6x − y + z = −3
4x − 3z = −13
2y + 5z = 15
Find the solutions?

Answer:

[tex]\theta =2\pi k,\ \ k\in Z\ \\\text{or}\ \\\theta=-\dfrac{2\pi}{3}+2\pi k,\ \ k\in Z[/tex]

Step-by-step explanation:

Given:

[tex]\cos \theta-\sqrt{3}\sin \theta=1[/tex]

Divide this equation by 2:

[tex]\dfrac{1}{2}\cos \theta-\dfrac{\sqrt{3}}{2}\sin \theta=\dfrac{1}{2}[/tex]

Note that

[tex]\cos \dfrac{\pi }{3}=\dfrac{1}{2}\\ \\\sin \dfrac{\pi }{3}=\dfrac{\sqrt{3}}{2}[/tex]

So, the previous equation is

[tex]\cos \dfrac{\pi}{3}\cdot \cos \theta-\sin \dfrac{\pi}{3}\cdot \sin \theta=\dfrac{1}{2}[/tex]

Remind that

[tex]\cos x\cos y-\sin x\sin y=\cos (x+y),[/tex]

then

[tex]\cos \left(\dfrac{\pi}{3}+\theta\right)=\dfrac{1}{2}[/tex]

The solution of this equation is

[tex]\dfrac{\pi}{3}+\theta=\pm \arccos \dfrac{1}{2}+2\pi k,\ \ k\in Z\\ \\\dfrac{\pi}{3}+\theta=\pm \dfrac{\pi}{3}+2\pi k,\ \ k\in Z\\ \\\theta =2\pi k,\ \ k\in Z\ \text{or}\ \theta=-\dfrac{2\pi}{3}+2\pi k,\ \ k\in Z[/tex]

Answer: 14/3 or 4 2/3

Step-by-step explanation:

you need 7 servings of 2/3 of a cup of cream

so, multiply the two values

7x2/3=14/3 or 4 2/3

Answer:

About 5 cups

Step-by-step explanation:

7×2/3

4.6666666667

you round up to 5 or put 4.6 repeated

Answer:

82g

Step-by-step explanation:

You multiply 9×8 to find the answer as each pencil is 9g and there is 8 pencils in a box.

72g because 8*9g is 72

Answer:

i think maybe 600k?

Step-by-step explanation:

To find out how much more interest is paid on a $195,000 home with a 20% down payment and a balance financed with a 30-year mortgage at 6% compared to 3.5%, one must calculate the total interest for each rate over the 30-year period and then compare the two totals.

To determine the difference in the amount of interest paid over the life of a 30-year mortgage because of different interest rates, we need to calculate the total interest for both scenarios. With a $195,000 home and a 20% down payment, the amount to be financed is $156,000 (since 20% of $195,000 is $39,000, and $195,000 - $39,000 = $156,000).

For the 6% mortgage rate, the monthly payment can be found using the formula for a fixed-rate mortgage, which is P[r(1+r)^n]/[(1+r)^n-1], where P is the principal ($156,000), r is the monthly interest rate (0.06/12), and n is the number of payments (360). Similarly, we calculate the monthly payment for the loan at 3.5% interest.

Once the monthly payments are determined, we can calculate the total payment over 30 years for each rate by multiplying the monthly payment by 360. The difference between the two total payments will give us the additional interest paid due to the higher interest rate.

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Answer:

D)99 degrees

Step-by-step explanation:

The answer is D. All the angles in any quadrilateral add up to 360 degrees. And since the figure is a paralledlogram, for each angle there is another angle with the same measurement. So 81 + 81 + x + x = 360 which gives us 162 + 2x = 360. If you solve for x you get 99 degrees.

Answer:

[tex]\displaystyle 4 > x[/tex]

Step-by-step explanation:

3 - 2x > −5

- 3 - 3

_________

[tex]\displaystyle \frac{-2x}{-2} > \frac{-8}{-2} \\ \\ x < 4[/tex]

Whenever you divide\multiply by a negative, you reverse the inequality symbol.

** The above answer is written in reverse, which is the exact same result.

I am joyous to assist you anytime.

Answer:

1/12

Step-by-step explanation:

1/2-[-1/4+2/3]

1/2-[-3/12+8/12]

1/2-[5/12]

1/2-5/12

6/12-5/12

1/12

Answer:

x=4, y=13. (4, 13).

Step-by-step explanation:

3x+2y=38

-6x+4y=28

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

simplify -6x+4y=28 into -3x+2y=14

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

3x+2y=38

-3x+2y=14

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

4y=52

y=52/4=13

3x+2(13)=38

3x+26=38

3x=38-26

3x=12

x=12/3=4

x=4, y=13.

Answer:

342 miles/6 hours = 57 mph

domain: {-5, 1, 3, 7}; Yes, it is a function ⇒ 3rd answer

Step-by-step explanation:

Let us revise the relation and the function

∵ The relation is {(-5, 2), (7, 7), (3, 6), (1, 7)}

∴ The inputs are: -5 , 7 , 3 , 1

∴ The outputs are: 2 , 7 , 6

∵ The domain is the set of the inputs

∴ The domain of the relation is {-5 , 1 , 3 , 7}

∵ The input -5 has the output 2

∵ The input 7 has the output 7

∵ The input 3 has the output 6

∵ The input 1 has the output 7

∴ There are one output for each input

∴ The relation is a function

domain: {-5, 1, 3, 7}; Yes, it is a function

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Answer:

$66.00

Step-by-step explanation:

first we need multiply :D I will use the formula I =prt

principal= $500

rate=3.3%

time= 4 years

but we need to convert the percent, 3.3%, to a decimal

3.3%=0.033 or .033

now we multiply !

500 × 0.033 × 4 = 66 = $66.00

:D

Answer:$66.00

Step-by-step explanation:

first we need multiply :D I will use the formula I =prt

principal= $500

rate=3.3%

time= 4 years

but we need to convert the percent, 3.3%, to a decimal

3.3%=0.033 or .033

now we multiply !

500 × 0.033 × 4 = 66 = $66.00

Answer: Kremena needs to deposit $1,101 today to earn a total of $1,250 in 3 years at 4.5% simple interest rate

Step-by-step explanation:

Interest = Principal (P) × Rate (R) × Time (T)

Amount = Principal (P) + Interest

Amount = P + PRT

Amount = P (1 + RT)

1250 = P (1 + (0.045 × 3))

1250 = P (1 + 0.135)

1250 = P (1.135)

Principal = 1250 ÷ 1.135

Principal = approx. 1101

Final answer:

Kremena needs to deposit approximately $1,101 today at a simple interest rate of 4.5% to have $1,250 in 3 years.

Explanation:

To find out how much Kremena needs to deposit today to have her account worth $1,250 in 3 years at a simple interest rate of 4.5%, we use the simple interest formula:

I = P * r * t, where:

Kremena wants the final amount (A) to be $1,250. The final amount is the sum of the principal and the interest earned: A = P + I. Therefore, we rearrange the simple interest formula to calculate P:

P = (A - (I = A * r * t)) / (1 + r * t)

Here, I = A - P, which means:

P = A / (1 + r * t)

Plugging in the numbers:

P = $1,250 / (1 + 0.045 * 3)

P = $1,250 / 1.135

P ≈ $1,101

Therefore, Kremena must deposit approximately $1,101 today to have $1,250 in 3 years at a 4.5% simple interest rate.

There are 152 men, 76 women and 38 children.

Step-by-step explanation:

Total persons = 266

Let,

men = m

women = w

children = c

According to given statement;

m+w+c=266    Eqn 1

m = 4c   Eqn 2

w = 2c   Eqn 3

Putting value of m and w from Eqn 2 and 3 in Eqn 1

[tex]4c+2c+c=266\\7c=266\\[/tex]

Dividing both sides by 7

[tex]\frac{7c}{7}=\frac{266}{7}\\c=38[/tex]

Putting c=38 in Eqn 2;

[tex]m=4(38)\\m=152[/tex]

Putting c=38 in Eqn 1;

[tex]w=2(38)\\w=76[/tex]

There are 152 men, 76 women and 38 children.

Keywords: linear equations, substitution method

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Answer:

61 ¹/₂₀

Step-by-step explanation:

You cannot get a proper fraction while converting 61.05 into a fraction. You will only end up with improper fractions. This is your answer as a simplified improper fraction. If you also want a mixed number answer, that is down below.

61   1/20

I hope this helps you! :)

The sales tax rate is 5%.

Step-by-step explanation:

Given,

Amount of sales tax = $49

Price of dining room set = $980

Sales tax rate = [tex]\frac{Amount\ of\ sales\ tax}{Price\ of\ dining\ room\ set}*100[/tex]

Sales taxa rate = [tex]\frac{49}{980}*100[/tex]

[tex]Sales\ tax\ rate=\frac{4900}{980}\\Sales\ tax\ rate=5\%[/tex]

The sales tax rate is 5%.

Keywords: percentage, division

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Answer:4/3

Step-by-step explanation:24/18

Divide both the numerator and the denominator by their highest common  factor. i.e 6

24÷6/18÷6

⇒ 4/3

The value of fraction for 24 divided by 18 is,

⇒ 4 / 3

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications.

For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

Given that;

The expression is,

⇒ 24 divided by 18

Now, We can write as;

⇒ 24 divided by 18

⇒ 24 ÷ 18

⇒ 24 / 18

⇒ 4 / 3

Thus, The value of fraction for 24 divided by 18 is,

⇒ 4 / 3

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Raise 2 to the 4th power:

√(16*9)

Multiply:

√144

Find square root:

Answer = 12

Answer:

Part 1) [tex]c=\sqrt{146}\ cm[/tex]

Part 2) [tex]c=8\sqrt{2}\ in[/tex]

Part 3) The diagonal of rectangle is [tex]d=3\sqrt{29}\ cm[/tex]

Part 4) The length of the diagonal of computer monitor is [tex]d=15\ in[/tex]

Part 5) The ramp is [tex]10.59\ ft[/tex] long

Part 6) The distance between their houses is [tex]=4\sqrt{2}\ mi[/tex]

Part 7) [tex]234.31\ mi[/tex]

Part 8) 120 feet of wire is required

Step-by-step explanation:

Part 1) we know that

To find the length of the hypotenuse in a right triangle apply the Pythagorean Theorem

[tex]c^2=a^2+b^2[/tex]

where

c is the hypotenuse (the greater side)

a and b are the legs

we have

[tex]a=11\ cm\\b=5\ cm[/tex]

substitute

[tex]c^2=11^2+5^2[/tex]

[tex]c^2=146[/tex]

[tex]c=\sqrt{146}\ cm[/tex]

Part 2) we know that

To find the length of the hypotenuse in a right triangle apply the Pythagorean Theorem

[tex]c^2=a^2+b^2[/tex]

where

c is the hypotenuse (the greater side)

a and b are the legs

we have

[tex]a=8\ in\\b=8\ in[/tex]

substitute

[tex]c^2=8^2+8^2[/tex]

[tex]c^2=128[/tex]

[tex]c=\sqrt{128}\ in[/tex]

simplify

[tex]c=8\sqrt{2}\ in[/tex]

Part 3) we know that

To find the length of the diagonal in a rectangle apply the Pythagorean Theorem

[tex]d^2=b^2+h^2[/tex]

where

d is the diagonal of rectangle

b and h are the base and the height of rectangle

we have

[tex]b=15\ cm\\hb=6\ cm[/tex]

substitute

[tex]d^2=15^2+6^2[/tex]

[tex]d^2=261[/tex]

[tex]d=\sqrt{261}\ cm[/tex]

simplify

[tex]d=3\sqrt{29}\ cm[/tex]

Part 4) we know that

To find the length of the diagonal of a computer monitor apply the Pythagorean Theorem

[tex]d^2=w^2+h^2[/tex]

where

d is the diagonal of computer monitor

w and h are the wide and the high of computer monitor

we have

[tex]w=12\ in\\h=9\ in[/tex]

substitute

[tex]d^2=12^2+9^2[/tex]

[tex]d^2=225[/tex]

[tex]d=\sqrt{225}\ in[/tex]

simplify

[tex]d=15\ in[/tex]

Part 5) we know that

To find out the length of the ramp apply the Pythagorean Theorem

[tex]L^2=x^2+y^2[/tex]

where

L is the length of the ramp

x  is the horizontal distance of the ramp

y  is the vertical distance of the ramp

we have

[tex]x=10\ ft\\y=3.5\ ft[/tex]

substitute

[tex]L^2=10^2+3.5y^2[/tex]

[tex]L^2=112.25[/tex]

[tex]L=10.59\ ft[/tex]

Part 6) we know that

To find the distance between their houses apply the Pythagorean Theorem

[tex]c^2=a^2+b^2[/tex]

where

c is the hypotenuse (distance between their houses)

a and b are the legs

we have

[tex]a=4\ mi\\b=4\ mi[/tex]

substitute

[tex]c^2=4^2+4^2[/tex]

[tex]c^2=32[/tex]

[tex]c=4\sqrt{2}\ mi[/tex]

Part 7) we know that

The speed is equal to divide the distance by the time

[tex]speed=distance/time[/tex]

so

the distance is equal to multiply the speed by the time

[tex]distance=speed*time[/tex]

First train

speed=60 mph

time=3 hours

distance=60(3)=180 miles

Second train

speed=50 mph

time=3 hours

distance=50(3)=150 miles

To find out how far apart are the trains at the end of 3 hours apply the Pythagorean Theorem

[tex]c^2=a^2+b^2[/tex]

where

c is the hypotenuse (distance between the trains)

a and b are the legs

we have

[tex]a=180\ mi\\b=150\ mi[/tex]

substitute

[tex]c^2=180^2+150^2[/tex]

[tex]c^2=54,900[/tex]

[tex]c=234.31\ mi[/tex]

Part 8)  we know that

For one tree is needed three wire

To find out the length of one wire apply the Pythagorean Theorem

[tex]c^2=a^2+b^2[/tex]

where

c is the hypotenuse (length of one wire)

a and b are the distance above the ground and the distance from the base

we have

[tex]a=3\ ft\\b=4\ ft[/tex]

substitute

[tex]c^2=3^2+4^2[/tex]

[tex]c^2=25[/tex]

[tex]c=5\ ft[/tex]

so

For one tree is required ----> (5)3=15 ft

therefore

For 8 trees is required

Multiply by 8

15(8)=120 ft

Answer:

The length of river frontage for each lot are 96.55 ft. 98.85 ft, 101.15 ft and 103.45 ft.

Step-by-step explanation:

See the attached diagram.

The river frontage of 400 ft will be divided into 84 : 86 : 88 : 90 for each lot as AP, BQ, CR, DS and ET all are parallel.

Therefore, PQ : QR : RS : ST = 84 : 86 : 88 : 90 = 42 : 43 : 44 : 45

Let, PQ = 42x, QR = 43x, RS = 44x and ST = 45x

So, (42x + 43x + 44x + 45x) = 400

⇒ 175x = 400

⇒ x = 2.2988.

So, PQ = 42x = 96.55 ft.

QR = 43x = 98.85 ft.

RS = 44x = 101.15 ft and  

ST = 45x = 103.45 ft

(Answer)

We can use the points (2, -2) and (4, -1) to solve.

Slope formula: y2-y1/x2-x1

= -1-2/4-(-2)

= -3/6

= -1/2

Point slope form: y - y1 = m(x - x1)

y - 2 = -1/2(x + 2)

______

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Answer:

The corresponding pair of sides are in the same ratio.

Step-by-step explanation:

For two triangles to be similar, the theorem that is required is the corresponding sides of the two triangles are in the same ratio.

Now, see the diagram of two triangles given with the question.

The ratios of the corresponding sides are [tex]\frac{20}{16} = \frac{19}{\frac{76}{5} } = \frac{5}{4}[/tex]

If two pairs corresponding sides are in 5 : 4 ratio, then the third pair of corresponding sides will also be in 5 : 4 ratio.

Therefore, the two triangles are similar. (Answer)

To prove that two triangles are similar, you can use the AA (Angle-Angle) Postulate, SSS (Side-Side-Side) Similarity Theorem, or SAS (Side-Angle-Side) Similarity Theorem. These methods involve comparing the angles and sides of the triangles to see if they meet the similarity conditions.

Triangle Similarity Theorem

To determine whether two triangles are similar, you can use several different theorems or postulates. These include the AA (Angle-Angle) Postulate, the SSS (Side-Side-Side) Similarity Theorem, and the SAS (Side-Angle-Side) Similarity Theorem.

AA (Angle-Angle) Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

SSS (Side-Side-Side) Similarity Theorem: If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. For example, if triangle ABC and triangle DEF have corresponding sides such that AB/DE = BC/EF = AC/DF, then the triangles are similar.

SAS (Side-Angle-Side) Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the sides containing these angles are proportional, then the triangles are similar. For example, if ∠A = ∠D and AB/DE = AC/DF, then triangles ABC and DEF are similar.

By using these theorems, you can determine if two triangles are similar by comparing their angles and sides as per the given conditions.

Equation:

(2.80 ÷ 7) + .75 (to find the original cost of the cookies)

2.80 ÷ 7 (to find the new cost of the cookies)

Answer:

$1.15 is the original cost of the cookies.

$0.40 is the new cost of the cookies.

Equation:

2.80 ÷ 7 (new cost of the cookies)

(2.80 ÷ 7) + .75 (original cost of the cookies)

Answer:

$1.15 is the original cost of the cookies.

$0.40 is the new cost of the cookies.

Answer:

6827 centimeters

Step-by-step explanation:

multiply length value by 100

Answer

x = 50 mph (level rate)

x-30 = 20 mph (mountain rate)

Explain:

time = time

160/(x-30) = 400/x

160x = 400(x-30)

160x = 400x - 12000

-240x = -12000

Answer:

The next two terms for the geometric sequence are 250 and - 1,250

Step-by-step explanation:

1. Let's find the next two terms for the geometric sequence which first three terms are as follows:

- 2, 10, - 50.....

Aₓ = Aₓ₋₁ * - 5 ; A₁ = - 2

A₁ = - 2

A₂ = A₁ * - 5 = -2 * - 5 = 10

A₃ = A₂ * - 5 = 10 * - 5 = - 50

A₄ = A₃ * - 5 = - 50 * - 5 = 250

A₅ = A₄ * - 5 = 250 * - 5 = - 1,250

The next two terms for the geometric sequence are 250 and - 1,250

Answer:

8

Step-by-step explanation:

5 + 7 = 12

48 / 12 = 4

5 x 4 = 20

7 x 4 = 28

There are 8 girls more than boys

Answer:

The number is 6.

Step-by-step explanation:

(78/x+10)*8=184

8(78/x+10)=184

78/x+10=184/8

78/x+10=23

78/x=23-10

78/x=13

x=78/13

x=6

Answer:The unknown number can be calculated by the following formula;

Step-by-step explanation:

Answer:

9x+15  is the simplified version.

   7

Step-by-step explanation:

Distribute 3 into the parenthesis.

3x+3+2

 2 1/3

Simplify the problem.

3x+5

2 1/3

Convert the mixed number into an improper fraction.

3x+5

7/3

Multiply the denominator to the numerators of the complex fraction.

3x(3)+5(3)

7  ×3

3

9x+15

   7

Answer:

-18/5 or - 3 3/5.

Step-by-step explanation:

-2/3 (x-3) - x = 8

-2/3x + 2 - x = 8

-2/3x - 3/3x  = 8 - 2

-5/3 x = 6

x =   6 * -3/5

x = -18/5