13. For what value of b would the following system of equations have an infinite number of solutions? 9x + 12y = 21
6x + 8y = 7b Please explain and show steps :)

Answer:

answer is a,b

Step-by-step explanation:

I have answered ur question

The mid-point of AC is (a, b). So, option D is correct. In a rectangle, the two diagonals bisect each other at their mid-point.

The given rectangle is ABCD

Its vertices have coordinates as

A - (0, 0)

B - (0, 2a)

C - (2a, 2b)

D - (2a, 0)

The diagonals are AC and BD.

Finding their mid-points:

Mid-point of the diagonal AC = ((0 + 2a)/2 , (0 + 2b)/2)

⇒ (2a/2, 2b/2)

⇒ (a, b) ... (1)

Mid-point of the diagonal BD = ((0 + 2a)/2, (2a+0)/2)

⇒ (2a/2, 2b/2)

⇒ (a, b)  ...(2)

From (1) and (2), the midpoints of both the diagonals are equal. So, the diagonals of the rectangle ABCD bisect each other.

Hence, proved.

Therefore, the mid-point of the diagonal AC is (a, b).

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[tex]\bf y=\stackrel{\downarrow }{\cfrac{4}{3}}x+1\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{\cfrac{4}{3}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{3}{4}}\qquad \stackrel{negative~reciprocal}{-\cfrac{3}{4}}}[/tex]

so then, we know this line has a slope of -3/4 and runs through (4 , 1)

[tex]\bf (\stackrel{x_1}{4}~,~\stackrel{y_1}{1})~\hspace{10em} slope = m\implies -\cfrac{3}{4} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-1=-\cfrac{3}{4}(x-4)\implies y-1=-\cfrac{3}{4}x+3[/tex]

[tex]\bf y=-\cfrac{3}{4}x\stackrel{\downarrow }{+4}\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}~\hfill \stackrel{\textit{y-intercept}}{(0,4)}[/tex]

Answer:

1 1/5 mi

Step-by-step explanation:

We need to add the three legs together

3/10 + 1/2 + 2/5

The common denominator is 10

3/10  =3/10

1/2 *5/5 = 5/10

2/5 *2/2 =4/10

3/10 + 5/10+4/10 = 12/10

10/10 = 10/10+2/10 = 1+2/10  = 1 1/5 mi

Answer:

Length of race = 1.2 miles

Step-by-step explanation:

Length of race is given by the perimeter of triangle.

Refer the given figure.

The first leg is 3/10 mi, the second leg is 1/2 mi, and the third leg is 2/5 mi.

[tex]\texttt{Perimeter =}\frac{3}{10}+\frac{1}{2}+\frac{2}{5}=\frac{3}{10}+\frac{5}{10}+\frac{4}{10}=\frac{3+5+4}{10}\\\\\texttt{Perimeter =}\frac{12}{10}=1.2 miles[/tex]

Length of race = 1.2 miles

Answer:

the school day ends at 1:45

Step-by-step explanation:

7:15   40 x 3 = 120

+2 hours =

9:15   35 x 4 = 140

+2 hours & 20 mins =

11:35   50 x 2 = 100

+1 hour & 40 mins

1:15   20 + 10 = 30

1:45

you're welcome((:

Answer:

(-1,-1)

The square needs to be all side with the same value when you graph the vertices on x-y plot, you obtain (x is the dot that correspond a vertice and y the forth vertice )

                            x       2 |                        x

                                      1 |

                                       |                                  

      -2                  -1         |          1              2

                            y        -1|                       x

                                     -2 |

If you draw a line between vertices the value it will be 2 +  (-1) so the forth vertice has to be (-1,-1)

Answer:

=11.155 in³

Step-by-step explanation:

Given data:

Radius= 1.75 inches

Height = 3.5 inches

Diameter= 0.5 inches

To find the volume of a cone we will apply the formula:

Volume of a cone = 1/3 πr²h

Substitute the values:

V= 1/3* 3.14 *(1.75)² * 3.5

V=1/3 *3.14*(3.0625)* 3.5

V= 33.66/3

V=11.22 in³

Now find the volume of the bubble gum:

Volume of bubble gum = 4/3 π*r³

Substitute the values:

V= 4/3*3.14*(0.5/2)³

V=4/3*3.14(0.25)³

V=4/3*3.14(0.015625)

V=0.19625/3

V=0.0654 in³

Now subtract the volume of bubble gum ball from the volume of a cone

=11.22 - 0.0654

=11.155 in³

Thus the closest approximation of the volume of the cone that can be filled with flavored ice is 11.155 in³....

Answer:

11.15 in³

Step-by-step explanation:

Answer:

D.

Step-by-step explanation:

First step: Identify the point for L.

L is at (-2,-5).

We are plugging this (x,y) into (x+8,y-3) to see where it takes us.

(-2+8,-5-3)=(6,-8).

The solution is D.

Answer: OPTION D

Step-by-step explanation:

You can observe in the figure that the coordinates of the point L are:

[tex]L(-2,-5)[/tex]

You know that the trapezoid rule applied for the translation of the trapezoid JKLM is:

[tex](x, y)[/tex]→[tex](x + 8, y - 3)[/tex]

Therefore, in order to find the coordinates of the point L', you need to add 8 to the x-coordinate of the point L and subtract 3 from the y-coordinate of the point L.

Then:

[tex]L'(-2 + 8, -5 - 3)\\\\L'(6,-8)[/tex]

Answer:

The equation is y = 2x. The slope is two and the line is a direct variation

Answer:

[tex](x+5)^2+(y+2)^2=625[/tex]

Step-by-step explanation:

The center-radius form a circle is [tex](x-h)^2+(y-k)^2=r^2[/tex]. Okay, it is called standard form.  But I like to call it center-radius form because it tells us the center (h,k) and the radius,r.

So we are given the following information r=25 and center=(h,k)=(-5,-2).

So we just plug this in like so:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

[tex](x--5)^2+(y--2)^2=25^2[/tex]

[tex](x+5)^2+(y+2)^2=625[/tex]

Answer:

The increase was exponential

After three months his expense was: $133.1

Step-by-step explanation:

The increase was exponential, because if we call x the initial price, then after the first month the new price p is:

[tex]p = x (1 + \frac{10\%}{100\%})\\\\p=x(1+0.1)[/tex]

After the second month, the new price is 10% of the price of the previous month, that is:

[tex]p = [x (1 + 0.1)](1 + 0.1)\\\\p = x (1 + 0.1) ^ 2[/tex]

After month n, the price is:

[tex]p = x (1 + 0.1) ^ n[/tex]

Note that the equation has the form of an exponential growth function, where x is the initial price and n is the number of months elapsed.

In this case [tex]x = 100[/tex] and [tex]n = 3[/tex]. So:

[tex]p = 100 (1 + 0.1) ^ 3\\\\p=\$133.1[/tex]

[tex]x^2 – 5x + 6 = 0\\x^2-2x-3x+6=0\\x(x-2)-3(x-2)=0\\(x-3)(x-2)=0\\x=3 \vee x=2\\A=\{-2,3\}\\\\x^2=9\\x=-3 \vee x=3\\B=\{-3,3\}\\\\A\cap B=\{3\}\\A\cup B=\{-3,-2,3\}[/tex]

Answer:

rs 10.5 .

Step-by-step explanation:

Simple Interest  = PRT/100     where P = sum invested, R = the rate per annum, T = the time in years.

14  weeks = 14/52 years so it is:

650* 6 * (14/52) / 100

= rs 10.5.

Answer:

10.50

Step-by-step explanation:

The simple interest (I) is calculated as

I = [tex]\frac{PRT}{100}[/tex]

where P is the principal ( amount deposited), R is the rate of interest and T the time in years.

note that 14 weeks = [tex]\frac{14}{52}[/tex] of a year, hence

I = [tex]\frac{650(6)}{100}[/tex] × [tex]\frac{14}{52}[/tex]

  = [tex]\frac{650(6)(14)}{100(52)}[/tex]

  = [tex]\frac{54600}{5200}[/tex] = 10.50

Answer:

16, 32, 48, 64

Step-by-step explanation:

Factors of a number are the numbers another number can be divided by. Multiples are numbers that can be divided by a number. Therefore, all the numbers in the list consist of 16 X 1, 16 X 2, 16 X 3, and so on.

The multiple of 16 is,

The multiples of 16 are 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, etc.

Multiple numbers:

It is a sequence where the difference between each next number and the preceding number, i.e. two consecutive multiples or products, is equal to 16. In simple words, multiples of a number are the products obtained by multiplying the given number by other natural numbers.

So, the required lists are,

16,32,48 and 64.

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Final answer:

When a 5 feet long rope is divided into 9 equal pieces, each piece is approximately 0.56 feet long after performing the division and rounding to the nearest hundredth.

Explanation:

To find the length of each piece of rope when a 5 feet long rope is divided into 9 equal pieces, we need to divide the total length of the rope by the number of pieces. This is a division problem in arithmetic.

Division: 5 feet ÷ 9 pieces = 0.5555... feet per piece.
Since we typically want to represent a length in a more practical way, we can round this number. Rounding to the nearest hundredth, we get approximately 0.56 feet for each piece of rope.

Final answer:

To determine the length of each rope piece when a 5-foot long rope is divided into 9 equal parts, divide the total length by the number of parts, resulting in pieces that are approximately 0.5556 feet long.

Explanation:

To find the length of each piece of rope when a 5-foot long rope is divided into 9 equal pieces, we need to divide the total length of the rope by the number of pieces. Therefore, we use the division:

Length of each piece = Total length of rope ÷ Number of pieces

Length of each piece = 5 feet ÷ 9

Upon doing the division, we find that the length of each piece is approximately 0.5556 feet.

Answer:

Shift +2 units to the right and +2 units up

Step-by-step explanation:

y = (x+3)² + 4 has a horizontal shift of -3 and a vertical shift of +4.

y = (x+1)² + 6 has a horizontal shift of -1 and a vertical shift of +6.

So do translate the first equation to the second, shift +2 units to the right and +2 units up.

Statement C can be combined with its converse to form a true biconditional because both statements are true.

In order for a statement and its converse to form a true biconditional, both statements must be true. Let's analyze the given statements:

A) If the measure of an angle is 30, then it is an acute angle.

B) If two lines intersect, then the two lines are not skew.

C) If the ray is the perpendicular bisector of the segment, then it divides the segment into two congruent segments.

D) If an angle is a straight angle, then its sides are opposite rays.

Out of these options, statement C can be combined with its converse to form a true biconditional because both statements are true:

If the ray is the perpendicular bisector of the segment, then it divides the segment into two congruent segments.

If the segment is divided into two congruent segments, then the ray is the perpendicular bisector of the segment.

Answer:

The equations represent circles that result in the same graph.

Step-by-step explanation:

we have

[tex]-10x^{2}-10y^{2}=-300[/tex]

Divide by -10 both sides

[tex]x^{2}+y^{2}=30[/tex] -----> equation A

This is the equation of a circle centered at origin with radius [tex]r=\sqrt{30} \ units[/tex]

and

[tex]5x^{2}+5y^{2}=150[/tex]

Divide by 5 both sides

[tex]x^{2}+y^{2}=30[/tex] -----> equation B

This is the equation of a circle centered at origin with radius [tex]r=\sqrt{30} \ units[/tex]

equation A and equation B are equal

therefore

The system has infinite solutions, because the equations represent circles that result in the same graph.

Answer:

Option B. The value of x is 20

Step-by-step explanation:

we know that

The intersecting chords theorem  states that the products of the lengths of the line segments on each chord are equal.

so

In this problem

[tex](x)(x-11)=(x-8)(x-5)\\x^{2}-11x=x^{2}-5x-8x+40\\-11x=-13x+40\\2x=40\\x=20[/tex]

The solutions to the linear equations in variable x and unknown constant [tex]\alpha[/tex] are [tex]\frac{-\alpha}{6}[/tex], [tex]\frac{3}{\alpha}[/tex] and [tex]\frac{-6}{\alpha}[/tex].

A linear equation is an algebraic equation of degree one. In general, the variable or the variables(in the case of a linear equation in two variables) the variables are x and y.

We are given linear equations in variable x with an unknown constant [tex]\alpha[/tex] and we have to solve for x.

[tex]4 = \frac{6}\alpha}x + 5\\\\\frac{6}{\alpha}x = - 1\\\\x = \frac{-\alpha}{6}[/tex].

[tex]7 + 2\alpha{x} = 13.\\\\2\alpha{x} = 6.\\\\x = \frac{3}{\alpha}[/tex]

[tex]-\alpha{x} - 20 = -14.\\\\-\alpha{x} = 6.\\\\x = \frac{-6}{\alpha}[/tex]

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Answer:

Only the function represented by graph 2 has an inverse function

Step-by-step explanation:

* Lets explain the inverse of the function

- The Function is a relation between x-coordinates and the y-

 coordinates of the order pairs under the condition every

 x-coordinate has only one y-coordinate

- Ex: R = {(2 , 3) , (1 , 5) , (-2 , -3)} is a function because every x-

 coordinate has only one y-coordinate and R = {(2 , 3) , (-1 , 4) ,

 (2 , 5)} not a function because the x-coordinate 2 has two y-

 coordinates 3 and 5

- We use the vertical line to test the graph is function or not, if

 the vertical line intersects the graph in one point then the

 graph is function if intersects it in more than one point then the

 graph is not function

- We have two types of function one-to-one function and

 many-to-one function

# one-to-one function means every x-coordinate has only 1 y-coordinate

# many-to-one function means some x-coordinates have only 1

  y-coordinate

- We find the inverse function by switching x and y, then one-to-

 one function has inverse but many-to-one has not inverse

 because when we switched x and y it will be one-to-many

 means one x-coordinate has many y-coordinates and this is not

 a function

- We use the horizontal line to test the graph of the function has

 inverse or not, if the horizontal line intersects the graph in one

 point then the function of the graph has inverse if it intersects

 the graph in more than one point ,then the function of the  

 graph has no inverse

* Now lets test all the graphs by using the horizontal line

# graph 1

∵ The horizontal line cuts the graph in more than 1 point

∴ The function of graph 1 has no inverse

# graph 2

∵ The horizontal line cuts the graph in just 1 point

∴ The function of graph 2 has inverse

# graph 3

∵ The horizontal line cuts the graph in more than 1 point

∴ The function of graph 3 has no inverse

# graph 4

∵ The horizontal line cuts the graph in more than 1 point

∴ The function of graph 4 has no inverse

* Only the function represented by graph 2 has an inverse function

Answer:

No

Step-by-step explanation:

The short answer is no. b stands for some number. Once that number is fixed, it does not change. So the equation would look something like

y = 2x + 6  Now if you look at that again, that 6 is fixed.

If you want to put something in for x like x = 5 you get

y = 2(5) + 6

y = 10 + 6

y = 16                   b = 6 did not change.

What does y = 2x equal

y = 2*5

y = 10

10 does not equal 16.

The only exception to what I've written is b = 0. Then both equations mean the same thing. But that is an exception.

Answer:

18,5

Step-by-step explanation:

middle numbers --> (17+20)/2 = 18,5

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Answer: 18.5

Step-by-step explanation:

Step 1) To find the median in a set of data values you must first arrange them from least greatest to greatest. Luckily, it's already done for you!

Step 2)  Then, take the two middle numbers (17 and 20) and add them.

17+20 = 37

Step 3) Now, just divide by two

37/2 = 18.5

Let me know if you have any more questions!

Answer:

Ar = ¼√3 As

Step-by-step explanation:

Area of an equilateral triangle is:

Ar = ¼√3 s²

Area of a square is:

As = s²

Substituting:

Ar = ¼√3 As

Answer:

The answer should be the last one.

Step-by-step explanation:

Answer:

=2√2 the fourth choice.

Step-by-step explanation:

We can use the Pythagoras theorem to calculate the value of y.

a²+b²=c²

a=1

b=y

c=3

Therefore substituting for the values in the theorem above we get:

1²+y²=3²

Leave y on one side.

y²=3²-1²

y²=9-1

y²=8

y=±2√2 in surd form.

Since we expressing length, a scalar quantity, we take the modulus of our answer. Thus y=2√2

Step-by-step explanation:

if a^2 +b^2 <c^2,

then abc is not a right triangle since for a right triangle a^2+ b^2 = c^2

The following features of the triangle are found: A. a² + b² - c² = 2 · a · b · cos θ, B. cos θ < 0, C. The triangle is not a right triangle.

How to analyze the features of a triangle

In this question we must infer all features from a triangle such that a² + b² < c². Then, the triangle is not a right triangles since relationship between side lengths is different from the relationship described by Pythagorean theorem. Then, triangle is described by law of cosine:

a² + b² - c² = 2 · a · b · cos θ

If a² + b² < c², then a² + b² - c² < 0 and 2 · a · b · cos θ < 0. Thus, we get the following result: cos θ < 0.

Answer: 94.99 in^2

Step-by-step explanation: The equation for the area of a circle is A=πr^2. To solve this, we need to find the radius. The diameter is the whole circle, whole the radius is half. So divide the diameter by 2.

11/2 = 5.5

The radius is 5.5 inches. Plug the radius into the equation.

A=π5.5^2

Square the 5.5 first. You will get 30.25.

A=π30.25

Plug in 3.14 for pi.

A=3.14 x 30.25

Multiply.

A=94.985

Round to the nearest hundredth.

A=94.99

The area of the circle is 94.99 in^2.

94.99 in^2

hope this helps somebody

Answer:

4/3 hours  or 1 hour 20 minutes.

Step-by-step explanation:

1/4 page takes 1/3 hour to write.

By proportion 1 page will take 1/3  / 1/4

=  1/3 * 4

= 4/3 hours.

Answer:

[tex]1\frac{1}{3}\text{ hours}[/tex]

Step-by-step explanation:

Given,

Time taken to write 1/4 of a page = [tex]\frac{1}{3}[/tex] hour,

i.e. the ratio of time taken and number of pages wrote = [tex]\frac{1/3}{1/4}=\frac{4}{3}[/tex]

Let x be the time taken to write a full page,

So, the ratio of time taken( in hours ) and page wrote = [tex]\frac{x}{1}[/tex]

[tex]\implies \frac{x}{1}=\frac{4}{3}[/tex]

[tex]x=1\frac{1}{3}[/tex]

Hence, the time taken to write 1 page is [tex]1\frac{1}{3}[/tex] hours.

Answer:

So the y-intercept is 10.

Step-by-step explanation:

So your table doesn't out right say the y-intercept.  If it did it would be (x=0,y=something).

So let's see if this is linear. I'm going to see if we have the same rise/run ratio per pair of points as shown in the attachment:

These ratios are all the same -6/3 = -2/1 = -4/2 = -6/3 . These are all equal to -2.

So this is a line.

Linear equations in the form y=mx+b is called slope-intercept form where m is the slope and b is the y-intercept.

We just found m to be -2.

So our equation is now in the form y=-2x+b.

We can find b, the y-intercept, by using a point on this line.  I like (4,2) from the table.

(x,y)=(4,2) with y=-2x+b will give us the information we need to find b.

2=-2(4)+b

2=-8+b

2+8=b

10=b

b=10

So the y-intercept is 10.

54 people owned either a mouse or a parrot

A Venn diagram is an illustration that uses circles to show the relationships among things or finite groups of things.

Number of people that owned only mice = 36 - 14 = 22

Number of people that owned only parrot = 32 - 14 = 18

Number of people that owned either a mouse or a parrot = 22 + 18 + 14 = 54

54 people owned either a mouse or a parrot

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By using the principle of inclusion-exclusion to find the total number of pet store customers who owned either a mouse or a parrot, we calculated that 54 customers owned at least one of the pets, taking into account that 14 customers owned both.

To determine the total number of store customers who owned either a mouse or a parrot, we need to apply the principle of inclusion-exclusion. This principle allows us to find the union of two sets without counting elements (customers) that are common to both sets more than once. The formula is ∑(A ∪ B) = ∑(A) + ∑(B) - ∑(A ∩ B), where ∑ denotes the number of elements in a set, A represents the group of mouse owners, B represents the group of parrot owners, and A ∩ B represents the group that owns both mice and parrots.

Applying this to our situation:

Using the principle of inclusion-exclusion:

∑(A ∪ B) = 36 + 32 - 14 = 54

This means that 54 customers owned either a mouse or a parrot or both.

To find the coordinates of point C, divide the x- and y-coordinates of AB in the ratio 2:1.

To find the coordinates of point C, we can use the concept of dividing a line segment in a given ratio. Given that AC:CB is 2:1, we can divide the x- and y-coordinates of the line segment AB in the same ratio.

The x-coordinate of point C is calculated by dividing the difference between the x-coordinates of points A and B by the sum of the ratio (2+1).

The y-coordinate of point C is calculated by dividing the difference between the y-coordinates of points A and B by the sum of the ratio (2+1).

Therefore, the coordinates of point C are (-2, 3).

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The correct option is d. [tex]\((4, 0)\)[/tex]. The coordinates of point [tex]\(C\)[/tex] are [tex]\((4, 0)\)[/tex].

To find the coordinates of point [tex]\(C\)[/tex] that partitions the segment [tex]\(AB\)[/tex] in the ratio [tex]\(AC:CB = 2:1\)[/tex], we use the section formula. Given points [tex]\(A(x_1, y_1)\)[/tex] and [tex]\(B(x_2, y_2)\)[/tex] with a ratio [tex]\(m:n\)[/tex], the coordinates [tex]\((x, y)\)[/tex] of the point dividing the segment in the ratio [tex]\(m:n\)[/tex] are given by:

[tex]\[x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}\][/tex]

For this problem:

- [tex]\(A(-8, 4)\)[/tex]

- [tex]\(B(10, -2)\)[/tex]

- Ratio [tex]\(m:n = 2:1\)[/tex]

Plugging in the values:

[tex]\[x = \frac{2 \cdot 10 + 1 \cdot (-8)}{2 + 1} = \frac{20 - 8}{3} = \frac{12}{3} = 4\][/tex]

[tex]\[y = \frac{2 \cdot (-2) + 1 \cdot 4}{2 + 1} = \frac{-4 + 4}{3} = \frac{0}{3} = 0\][/tex]

Thus, the coordinates of point [tex]\(C\)[/tex] are [tex]\((4, 0)\)[/tex].

The complete question is:

What are the coordinates of point C on the directed segment from A(−8,4) to B(10,−2) that partitions the segment such that AC:CB is 2:1 ?

A. (1,1)

B. (−2,2)

C. (2,−2)

D. (4,0)