Find the slope of the line through the points (13,14) and (0,−1)

See below for the proof of the equation [tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex]

The equation is given as:

[tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex]

Take the LCM

[tex]\frac{(ax + by)(x - y) -(ax - by)(x + y)}{(x + y)(x - y)}= 2(a - b)[/tex]

Expand

[tex]\frac{ax^2 - axy + bxy - by^2 -ax^2 - axy + bxy + by^2}{(x + y)(x - y)}= 2(a - b)[/tex]

Evaluate the like terms

[tex]\frac{-2axy + 2bxy }{(x + y)(x - y)}= 2(a - b)[/tex]

Rewrite as:

[tex]\frac{-2axy + 2bxy }{x^2 - y^2}= 2(a - b)[/tex]

Factorize the numerator

[tex]\frac{2(a - b)(x^2 - y^2)}{x^2 - y^2}= 2(a - b)[/tex]

Divide

2(a - b)= 2(a - b)

Both sides are equal

Hence, the equation [tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex] has been proved

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The graph is in a reciprocal relationship with Option C

A function is a mathematical statement used to relate a dependent and n independent variable.

The function is represented by the graph

The nature of the graph is exponential

and the equation is determined by the points on the graph

( 1,1) ( 2,4) '

The equation is

y = ( 1/4) 4ˣ

The inverse of the function will be found by replacing the y variable with x and vice versa and then solving to bring the function into the form of x.

x = ( 1/4) [tex]\rm 4^y[/tex]

Plotting this gives the answer as Option C

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Answer:

A.28 minutes

B.40 minutes

C.56 minutes

D.70 minutes

Step-by-step explanation:

Answer:

False

Step-by-step explanation:

I'm not a 100 percent though

Answer:

-14 or (6,20)

This graph represents Option D = [tex]-3(x+2)^{2} + 4[/tex]

As the given figure is a parabola, it will have quadratic equation because the general equation of parabola corresponds  to [tex]x^{2} = 4ay[/tex] which has highest power of the variable equals to 2, thus its a quadratic equation.

The maximum value of a quadratic equation [tex]ax^{2} + b x +c[/tex]  is at points

[tex](-b/2a, c-\frac{b^{2} }{4a} )[/tex]

By looking at the graph we can observe the point at which the graph has achieved maxima is (-2,4).

∴ ([tex]-b/2a = -2[/tex] , [tex]c-\frac{b^{2} }{4a} = 4[/tex]

The coefficient of [tex]x^2[/tex] should be negative the graph opens downwards.

∴ Option B and C are eliminated.

On putting x = -2 only option D yields 4.

Thus this graph represents Option D = [tex]-3(x+2)^{2} + 4[/tex]

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What you can do that can be set up as a two- step equation is to follow the form which is  [tex]ax + b = c[/tex] and should be noted that  a, b, c are real numbers.

Two- step equation can be solved by following these

Two step equations serves as  algebraic equations which can be solved in exactly two steps to get the  final value of the variable .

Example of this is [tex]2x + 3 = 7.[/tex]

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wth is that wdym ???

Question 4

1) [tex]\overline{BD}[/tex] bisects [tex]\angle ABC[/tex], [tex]\overline{EF} \perp \overline{AB}[/tex], and [tex]\overline{EG} \perp \overline{BC}[/tex] (given)

2) [tex]\angle FBE \cong \angle GBE[/tex] (an angle bisector splits an angle into two congruent parts)

3) [tex]\angle BFE[/tex] and [tex]\angle BGE[/tex] are right angles (perpendicular lines form right angles)

4) [tex]\triangle BFE[/tex] and [tex]\triangle BGE[/tex] are right triangles (a triangle with a right angle is a right triangle)

5) [tex]\overline{BE} \cong \overline{BE}[/tex] (reflexive property)

6) [tex]\triangle BFE \cong \triangle BGE[/tex] (HA)

Question 5

1) [tex]\angle AXO[/tex] and [tex]\angle BYO[/tex] are right angles, [tex]\angle A \cong \angle B[/tex], [tex]O[/tex] is the midpoint of [tex]\overline{AB}[/tex] (given)

2) [tex]\triangle AXO[/tex] and [tex]\triangle BYO[/tex] are right triangles (a triangle with a right angle is a right triangle)

3) [tex]\overline{AO} \cong \overline{OB}[/tex] (a midpoint splits a segment into two congruent parts)

4) [tex]\triangle AXO \cong \triangle BYO[/tex] (HA)

5) [tex]\overline{OX} \cong \overline{OY}[/tex] (CPCTC)

Question 6

1) [tex]\angle B[/tex] and [tex]\angle D[/tex] are right angles, [tex]\overline{AC}[/tex] bisects [tex]\angle BAD[/tex] (given)

2) [tex]\overline{AC} \cong \overline{AC}[/tex] (reflexive property)

3) [tex]\angle BAC \cong \angle CAD[/tex] (an angle bisector splits an angle into two congruent parts)

4) [tex]\triangle BAC[/tex] and [tex]\triangle CAD[/tex] are right triangles (a triangle with a right angle is a right triangle)

5) [tex]\triangle BAC \cong \triangle DCA[/tex] (HA)

6) [tex]\angle BCA \cong \angle DCA[/tex] (CPCTC)

7) [tex]\overline{CA}[/tex] bisects [tex]\angle ACD[/tex] (if a segment splits an angle into two congruent parts, it is an angle bisector)

Question 7

1) [tex]\angle B[/tex] and [tex]\angle C[/tex] are right angles, [tex]\angle 4 \cong \angle 1[/tex] (given)

2) [tex]\triangle BAD[/tex] and [tex]\triangle CAD[/tex] are right triangles (definition of a right triangle)

3) [tex]\angle 1 \cong \angle 3[/tex] (vertical angles are congruent)

4) [tex]\angle 4 \cong \angle 3[/tex] (transitive property of congruence)

5) [tex]\overline{AD} \cong \overline{AD}[/tex] (reflexive property)

6) [tex]\therefore \triangle BAD \cong \triangle CAD[/tex] (HA theorem)

7) [tex]\angle BDA \cong \angle CDA[/tex] (CPCTC)

8) [tex]\therefore \vec{DA}[/tex] bisects [tex]\angle BDC[/tex] (definition of bisector of an angle)

Answer:

6

Step-by-step explanation:

9 x 2/3

9/1   x 2/3 =   18 / 3  = 6

Since it has a negative discriminant, the quadratic equation that does not have a real solution is given by:

−4x2 − 4x − 6 = 0.

A quadratic equation is modeled by:

[tex]y = ax^2 + bx + c[/tex]

The discriminant is:

[tex]\Delta = b^2 - 4ac[/tex]

The solutions are as follows:

We want [tex]\Delta < 0[/tex], hence, for equation -4x² - 4x - 6, we have that a = -4, b = -4, c = -6, hence:

[tex]\Delta = (-4)^2 - 4(-4)(-6) = 16 - 96 = -80[/tex]

Negative discriminant, hence it does not have a real solution.

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Answer:

The answer above is correct. The answer was confirmed by a teacher, and was marked as correct on the test

Step-by-step explanation:

[tex]f( - 1) = - 3[/tex]

[tex]f(x) = 0 \\ for \: x = 2 \: or \: x = - 2[/tex]

Question:

The perimeter of a square field is 336 yd. How long is each side?

Answer:

1,344 yd

Answer:

x = -6-2y

Step-by-step explanation:

The scale factor of the dilation shown in the image is: C. 4.

Scale factor = New dimension/corresponding old dimension.

From the image given, we have:

Scale factor = OA'/OA

OA' = 3 + 9 = 12 units

OA = 3 units

Scale factor of dilation = 12 units/3 units

Scale factor of dilation = 4

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C. 4

Next one is 6 if you have the option on edge

The number of 1 foot cubic blocks that can be stuffed in the pyramid are; 10,583 cubes

The equation for the surface area of a square pyramid is;

Surface area = A + 1/2·p·s

where;

A = area of base

p = perimeter

s = slant height

The volume of the pyramid = (1/3) × A × h

Where:

A = Area of the base = 20 × 20 = 400 ft²

h = The height of the pyramid = √((slant height)² - ((base side length)/2)²)

h = √(80² - (20/2)²) = 30√7

The volume of the pyramid = 1/3*400*30·√7 = 4000·√7 ft³ = 10,583.005 ft³

If the cubes are flexible, then there are approximately 10,583 cubes

For rigid cubes we have;

Given that the height of the pyramid = 30·√7

The slope of the pyramid = (30/10)√7 = 3·√7

A increase in height of 1 foot gives a reduction in width of 2/(3√7)

The bottom can hold 400 cubes, Then next layer can hold;

19 × 19 = 361

If we continue to iterate, we will get a total of approximately 9915 rigid cubes in the pyramid.

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The ways to collect data in a sample include:

It should be noted that a sample simply means a subset of a population that has the characteristics of the entire population.

In this case, the ways to collect data in a sample include the use of questionnaire, through observations, and through surveys.

l

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Answer:

10+1=11 but the 55 so it is now 11:55+15min so it makes it 12:10

Answer:

Hello! The answer to your question is 12:10.

Step-by-step explanation:

1 hour after 10:55 is equivalent to 11:55. Now, we have 15 minutes remaining. We can add in increments of 5 to get to 15 in terms of time:

11:55 + 5 minutes

= 12:00 + 5 minutes

= 12:05 + 5 minutes

= 12:10

5 + 5 + 5 = 15, so we reached 15.

The teacher collected the answer scripts at 12:10.

Answer:

[tex]a = (h)(2h - 5)[/tex]

Answer:

Area of the rectangle = 2h² - 5

Step-by-step explanation:

• base length l = 5 less than twice the height h

                        = 2h - 5

• height = h

Area of rectangle = base × height

                             ⇒ l × h

                             ⇒ (2h - 5)(h)

                              ⇒ 2h² - 5

The margin of error for a 95% confidence interval for the population proportion is 0.0815.

Given sample size=60 and confidence interval of 95%.

We have to calculate margin of error for a confidence interval of 95%.

Margin of error is the difference between actual values and calculated values. Margin of error is a part of z test.

We have been given sample size=60.

8 people have received help from their parents from the sample.

p=8/60=0.13

which is sample proportion.

z=1-0.13

=0.87

To calculate standard error=[tex]\sqrt{p*z/n}[/tex]

=[tex]\sqrt{0.13*0.8/60}[/tex]

=0.0416

at 95% confidence interval

z(α/2)=1.96

therefore margin of error=1.96*0.0416

=0.081536

=0.0815

Hence the margin of error for 95% confidence interval is 0.0815

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Answer:

[tex]94.6cm^{2}[/tex]

Step-by-step explanation:

First we will find the area of Square PQRS.

Area of Square PQRS = Length x Width = 21 x 21

= [tex]441cm^{2}[/tex]

Next we will found the Area of Semicircles PS and QR.

Note: Area of Semicircle PS = Area of Semicircle QR

Area of Semicircle = [tex]\frac{1}{2} \pi r^{2}[/tex]

Total Area of Semicircles PS and QR combined = [tex]2(\frac{1}{2} \pi r^{2} )\\=\pi r^{2}[/tex]

We know that the diameter of PS = QR = 21 cm (due to the length of the square)

Radius = Half of Diameter = 0.5 x 21cm = 10.5cm

Total Area of Semicircles PS and QR = [tex]\pi (10.5)^{2} \\=110.25\pi cm^{2}[/tex]

Finally,

Area of Shaded Region = Area of Rectangle PQRS - Total Area of Semicircles PS and QR

= [tex]441 - 110.25\pi\\= 94.6cm^{2} (1dp)[/tex]

In this case , you can choose the nearest answer as there might be some rounding differences.

Answer:

-1/3

Step-by-step explanation:

When x is 1, y is 5

When x is 4, y is 4

5-4=1

1-4=-3

1/-3=-1/3

Using an exponential function, it would take 824 minutes before the amount of medication in Aldi's bloodstream is effectively 0.

A decaying exponential function is given as;

[tex]A(t) = A(0)e^-kt[/tex]

Where A = initial value

k = decay constant

We have the half-life of amoxicillin as 62minutes, to determine k, we get

[tex]A(62) = 0. 5(0)[/tex]

[tex]A(62) = 0.5A(0)[/tex]

[tex]0. 5A(0) = A(0) e^{-62k}[/tex]

Solve the exponential function thus

[tex]e^{-62k} = 0. 5[/tex]

㏑ [tex]e^{-62k}[/tex] = ㏑ 0. 5

- 62k = ㏑ 0.5

Make k subject of formula

k = -㏑[tex]\frac{0. 5}{62}[/tex]

k = 0. 011179

The equation is given by

[tex]A(t) = A(0)e^-0.011179[/tex]

We have

[tex]0. 0001A(0) = A(0)e^-0.01179[/tex]

[tex]e^-0.011179t = 0. 0001[/tex]

㏑ [tex]e^-0.011179[/tex] = ㏑ 0.0001

[tex]-0.011179t = In 0.0001[/tex]

t = [tex]-\frac{In 0. 0001}{0. 011179}[/tex]

t = [tex]824[/tex]

Therefore, it would take 824 minutes before the amount of medication in Aldi's bloodstream is effectively 0.

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Answer:

x = 69 v = 103

Step-by-step explanation:

Angles in a triangle add up to 180 degrees:

34 + 77 + x = 180

x = 69

Angles on a straight line add up to 180:

77 + v = 180

v = 103

Answer:

Step-by-step explanation:

the sum of the interior angles in a triangles is 180°, take away from 180 the know angles (34° and 77°) and you will have your answer

180 - 34 - 77 = 69°

Now we find angle V

The angle V and the angle of 77 ° are on a straight line, so the sum is 180°, from 180° you subtract 77 ° and you have the value of V

180 - 77 = 103 °

Answer:

the third option

Step-by-step explanation:

Our equation is

[tex] {y}^{2} = 36(x - 10)[/tex]

Equation of a vertical parabola is

[tex]( {y - k)}^{2} = 4p(x - h)[/tex]

where (h,k) is the center

The focus is

(h+p, k)

Equation of directrix is

x= h-p,

Here the center is (10,0)

Next, we factor out 4 in the original equation

[tex]4(9)[/tex]

So we have

[tex] {y}^{2} = 4(9)(x - 10)[/tex]

So our p=9,

So our focus is

(10+9,0) or (19,0)

The third option is the answer

The location of the light bulb is at point A. (10, 0).

Option A is the correct answer.

We have,

To find the location of the light bulb, we need to identify the coordinates (x, y) at which the light bulb is located.

In the given parabolic mirror equation, y² = 36(x - 10), the vertex form of the equation for a parabola with a vertical axis is (h, k), where h is the

x-coordinate of the vertex and k is the y-coordinate of the vertex.

Comparing the given equation with the standard form

(y - k)² = 4a(x - h),

we can see that h = 10 and k = 0.

So, the location of the light bulb is at the point (10, 0).

Therefore,

The location of the light bulb is at point A. (10, 0).

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A searchlight uses a parabolic mirror to cast a beam of light in parallel rays where the light bulb is located at the focus of the parabola in order to give the best illumination.

The mirror is modeled by y² = 36 (x-10), where the measurements are in cm.

What is the location of the light bulb?

A. (10, 0)

B. (14, 0)

C. (19, 0)

D. (36, 0)

The values of a, b, and c are -4,0 and 0 and the parabola opens downward with y-intercept (0,0).

we have

y=-4x^2

The quadratic equation in standard form is equal to

ax^2+bx+c=0

So,In this problem

a=-4, b=0 and c=0

The y-intercept is the value of y when the value of x is equal to zero

y=-4(0)^2=0

The y-intercept is the point (0,0)

The coefficient a is negative, therefore the parabola opens down.

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