex]1.35x + 1.2y \leq 175[/tex]

Hence, Option D is correct.

The perimeter of the rectangle is 75 cm, then the length of the rectangle will be equal to 25.25 cm.

A shape's perimeter is the whole distance encircling it. It is the perimeter or length of the contour of any geometric two-dimensional shape. The circumference of several figures could be comparable based on the measurements.

As per the information provided by the question,

The perimeter of the rectangle = 75 cm

Let the width of the rectangle is w.

Then, length (l) is 13 more than,

l = w + 13   (i)

As per the formula of the perimeter of a rectangle,

[tex]P=2(l+w)[/tex]

75 = 2(w + 13 + w)

75 = 4w + 26

4w = 49

w = 12.25 cm

Substitute w = 12.25 cm in equation (i)

l = 12.25 + 13

l = 25.25 cm.

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Let

b------> the length side of a square

we know that

the area of a square is equal to

[tex]A=b^{2}[/tex]

Find the length side of the diagonal applying the Pythagoras Theorem

[tex]d^{2}=b^{2}+b^{2}[/tex]

[tex]d^{2}=2b^{2}[/tex]

[tex]d=b\sqrt{2}\ units[/tex]

Remember that

[tex]d=x\ units[/tex] -----> given problem

substitute

[tex]d=b\sqrt{2}\ units[/tex]

[tex]x=b\sqrt{2}\ units[/tex]

Solve for b

[tex]b=\frac{x\sqrt{2}}{2}\ units[/tex]

Substitute in the formula of area

[tex]A=(\frac{x\sqrt{2}}{2})^{2}[/tex]

[tex]A=\frac{x^{2}}{2}\ units^{2}[/tex]

therefore

the answer is

[tex]\frac{x^{2}}{2}\ units^{2}[/tex]

The area of the square in terms of x is [tex]\( \frac{x^2}{2} \),[/tex] obtained by using the Pythagorean theorem and the formula for the area of a square.

To find the area of the square in terms of [tex]\(x\),[/tex]  we need to first understand the relationship between the diagonal and the side length of a square.

In a square, each diagonal divides the square into two congruent right triangles. Using the Pythagorean theorem, we can find the relationship between the diagonal [tex](\(x\))[/tex] and the side length of the square [tex](\(s\)).[/tex]

The Pythagorean theorem states:

[tex]\[ \text{Hypotenuse}^2 = \text{Adjacent side}^2 + \text{Opposite side}^2 \][/tex]

For a square:

[tex]\[ x^2 = s^2 + s^2 \][/tex]

[tex]\[ x^2 = 2s^2 \][/tex]

Now, we can solve for [tex]\(s\)[/tex] in terms of [tex]\(x\):[/tex]

[tex]\[ s^2 = \frac{x^2}{2} \][/tex]

[tex]\[ s = \sqrt{\frac{x^2}{2}} \][/tex]

[tex]\[ s = \frac{x}{\sqrt{2}} \][/tex]

The area [tex](\(A\))[/tex] of a square is given by:

[tex]\[ A = s^2 \][/tex]

Substituting the expression for s in terms of x :

[tex]\[ A = \left(\frac{x}{\sqrt{2}}\right)^2 \][/tex]

[tex]\[ A = \frac{x^2}{2} \][/tex]

Therefore, the area of the square in terms of x is [tex]\( \frac{x^2}{2} \).[/tex]

The question probable may be:

The diagonal of a square is x units. What is the area of the square in terms of x?

Answer:idek tbh

Step-by-step explanation:

Answer:

July has 31 days.

Step-by-step explanation:

The diagram shows that april has 30 days along with June, while May and July have 31 days.

The temperature decreased at a constant rate of 5 degrees Fahrenheit per hour.

The question is asking for the rate at which the temperature is decreasing per hour. The temperature started at 0oF and decreased to -25oF. That's a total change in temperature of 25oF (since we're counting how much it dropped below 0o). This change happened over the course of 5 hours. Therefore, to find the rate of change, we divide the total change (25 degrees) by the total time (5 hours). This gives us 5oF per hour. So, the temperature decreased at a rate of 5oF per hour.

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Answer: The scale factor of the dilation is 0.5.

Explanation:

It is given that ABCD is polygon which is reflected and dilated , then we get PQRS.

The vertices of ABCD are (2, 2), (6, 8), (12, 8), and (16, 2) respectively. The vertices of image  PQRS are (11, 15),(9, 12), (6, 12), and (4, 15) respectively.

The reflection affects the coordinates but does not affect the length of the sides.

But when we talk about dilation it affects the length of sides according to the scale factor or a constant factor.

If a line segment AB is dilated by scale factor k then length of A'B' is k times length of AB.

[tex]|A'B'|=k\times |AB|[/tex]   .... (1)

So we have to find any side length of preimage and the side length of that side in image. it means we have to find AB and PQ.

Use distance formula to find the side length.

[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

[tex]AB=\sqrt{(8-2)^2+(6-2)^2}=\sqrt{36+16}=2\sqrt{13}[/tex]

[tex]PQ=\sqrt{(9-11)^2+(12-15)^2}=\sqrt{4+9}=\sqrt{13}[/tex]

It is noticed that the size of side in preimage is [tex]2\sqrt{13}[/tex] and the size of same side in image is [tex]\sqrt{13}[/tex].

Using equation (1), we get

[tex]\sqrt{13}=k \times 2\sqrt{13}[/tex]

[tex]k=\frac{1}{2}[/tex]

[tex]k=0.5[/tex]

Hence, the value of scale factor is 0.5.

The number of permutations of ABCDEFG containing BCD, CFGA, and BA & GF are 120, 24, and 120, respectively.

In permutations, the order of the items is important. For all these problems, we're asked how many permutations of the letters ABCDEFG contain a specific string. To solve this, we can initially treat the required string as a single item.

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The word product is the answer to a multiplication problem.

7 x 2/3

7/1 x 2/3

(7 x 2)/(1 x 3)

Answer: 14/3

Answer:

1.50 pounds

Step-by-step explanation:

Given : A retailer charges a flat handling fee of $5.00, plus $0.75 per quarter pound, to ship an item.

Bailey pays $9.50 to have an item shipped from the retailer.

To Find: What is the weight of the item?

Solution:

He paid total amount = $9.50

A flat handling fee = $5.00

Amount he paid without handling fee = $9.50-$5.00

                                                              =$4.50

Now we are supposed to find what is the weight of an item that costs $4.50

So, we are given that it costs $0.75 per quarter pound, to ship an item.

So, cost of 1 pound = [tex]\frac{0.75}{\frac{1}{4}}=3[/tex]

So, it cost $3 for 1 pound

So, weight of an item that costs $1 = [tex]\frac{1}{3}[/tex]

So, weight of an item that costs $4.50=[tex]\frac{1}{3}\times 4.50[/tex]

                                                               =1.50 pounds

Hence the weight of the item is 1.50 pounds.                

Answer:

1.50 pounds

Step-by-step explanation:

el de arriba lo dijo

Kate uses 1 cup of lemon juice and 2 cups of sugar to make lemonade. She requires 5 cups of lemon juice if she uses 5 cups of sugar to produce a larger batch of lemonade.

It is defined as the comparison of two quantities to ascertain how many times one yields the other. The ratio can be written as a fraction or as a sign: between two numbers.

It is given that, Kate uses 2 cups of sugar and 1 cup of lemon juice to make lemonade. If she uses 10 cups of sugar to create more lemonade.

The ratio of lemon juice to sugar is 1:2.

For 10 cups of sugar, the number of cups of lemon juice needed as,

We have to maintain the ratio as 1:2.So for 10 cups of sugar, the ratio is maintained as,

=5:10

=1:2

Thus, she needs 5 cups of juice to make lemonade.

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Answer:

Step-by-step explanation:

alright lets get started.

the linear combination is given by :

[tex]4(-2,3)-2(6,-7)[/tex]

foiling 4 and 2 into the parenthesis

[tex](4*-2, 4*3)+(-2*6,-2*-7)[/tex]

[tex](-8, 12)+(-12,14)[/tex]

Combinig x and y terms together

[tex](-8-12, 12+14)[/tex]

[tex](-20,26)[/tex]    :     answer

Hope it will help. :)

                    5*x^2-23*x-(-26)=0 

1. x = 2 

Answer:

Option B: 42 feet is the answer.

Step-by-step explanation:

The length of the shadow of the building is = 50 feet

Let the height of the building be = L

Height of Alex = 5 feet

Length of Alex's shadow = 6 feet

Now we will equate the ratios.

[tex]\frac{5}{6}= \frac{L}{50}[/tex]

[tex]6L=250[/tex]

[tex]L=250/6[/tex]

L = 41.67 ≈ 42 feet

So, the height of the building is approx 42 feet.