Please help!!!!!!!!!!!asap

Please Help!!!!!!!!!!!asap

Answers

Answer 1

By solving y = 25 · x² - 100 through the square root method, we conclude that the two roots of the quadratic formula are y₁ = + 2 and y₂ = - 2.

How to obtain the roots of a second order polynomial

Herein we have a polynomials of the form y = a · x² - b. By algebra we find that two roots exists, which are described below according to the square root method:

[tex]y_{1} = +\sqrt{\frac{b}{a} }[/tex], [tex]y_{2} = -\sqrt{\frac{b}{a} }[/tex]

If we know that a = 25 and b = 100, then the two roots of the equation are:

y₁ = + 2, y₂ = - 2

By solving y = 25 · x² - 100 through the square root method, we conclude that the two roots of the quadratic formula are y₁ = + 2 and y₂ = - 2.

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Related Questions

La siguiente operación 147.3 = 441 es una:
División b- Sustracción c- Multiplicación
plis necesito la respuesta ahora dare estrellas lo juro

Answers

La operación que representa 147.3 = 441 es dada por una multiplicación, por eso opción c es correcta.

¿Qué operación está representada por 147 . 3 = 441?

El resultado del producto entre 147 y 3 es de 441, por lo que se hace una operación de multiplicación.

Así, hay que la opción c es correcta.

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In a certain region of space, the potential is given by v=2x−5x2y 3yz2. How much is the magnitude of the electric field at point (-2, 2, 0)?

Answers

The magnitude of the electric field at the point (-2, 2, 0) is 46.52 V/m

How to calculate the electric field?

Since in a certain region of space, the potential is given by v = 2x − 5x²y 3yz²,

The electric field, E = -grad(V) where grad(V) = (dV/dx)i + (dV/dy)j + (dV/dz)k

Since V = 2x − 5x²y + 3yz²

dV/dx = d(2x − 5x²y + 3yz²)/dx

= d2x/dx - d5x²y/dx + d3yz²/dx

= 2 - 10xy + 0

= 2 - 10xy

dV/dy = d(2x − 5x²y + 3yz²)/dy

= d2x/dy - d5x²y/dy + d3yz²/dy

= 0 - 5x² + 3z²

=  - 5x² + 3z²

dV/dz = d(2x − 5x²y + 3yz²)/dz

= d2x/dz - d5x²y/dz + d3yz²/dz

= 0 - 0 + 6z

=  6z

The electric field

So, E = -grad(V)

= -[(dV/dx)i + (dV/dy)j + (dV/dz)k]

= -[(2 - 10xy)i + (- 5x² + 3z²)j + (6z)k]

So, the electric field at (-2, 2, 0) is

E = -[(2 - 10xy)i + (- 5x² + 3z²)j + (6z)k]

E = -[(2 - 10(-2)(2))i + (- 5(2)² + 3(0)²)j + (6(0))k]

E = -[(2 + 40)i + (- 5(4) + 0)j + (0)k]

E = -[42i + (-20 + 0)j + (0)k]

E = -[42i - 20j + 0k]

E = -42i + 20j - 0k

The magnitude of the electric field

So, the magnitude of E at (-2, 2, 0) is

|E| = √[(-42)² + 20² + 0²]

=  √[1764 + 400 + 0]

= √2164

= 46.52 V/m

So, the magnitude of the electric field at the point (-2, 2, 0) is 46.52 V/m

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In the square pyramid shown below, the slant height is 12 centimeters and the height of the pyramid is 7 centimeters.

To the nearest whole number, find the diagonal length of the base of the pyramid. In your final answer, include your calculations with reasoning for each step.

Answers

The diagonal of the square pyramid will be equal to 33.10 centimetres.

What is diagonal?

The lines which connect the two opposite corners of any quadrilateral are called diagonal. here we have a square pyramid whose slant height and the vertex height are given and we have to find the diagonal length of the base.

Let the side of the square base of the pyramid is "a". Now the side will be calculated by the following relation.

[tex](\dfrac{a}{2})^2[/tex] =  s² - H

[tex](\dfrac{a}{2})^2[/tex]  = 12² - 7 = 137

a²   = 137 x 4  = 548

a  =  √548 = 23.40 cm

Now the diagonal will be calculated by using the Pythagorean theorem

D² = 23.04²  + 23.40²

D² = 548 + 548

D  = √1096

D  = 33.10 centimeter

Therefore the diagonal of the square pyramid will be equal to 33.10 centimetres.

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Could someone help me with this question real quick?

Answers

Answer:

Width is 46 meter

Length is 86 meter

Given:

width = wlength = (w + 40)

Formula:

Area of rectangle = Length × Width

Trial and improvement:

1st try with w = 50

w(w + 40) = 4000

50(50 + 40) = 4000

4500 ≠ 4000

Value too high.

2nd try with w = 45

w(w + 40) = 4000

45(45 + 40) = 4000

3825 ≠ 4000

Value too small

3rd try with w = 46

w(w + 40) = 4000

46(46 + 40) = 4000

3956 ≠ 4000

This value is somewhat near to the area value.

It is found that width = 46 m then length = (w + 40) = (46 + 40) = 86 m

Lines l1 and l2 are parallel and l3 is a transversal.



Which statement is true?

A
m∠2+m∠4=180°

B
m∠2+m∠3=180°

C
m∠1=m∠4

D
m∠1=90°

Answers

Answer:

C. m∠1 = m∠4

Step-by-step explanation:

A. m∠2 + m∠4 = 180°

∠2 and ∠4 are vertically opp. angles which means m∠2 = m∠4.

Hence, it's a false statement.

B. m∠2 + m∠3 = 180°

∠2 and ∠3 are corresponding angles which means m∠2 = m∠3.

It's a false statement.

C. m∠1 = m∠4

∠1 and ∠4 are another pair of corresponding angles which clearly states that m∠1 = m∠4.

Therefore, it's a true statement.

D. m∠1 = 90°

As clearly visible in the diagram;

It's a false statement.

Two planes that do not intersect are _____ parallel.
sometimes always never

Answers

Two planes that do not intersect are always parallel.

ALWAYS………………………………..

Matching a Table to a Graph On a coordinate plane, a line goes through points (negative 1, 0), (0, 1), and (1, 2). Which table goes with the graph? A 2-column table with 4 rows. Column 1 is labeled x with entries negative 1, 0, 1, 2. Column 2 is labeled y with entries 0, 4, 6, 8. A 2-column table with 4 rows. Column 1 is labeled x with entries negative 1, 0, 1, 2. Column 2 is labeled y with entries 0, 2, 4, 6. A 2-column table with 4 rows. Column 1 is labeled x with entries negative 1, 0, 1, 2. Column 2 is labeled y with entries 0, 1, 2, 3.

Answers

At x = -1 , 0 ,1 ,2 , the y value = 0, 1, 2, 3, Option C is the correct answer.

What is a Function ?

A function is a law that relates a dependent variable to an independent variable.

The points of line are

(- 1, 0), (0, 1),(1, 2).

The equation of the line formed will be

y = mx +c


[tex]\rm m = \dfrac{y_2-y_1}{x_2-x_1}[/tex]

m = 1

y = x +c

at x = 0 , y =1

c = 1

The equation is

y = x+1

The option 1 is

x = - 1, 0, 1, 2.

y1 = 0, 4, 6, 8.

y2= 0, 2, 4, 6

y3 =0, 1, 2, 3.

From the equation

at x = -1 , 0 ,1 ,2

the y value = 0, 1, 2, 3

Therefore Option C is the correct answer.

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What is the domain of f(x) = 5x – 7?

{x | x > –7}
{x | x < –7}
{x | x > 0}
{x | x is a real number}

Answers

Answer: Choice D) x is a real number

Reason:

We can replace x with any number we want without any restriction. There's no issues dealing with division by zero errors for instance, and we also don't have negatives under square roots to worry about. The domain of any linear function is always "the set of all real numbers".

Extra info: The domain in interval notation is [tex](-\infty, \infty)[/tex] which could be thought of as [tex]-\infty < x < \infty[/tex], i.e. x is between negative infinity and positive infinity.

Answer:

d

Step-by-step explanation:

HURRRRYYY
Hurry pleaseee use elimination show your work


15y - 9x =-9 -8y+9x=12

Answers

Add the two functions together to get

7y=-3
y=-3/7

Then plug in y to either function to solve for x, I’ll just use the left one

15(-3/7)-9x=-9
-45/7-9x=-9
-9x=-18/7
x=18/7/9

x=2/7

Anne is visiting a friend who lives 18.5 km away. She travels 10 km by
train, 1.5 km by bus, and walks the rest of the way. If she walks at a rate
of 3.5 km per hour, for how long will Anne walk?

Answers

Answer:

Anne will walk for 2 hours.

Step-by-step explanation:

Distance formula = velocity × time

Total distance is 18.5 km

Anne travels by train = 10 km

and by bus = 1.5 km

let the distance covered by anne by walking be x

10 + 1.5 + x = 18.5

x = 18.5 - 10 - 1.5

x = 7km

Rate of walking is 3.5 km per hour

the time she will take while walking = 7 ÷ 3.5 = 2 hours.

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True or false for 1 and 3

Answers

Step-by-step explanation:

I think 1 is true.

3 is false. it's $10 per hour

1 = true
3 = false ……

What are the domain and range of the function below?
A. Domain: (-4,0)
Range: [-2,∞)
B. Domain: (-3,0)
Range: [-2,∞)
C. Domain: (-∞,∞)
Range: [-2,∞)
D. Domain:
Range: (-∞,∞)
PS don't mind that dot on the graph, didn't mean to put it there.

Answers

The domain is (-∞,∞) and the range is [-2,∞)

How to determine the domain and the range?

From the graph, we can see that the x values extend at both sides of the graph.

This means that the domain is the set of real values i.e. (-∞,∞)

However, the minimum y value is -2 and it increases from their

This means that the range is [-2,∞)

Hence, the domain is (-∞,∞) and the range is [-2,∞)

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Peter's dog weighs 12 pounds more than Joe's dog. The dogs weigh 36
pounds all together. Write an equation to find the weight of Joe's dog.

Answers

Answer: x + x + 12 = 36, 12 pounds

Step-by-Step Explanation:

Let Joe’s dog’s weight be ‘x’ pounds
Peter’s dog’s weight be ‘x + 12’ pounds
Both dogs weigh 36 pounds together.

Therefore,
x + x + 12 = 36
2x + 12 = 36
2x = 36 - 12
2x = 24
x = 24/2
=> x = 12

Joe’s Dog’s Weight = x => 12
Peter’s Dog’s Weight = x + 12 => 24

(08.01)
A new video game is expected to sell 120 copies the first hour at a local game store. After that, the sales will follow the function s(x) = 15(=
1) where x is the number of hours. What is the function that shows total sales, including the first hour?

Answers

Answer:

f(x)=15x+120

Step-by-step explanation:

trust

For her job, natasha spent a total of n minutes processing id card applications and driver's license applications. it takes natasha 15 minutes to process and id card aplication and 20 minutes to process a driver's license application. what is the value of n

Answers

Step-by-step explanation:

if I understand the question, write an equation for n in terms of i for number of Id cards an d for number driver license.

n = 15i + 20d

The value of n for a total minutes processing id card applications and driver's license applications would be 35 minutes.

Used the concept of addition that states,

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

Given that,

For her job, Natasha spent a total of n minutes processing id card applications and driver's license applications.

And, it takes Natasha 15 minutes to process an id card application and 20 minutes to process a driver's license application.

Now by the given condition, we get;

n = 15 minutes + 20 minutes

n = 35 minutes

Therefore, the value n is 35 minutes.

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which of the following is a rational number
a.√5
b.π
c.1/2
d.√(77)

Answers

c. 1/2

the quotient of two integers

Maria is on a hike. if she hikes to the scenic lookout on the following map first, she will have to hike farther than if she went straight to the end of the hike. 3 connected points on a coordinate plane. the point labeled, "maria," is at 4, -4. the point labeled, "end," is at -5, 5. a solid line connect the points "maria" and "end." the point labeled, "scenic lookout," is at 2, 3. dashed lines connect "maria" to "scenic lookout" and connect "scenic lookout" to "end." coordinate values on the map are in kilometers. how much shorter is the path straight to the end of the hike than past the scenic lookout? round your final answer only to the nearest kilometer. \text{km}kmstart text, k, m, end text

Answers

The path straight to the end of the hike is 6 km farther than past the scenic lookout.

How to determine the difference in the parts?

The map is not given, but the question can still be answered

From the question, the positions are given as::

Maria = (4, -4)

Scenic = (2, 3)

End = (-5, 5)

The distance between Maria and the end point is calculated using:

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

So, we have:

[tex]Route\ 1 = \sqrt{(4 + 5)^2 + (-4 -5)^2}[/tex]

[tex]Route\ 1 = 13[/tex]

The distance between Maria and the scenic lookout is calculated using:

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

So, we have:

[tex]Route\ 2 = \sqrt{(4 - 2)^2 + (-4 -3)^2}[/tex]

[tex]Route\ 2 = 7[/tex]

The distance between both routes is:

Distance = 13 - 7

Evaluate

Distance = 6

Hence, the path straight to the end of the hike is 6 km farther than past the scenic lookout.

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If f(x) = 4x² + 1 and g(x)=x2-5, find (f- g)(x).

Answers

Answer:

3x² +6

Step-by-step explanation:

(f-g)(x) = f(x) -g(x) = (4x²+1) -(x²-5) . . . . . substitute the function definitions

... = 4x² +1 -x² +5 . . . . . . . . . . . eliminate parentheses

... = (4-1)x² +(1+5) . . . . . . . . . . . collect like terms

... = 3x² +6

How much is the bill for a person who used 700 kWh in a month?

Answers

Answer:

700 kWh which is more than 200 so

0.10(x - 200) + 30

0.10( 700 -200)+ 30

0.10( 500) + 30

50 + 30

80

$80

Linear equations in two variables
Graph y = -4 then explain why it is not a function, can somone ELI5

Answers

Answer:

y=-4 is a function

Step-by-step explanation:

It passes the vertical line test, and each x has only one y, so it is a function.

x = -4 is NOT a function because one x has multiple y values. It does not pass the vertical line test.

Please help me on this question!

Answers

Answer:

u = 10 ft

Step-by-step explanation:

cuboid volume = w × h × l

width * hight * length

560 = 7 × 8 × u

u = 10 ft

Answer:

u = 10 ft

Step-by-step explanation:

Volume of a rectangular prism = lwh

where:

l = lengthw = widthh = height

Given:

Volume = 560 ft³width = 7 ftlength = 8 ftheight = u

Substitute the given values into the formula and solve for h:

⇒ 560 = 8 × 7 × u

⇒ 560 = 56u

⇒ 56u ÷ 56 = 560 ÷ 56

⇒ u = 10 ft

Therefore, the height of the prism is 10 ft.

Drag each expression to the correct location on the table.
Simplify each exponential expression using the properties of exponents and match it to the correct answer.

Answers

The solution to the exponential expression is as follows:

[tex]\mathbf{\dfrac{(2\times 3^{-2})^3(5\times 3^2)^2}{(3^{-2})(5\times 2)^2} = 2}[/tex][tex]\mathbf{=(3^3)(4^0)^2(3\times 2)^{-3}(2^2)=\dfrac{1}{2}}[/tex][tex]\mathbf{\dfrac{(3^7\times 4^7)(2 \times 5)^{-3}(5^2)}{12^7\times 5^{-1}\times 2^{-4}} = 2}[/tex][tex]\mathbf{\dfrac{(2\times 3)^{-1}\times 2^0}{(2\times 3)^{-1}}= 1}[/tex]

What are exponential expressions?

Exponential expressions are convenient ways to express the mathematical power of a number in short forms.

Simplifying the following expressions given:

1.

[tex]\mathbf{\dfrac{(2\times 3^{-2})^3(5\times 3^2)^2}{(3^{-2})(5\times 2)^2} }[/tex]

Applying the exponent rule: [tex]\mathbf{(a\times b)^n = a^n b^n}[/tex]

[tex]\mathbf{\implies \dfrac{(2\times 3^{-2})^3\times 5^2\times 81}{3^{-2}\times 5^2\times2^2} }[/tex]

cancel out the common factor, we have:

[tex]\mathbf{\implies \dfrac{(2\times 3^{-2})^3\times 81}{3^{-2}\times2^2} }[/tex]

[tex]\mathbf{\implies \dfrac{(2\times 3^{-2})^3\times 3^4}{3^{-2}\times2^2} }[/tex]

[tex]\mathbf{\implies \dfrac{(2\times 3^{-2})^3\times 3^6}{2^2} }[/tex]

[tex]\mathbf{\implies \dfrac{2^3 \times \dfrac{1}{729} \times 3^6}{2^2} }[/tex]

[tex]\mathbf{\implies \dfrac{2^3 \times \dfrac{1}{729} \times729}{2^2} }[/tex]

[tex]\mathbf{\implies \dfrac{2^3 }{2^2} = 2}[/tex]

2.

[tex]\mathbf{=(3^3)(4^0)^2(3\times 2)^{-3}(2^2)}[/tex]

= [tex]\mathbf{\dfrac{1}{2}}[/tex]

3.

[tex]\mathbf{\dfrac{(3^7\times 4^7)(2 \times 5)^{-3}(5^2)}{12^7\times 5^{-1}\times 2^{-4}} = 2}[/tex]

4.

[tex]\mathbf{\dfrac{(2\times 3)^{-1}\times 2^0}{(2\times 3)^{-1}}= 1}[/tex]

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The function f(x) = -√x is shown on the graph.
-6
43
4-
N
-7-6-5-4-3-2-₁ 1 2 3 4 5 6 7 x
-2+
-3-
-4
Which statement is correct?
The domain of the function is all real numbers less
than or equal to -1.
The range of the function is all real numbers greater
than or equal to 0.
O The range of the function is all real numbers less
than or equal to 0.
The domain of the function is all real numbers less
than or equal to 0.

Answers

Answer: The range of the function is all real numbers less

than or equal to 0.

The correct statement for the given function f(x) = -√x is "The range of the function is all real numbers less than or equal to 0." So, the correct option is C.

The domain of the function is all real numbers that are greater than or equal to 0 but we know that the square root of a negative number is in the real number system.

The graph of the function shows that for each positive input i.e. the range values we get the output values that are negative or zero.

Therefore, the range of the function is all real numbers less than or equal to 0, which is option C.

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find the area of the rectangle below
(3x+4) feet x
(x+3) feet
select the correct answer

Answers

Answer:

3x^2+ 13x + 12

Step-by-step explanation:

Multiply the length and width:
(3x+4) * (x+3)
FOIL method:
3x^2 + 9x + 4x + 12
Combine like terms
3x^2+ 13x + 12

Can I have the solution with steps

Answers

Answer:

1/6

Step-by-step explanation:

As usual in trigonometry, there are so many relationships between the trigonometric functions, there is bound to be more than one method to arrive at the answer.

Knowing values for the sine and cosine function for certain angles on the unit circle proves to be quite helpful.  In general, I recommend 0°, 30°, 45°, 60°, and 90° at a minimum.

Recall the following 6 things:

[tex]\cos(30^o)=\frac{\sqrt{3}}{2}[/tex][tex]\cos(60^o)=\frac{1}{2}[/tex][tex]\sin(30^o)=\frac{1}{2}[/tex][tex]\sin(60^o)=\frac{\sqrt{3}}{2}[/tex][tex]\tan(\theta)=\dfrac{\sin(\theta)}{\cos(\theta)}[/tex][tex]\tan^2(\theta)=\tan(\theta)\tan(\theta)[/tex]

Knowing these 6 things (4 things from the unit circle, a way to write the tangent function in terms of sine and cosine, and the definition of a squared trig function), we can simplify the given expression down to a single number:

The definition of the tangent squared function is that the output of the tangent function is squared...

[tex]\dfrac{1-\tan^2(30^o)}{1+\tan^2(60^o)}=\dfrac{1-(\tan(30^o))^2}{1+(\tan(60^o))^2}[/tex]

Using fact 5, and turning the tangent function into sines and cosines...

[tex]=\dfrac{1-\left(\frac{\sin(30^o)}{\cos(30^o)}\right)^2}{1+\left(\frac{\sin(60^o)}{\cos(60^o)}\right)^2}[/tex]

Using facts 1-4, and substituting known values for sines and cosines...

[tex]=\dfrac{1-\left(\frac{(\frac{1}{2})}{(\frac{\sqrt{3}}{2})}\right)^2}{1+\left(\frac{(\frac{\sqrt{3}}{2})}{(\frac{1}{2})}\right)^2}[/tex]

Division is multiplication by the reciprocal...

[tex]=\dfrac{1-\left(\frac{1}{2}*\frac{2}{\sqrt{3}}\right)^2}{1+\left(\frac{\sqrt{3}}{2}*\frac{2}{1}\right)^2}[/tex]

Reducing/simplifying common factors of 2...

[tex]=\dfrac{1-\left(\frac{1}{\sqrt{3}}\right)^2}{1+(\sqrt{3})^2}[/tex]

Squaring a square root...

[tex]=\dfrac{1-\left(\frac{1}{3}\right)}{1+(3)}[/tex]

Finding a common denominator...

[tex]=\dfrac{\frac{3}{3}-\frac{1}{3}}{4}[/tex]

Definition of subtracting fractions...

[tex]=\dfrac{\frac{3-1}{3}}{4}[/tex]

Arithmetic...

[tex]=\dfrac{\frac{2}{3}}{4}[/tex]

Division is multiplication by a reciprocal...

[tex]=\dfrac{2}{3}*\dfrac{1}{4}[/tex]

Simplification and reducing the fraction...

[tex]=\dfrac{1}{6}[/tex]

So, [tex]\dfrac{1-\tan^2(30^o)}{1+\tan^2(60^o)}=\dfrac{1}{6}[/tex]

Which equation represents the graph below?

Answers

Answer:

D

Step-by-step explanation:

The slope-intercept form is the equation of a straight line (y=mx+b)

m= gradient and b= y-intercept

m=2/3 (rise over run)  b= 4 (it goes through the point (0,4)

y=(2/3)x+4

Get rid of the fraction by multiplying everything by 3

3y=2x+12

which is also -2x+3y=12

Which equation represents a side that is perpendicular to side d? O y = 2x+5 Oy=-2x+5 NG 0 Y=2x-5/2 OY--12-12 = Which equation represents a side that is perpendicular to side d ? O y = 2x + 5 Oy = -2x + 5 NG 0 Y = 2x - 5 / 2 OY -- 12-12 =​

Answers

The equation represents a side that is perpendicular to side d is y = -2x+5. Option B is correct.

What is the slope?

A numerical assessment of a line's angle relative to the ground is known as the slope.

The inclination of any line, ray, or line segment is the fraction of the perpendicular to the horizontal distance between any two positions on it.

The given linear equation in the problem is;

y = 2x+5

Compare the equation with the standard equation;

[tex]\rm y = mx+ c[/tex]

where,

m is the slope and c is the intercept. After comparing, we will get; Hence, the slope and y-intercept will be;

m = 2

The product of the slope of the two-line perpendicular to each will be -1.

m₁×m₂ = -1

The equation represents a side that is perpendicular to side d is;

y = -2x+c.

m₁=2

m₂=-2

m₁×m₂ = -1

The equation represents a side that is perpendicular to side d is y = -2x+5.

Hence, option B is correct.

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Adam wins $30 in a "spot the ball" competition. He decides to save the money and opens a saving account with the $30. After his win, he continues to put $8 into his savings account each week. Identify a rule for the linear function that describes Adam's savings. Use the rule to find the amount Adam saved during the first 5 weeks.

Answers

Answer:

$110

Step-by-step explanation:

$30-$8=$22 each week

$22×5weeks=$110


What is the area of rectangle ABCD?

coordinate plane with rectangle ABCD at A 0 comma negative 1, B 0 comma 4, C 4 comma 4, and D 4 comma negative 1

Answers

Answer:

Area = 20 units

Step-by-step explanation:

A (-1,0)

B (0,4)

C (4,4)

D (4,-1)

i hope this is right

Answer:

Area = 20

Step-by-step explanation:

A (0,-1)

B (0, 4)

C (4, 4)

D (4, -1)

Please answer both questions math

Answers

Answer:

[tex]3[/tex] and perpendicular

Step-by-step explanation:

It's important to remember that equations of lines are always set up in the format of [tex]y=mx+c[/tex].

[tex]y[/tex] is the [tex]y[/tex] value, [tex]m[/tex] is the gradient/slope, [tex]x[/tex] is the [tex]x[/tex] value and [tex]c[/tex] is the y-intercept.

Therefore, we must always rearrange our line equations so that [tex]y[/tex] is the subject.

In the first question, we are given the equation [tex]2x+6y=12[/tex].

We rearrange to make [tex]y[/tex] the subject:

[tex]6y = 12-2x[/tex]

Then, we divide both sides by 6 to have the [tex]y[/tex] isolated:

[tex]\frac{6}{6} y = \frac{12}{6} - \frac{2}{6}x\\\\ y = 2 - \frac{1}{3}x[/tex]

To find a perpendicular slope, we use the negative reciprocal of the original slope. The negative part of that means we negate the value (it becomes inverse) and the reciprocal part means that we flip the fraction.

So here, the slope is [tex]-\frac{1}{3}[/tex].

So, the negative reciprocal of that would be [tex]3[/tex] because we inverse the sign (the negative becomes positive) and the fraction is flipped to create [tex]\frac{3}{1}=3[/tex].

So, the perpendicular slope is [tex]3[/tex].

In the second question, we have two equations. They both need to be rearranged to isolate the [tex]y[/tex].

[tex]4x-y=5\\4x = 5 + y\\y = 4x - 5\\\\4y = -x-12\\y = -\frac{1}{4}x - 3[/tex]

Look at the slopes of the two equations. If you look carefully, you will see that they are inverse of each other (one is positive and one is negative) and if you imagine [tex]4 = \frac{4}{1}[/tex], the fraction is flipped.

This means that the values are negative reciprocals of each other, meaning the lines are perpendicular.

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