Which situation can be represented by the inequality?
x<3


The child is 3 years old.

The movie is longer than 3 h.

The backpack is heavier than 3 kg.

The ceiling is lower than 3 m.
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Answer:

Step-by-step explanation:

8h – 3 = 11h + 12

Collecting like terms

8h - 11h = 12 + 3

- 3h = 15

h = 15/-3

h = -5

Answer:

y-y/x-x

1--1/3-4

1+1/3-4

2/-1

= -2

then you plug in with any one of the points

y=mx+b

1= -2(3) + b

1 = -6 + b

b = 1+6

b = 7

so the equation is: y = -2x+7

D. x = –2, x = 2, and x = 3

Please correct me If I'm wrong, I did the math myself and this is what I got.

Answer:

11

Step-by-step explanation:

2x+3y=27

3x-2y=8

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

3(2x+3y)=3(27)

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

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

6x+9y=81

-6x+4y=-16

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

13y=65

y=65/13

y=5

2x+3(5)=27

2x+15=27

2x=27-15

2x=12

x=12/2

x=6

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

x+y=6+5=11

Answer:

-7 feet above the ground

Step-by-step explanation:

Answer:

y=2/3x+2

Step-by-step explanation:

2x-3y=12

3y=2x-12

y=2/3x-12/3

y=2/3x-4

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

y-y1=m(x-x1)

y-(-2)=2/3(x-(-6))

y+2=2/3(x+6)

y=2/3x+12/3-2

y=2/3x+4-2

y=2/3x+2

Final answer:

The question involves an algebraic expression defining Y in terms of x in Mathematics.

Explanation:

The subject of this question is Mathematics.

To define the expression given, Y is equal to 2 plus the product of 3 and x, it can be represented as: Y = 2 + 3x.

This is a basic algebraic expression where Y depends on the value of x.

Answer:

508.8 seconds (8.48 minutes)

Step-by-step explanation:

Answer:

8.

Step-by-step explanation:

If you mean the GCF of 16 and 24 , from the prime factors, it is 2*2*2

= 8.

Answer:

8

Step-by-step explanation:

Answer:

One (1) significant figure.

Step-by-step explanation:

All non zero digits are always significant zeros. Since there is no decimal point, the 0 is a trailing zero, and does not count as a significant figure.

Answer:

Karoun bought 12n + 15 numbers of pencils.

40 pencils

Step-by-step explanation:

If there are n numbers of pencils in each box of the pencil then there are 4n numbers of pencils in each package of pencils.

Now, Karoun bought 3 packages of pencils and get (5 × 3) = 15 pencils free of cost.

Therefore, Karoun bought (4n × 3) + 15 = 12n + 15 numbers of pencils. (Answer)

Now, if n = 10 pencils then in each package of pencils there will be 4n i.e. (4 × 10) = 40 pencils. (Answer)

Answer:

The slope will be 2/2 which is equal to 1.

Answer:

Step-by-step explanation:

The slope is 2/2 which equals to 1.

Answer:

Nine less than two-thirds of number is less than the number plus one.

Step-by-step explanation:

I jus got it right on edge.

Nine less than two-thirds of the number is less than the number plus one.

Algebra is the study of mathematical symbols, and the rule is the manipulation of those symbols.

The expression is given below.

2/3y − 9 < y + 1

Then complete the sentence.

Then we have

Nine less than two-thirds of the number is less than the number plus one.

More about the Algebra link is given below.

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Answer:

Hence Cos X = 40/41 .

Step-by-step explanation:

Given-   X Y = 41 units, Y Z = 9 units, X Z = 40 units, ∠ Z= 90°,

∵ ∠ Z = 90°, so Δ XYZ is a right triangle.

∴ By Pythagorean Theorem-

Cos X = [tex]\frac{Base}{Hypotenuse}[/tex]

Cos X = [tex]\frac{X Z}{X Y}[/tex]

Cos X = [tex]\frac{40}{41}[/tex]

Final answer:

To find the inverse of a matrix, follow these steps: 1. Check if the matrix is square and has a non-zero determinant. 2. Use the formula for finding the inverse of a 2x2 matrix. 3. For larger matrices, use row reduction or the adjugate matrix.

Explanation:

To find the inverse of a matrix, follow these steps:

Check if the matrix is square (the number of rows equals the number of columns). If it is not, then the matrix does not have an inverse.

Calculate the determinant of the matrix. If the determinant is 0, then the matrix does not have an inverse.

If the matrix is square and has a non-zero determinant, use the formula for finding the inverse of a matrix. For a 2x2 matrix:

A-1 = 1/det(A) * [d -b;
-c a]

where A is the original matrix, A-1 is the inverse matrix, det(A) is the determinant of A, and a, b, c, and d are the entries of A.

If the matrix is larger than 2x2, you can use methods such as row reduction or the adjugate matrix to find the inverse.

Answer:

Step-by-step explanation:

Use Newton's Law of Cooling for this one.  It involves natural logs and being able to solve equations that require natural logs.  The formula is as follows:

[tex]T(t)=T_{1}+(T_{0}-T_{1})e^{kt}[/tex] where

T(t) is the temp at time t

T₁ is the enviornmental temp

T₀ is the initial temp

k is the cooling constant which is different for everything, and

t is the time (here, it's in minutes)

If we are looking first for the temp after 20 minutes, we have to solve for the k value.  That's what we will do first, given the info that we have:

T(t) = 80

T₁ = 30

T₀ = 100

t = 5

k = ?

Filling in to solve for k:

[tex]80=30+(100-30)e^{5k}[/tex] which simplifies to

[tex]50=70e^{5k}[/tex] Divide both sides by 70 to get

[tex]\frac{50}{70}=e^{5k}[/tex] and take the natural log of both sides:

[tex]ln(\frac{5}{7})=ln(e^{5k})[/tex]

Since you're learning logs, I'm assuming that you know that a natural log and Euler's number, e, "undo" each other (just like taking the square root of something squared).  That gives us:

[tex]-.3364722366=5k[/tex]

Divide both sides by 5 to get that

k = -.0672944473

Now that we have a value for k, we can sub that in to solve for T(20):

[tex]T(20)=30+(100-30)e^{-.0672944473(20)}[/tex] which simplifies to

[tex]T(20)=30+70e^{-1.345888946}[/tex]

On your calculator, raise e to that power and multiply that number by 70:

T(20)= 30 + 70(.260308205) and

T(20) = 30 + 18.22157435 so

T(20) = 48.2°

Now we can use that k value to find out when (time) the temp of the object cools to 35°:

T(t) = 35

T₁ = 30

T₀ = 100

k = -.0672944473

t = ?

[tex]35=30+100-30)e^{-.0672944473t}[/tex] which simplifies to

[tex]5=70e^{-.0672944473t}[/tex]

Now divide both sides by 70 and take the natural log of both sides:

[tex]ln(\frac{5}{70})=ln(e^{-.0672944473t})[/tex] which simplifies to

-2.63905733 = -.0672944473t

Divide to get

t = 39.2 minutes

The temperature of the object after 20 minutes is 48.2° and temperature of body will be 35° after 39.2 minutes.

The formula can be expressed as:

[tex]\[ \frac{dT}{dt} = -k(T - T_a) \][/tex]

where:

[tex]\( T \)[/tex] is the temperature of the object at time [tex]\( t \)[/tex],

[tex]\( T_a \)[/tex] is the ambient temperature,

[tex]\( k \)[/tex] is a positive constant that depends on the characteristics of the object and the environment.

First, we need to find the constant [tex]\( k \)[/tex]. We have the following data:

Initial temperature of the object, [tex]\( T_0 = 100^\circ \)[/tex],

Temperature of the object after 5 minutes, [tex]\( T_1 = 80^\circ \)[/tex],

Ambient temperature, [tex]\( T_a = 30^\circ \)[/tex],

Time [tex]\( t_1 = 5 \)[/tex] minutes.

Using the integrated form of Newton's law of cooling, we have:

[tex]\[ T = T_a + (T_0 - T_a)e^{-kt} \][/tex]

Plugging in the values for [tex]\( T_1 \)[/tex] and [tex]\( t_1 \)[/tex], we get:

[tex]\[ 80 = 30 + (100 - 30)e^{-k \cdot 5} \][/tex]

Solving for [tex]\( k \)[/tex], we find:

[tex]\[ 50 = 70e^{-5k} \][/tex]

[tex]\[ e^{-5k} = \frac{50}{70} \][/tex]

[tex]\[ -5k = \ln\left(\frac{50}{70}\right) \][/tex]

[tex]\[ k = -\frac{1}{5}\ln\left(\frac{50}{70}\right) \][/tex]

Now that we have [tex]\( k \)[/tex], we can find the temperature after 20 minutes [tex]\( t_2 = 20 \)[/tex] minutes:

[tex]\[ T_2 = 30 + (100 - 30)e^{-k \times 20} \][/tex]

Substituting [tex]\( k \)[/tex] into the equation, we get:

[tex]\[ T_2 = 30 + (100 - 30)e^{\frac{1}{5}\ln\left(\frac{50}{70}\right) \times 20} \][/tex]

[tex]\[ T_2 = 30 + 70e^{\frac{20}{5}\ln\left(\frac{50}{70}\right)} \][/tex]

[tex]\[ T_2 = 30 + 70e^{4\ln\left(\frac{50}{70}\right)} \][/tex]

[tex]\[ T_2 = 30 + 70\left(\frac{50}{70}\right)^4 \][/tex]

[tex]\[ T_2 = 48.2^\circ[/tex]

Now, we need to solve for the time [tex]\( t_3 \)[/tex] when the temperature of the object is [tex]\( 35^\circ \)[/tex]:

[tex]\[ 35 = 30 + (100 - 30)e^{-kt_3} \][/tex]

[tex]\[ 5 = 70e^{-kt_3} \][/tex]

[tex]\[ e^{-kt_3} = \frac{5}{70} \][/tex]

[tex]\[ -kt_3 = \ln\left(\frac{5}{70}\right) \][/tex]

[tex]\[ t_3 = -\frac{1}{k}\ln\left(\frac{5}{70}\right) \][/tex]

Substituting [tex]\( k \)[/tex] into the equation, we get:

[tex]\[ t_3 = -\frac{5}{\ln\left(\frac{50}{70}\right)}\ln\left(\frac{5}{70}\right) \][/tex]

[tex]\ln\left(\frac{50}{70}\right)} = -0.336[/tex]

[tex]\ln\left(\frac{5}{70}\right) = -2.639[/tex]

[tex]|\[ t_3 = -\frac{5}{(-0.336)\right)} \times\ -2.639}|[/tex]

[tex]\ t_3 = 39.2 \text{minutes}[/tex]

s = 3x - 15 is the required expression for perimeter of table top

Perimeter of square tabletop is 240 cm

Solution:

A square tabletop has an area given as:

[tex](9x^2 - 90x+225) cm^2[/tex]

The dimensions  of the tabletop have the form cx - di  ,where cand d are whole numbers

To find perimeter of tabletop when x = 25 centimeters

Let us first find the length of each side of square

Given area is:

[tex]area = (9x^2 - 90x+225)[/tex]

We know that,

[tex]area = (side)^2 = s^2[/tex]

Therefore,

[tex]s^2 = (9x^2 - 90x+225)\\\\s^2 = (3x - 15)(3x - 15)\\\\s^2 = (3x - 15)^2[/tex]

Taking square root on both sides,

s = 3x - 15

The above expression is the required expression for perimeter of table top

To find perimeter when x = 25 centimeter

The perimeter of square is given as:

[tex]perimeter = 4s[/tex]

perimeter = 4(3x - 15)

Substitute x = 25

perimeter = 4(3(25) - 15)

perimeter = 4(60) = 240

Therefore perimeter of square tabletop is 240 cm

Step-by-step explanation:

I hope you are able to understand the answer

A student asked to verify a Pythagorean identity, which typically involves applying the Pythagorean Theorem, a² + b² = c², to trigonometric functions. However, the specific identity to be verified was not provided in the question.

The question involves verifying a Pythagorean identity, which likely refers to an equation involving the use of the Pythagorean Theorem. The Pythagorean Theorem states that for a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). The theorem is expressed mathematically as a² + b² = c².

However, the student's question seems incomplete as it does not provide the specific identity they wish to verify. Common Pythagorean identities include trigonometric identities that relate sine, cosine, and tangent to one another using the Pythagorean Theorem, such as sin²(x) + cos²(x) = 1 for any angle x. To verify a Pythagorean identity, one typically substitutes the trigonometric functions with their corresponding right triangle side ratios and demonstrates that the equation holds true.

Answer:

all work is shown and pictured

Answer:

$6770.65

Step-by-step explanation:

The purchase price is $20,300

The depreciation rate is 9.5% per year, in decimal, that is:

9.5/100 = 0.095

We want to find the value of car after 11 years. We will use the compound decay formula:

[tex]F=P(1-r)^t[/tex]

Where

F is the future value (what we want to find after 11 years)

P is the present value, purchase price (20,300 given)

r is the rate of depreciation (r = 0.095)

t is the time in years ( t = 11)

Substituting, we solve for F:

[tex]F=P(1-r)^t\\F = 20,300(1-0.095)^{11}\\F=20,300(0.905)^{11}\\F=6770.65[/tex]

Thus,

The value of the car would be around $6770.65, after 11 years

Answer:

f(x) = (x - 2)²

Step-by-step explanation:

f(x) = x² - 4x + 4 is a perfect trinomial.

You know a trinomial is perfect when double the square root of the first term multiplied by the square root of the last term equals the middle term.

As an equation, a trinomial in the form ax² + bx + c = 0:

2√a√c = b is a perfect trinomial.

Perfect trinomial are factored in this form:

(√a ± √c )(√a ± √c ) = (√a ± √c )²

Whether the sign is + or - depends on if the middle term is positive or negative.

In f(x) = x²-4x+4, the middle term is a negative.

The square root of the first term is 1.

The square root of the second term is 2.

The factored form is (x - 2)²

The polynomial equation with zeroes 2i, -2i, 2 is [tex]x^3 -2x^2 + 4x - 8 = 0[/tex]

Given that zeros of polynomial are 2i, -2i, 2

To find: polynomial equation in standard form

zeros of polynomial are 2i, -2i, 2. So we can say,

x = 2i

x = -2i

x = 2

Or x - 2i = 0 and x + 2i = 0 and x - 2 = 0

Multiplying the above factors, we get the polynomial equation

[tex](x - 2i)(x + 2i)(x - 2) = 0\\[/tex] ------- eqn 1

Using a algebraic identity,

[tex](a - b)(a + b) = a^2 - b^2[/tex]

Thus [tex](x - 2i)(x + 2i) = x^2 - (2i)^2[/tex]

We know that [tex]i^2 = -1[/tex]

[tex]Thus (x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 -4(-1) = x^2 + 4[/tex]

Substitute the above value in eqn 1

[tex](x^2 + 4)(x - 2) = 0[/tex]

Multiply each term in first bracket with each term in second bracket

[tex]x^3 -2x^2 + 4x - 8 = 0[/tex]

Thus the required equation of polynomial is found

Answer:

-(x^2+4)

Step-by-step explanation:

Answer:

(-x-2i)(x-2i)

Step-by-step explanation:

Answer:

B.

Step-by-step explanation:

Try solving:

13x - 5 = - 5 + 13x

13x - 13x = -5 + 5

0 = 0.

Therefore it is an identity.

We could make x any real value and the identity would hold.

The given equation 13x - 5 = -5 + 13x is an identity with infinite solutions. The option B is correct.

The given equation is 13x - 5 = -5 + 13x. To determine what kind of solution this equation has, we should simplify and solve for x. Starting by adding 5 to both sides of the equation to cancel out the -5, we get 13x = 13x. This results in the terms involving x on both sides of the equation being identical. Hence, no matter what value of x we substitute, the equation will always hold true.

Therefore, the equation represents an identity, meaning that it is true for all real numbers and there are infinite solutions. This is because subtracting or adding equal amounts on both sides of an equation does not change its solutions, leading to an expression of an identity.

The person has to travel 40 feet from the bottom to  the top of the escalator

Solution:

Given that escalator lifts people to the second floor of a building, 20 ft

above the first floor

The escalator rises at a 30° angle

To find: Distance person travel from the bottom to  the top of the escalator

The above scenario forms a right angled triangle where the escalator follows the path of the hypotenuse.

The diagram is attached below

In the right angled triangle ABC,

AC represents Distance person travel from the bottom to  the top of the escalator

AB = 20 feet

angle ACB = 30 degree

We know that,

[tex]sin \theta = \frac{opposite}{hypotenuse}[/tex]

[tex]sin 30 = \frac{20}{AC}[/tex]

[tex]\frac{1}{2} = \frac{20}{AC}\\\\AC = 20 \times 2 = 40[/tex]

So the person has to travel 40 feet from the bottom to  the top of the escalator

Answer:

Correct Pairs:

E(4, 4), F(8, 3), G(10, 6), and H(8, 6) ⇒ a translation 7 units right

E(3,4), F(-1,3), G(-3, 6), and H(-1,6) ⇒ a reflection across the y-axis

E(-3,-4), F(1,-3), G(3,-6), and H(1,-6) ⇒ a reflection across the x-axis

E(-3,-1), F(1,-2), G(3, 1), and H(1,1) ⇒ a translation 5 units down

Step-by-step explanation:

In Mathematics Geometry, translation just means 'moving'. A move without altering the size, rotation or anything else. When you perform a translation on a shape, the coordinates of that shape will change.

Translation right means you would add the translated unit to the x-coordinates of the of the point, let say P(x, y), in the original object.

Translation down means you would subtract the translated unit from the y-coordinates of the of the point, let say P(x, y), in the original object.

In Mathematics, a reflection just means a 'flip' over a line.

A reflection across x axis means, if a point P(x, y) is reflected across the x-axis, the x coordinate remains the same, while y coordinate changes its sign. i.e. the point (x, y) is changed to (x, -y).

A reflection across y axis means, if a point P(x, y) is reflected across the y-axis, the y coordinate remains the same, while x coordinate changes its sign. i.e. the point (x, y) is changed to (-x, y).

Now, lets head towards the solution:

Analyzing "a translation 7 units right"

As the given ABCD Quadrilateral has vertices as A (-3, 4),  B (1, 3), C (3, 6) and D (1, 6).

As EFGH is congruent to ABCD.

And

A translation of ABCD 7 units right would bring the following transformation:

A (-3, 4),  B (1, 3), C (3, 6) and D(1, 6) ⇒ A'(4, 4), B'(8, 3), C'(10, 6), and D'(8, 6)

As EFGH ≅ ABCD

So,

Here are the matching vertices when a translation 7 units right is made:

E(4, 4), F(8, 3), G(10, 6), and H(8, 6) ⇒ a translation 7 units right

Analyzing "a reflection across the y-axis"

As the given ABCD Quadrilateral has vertices as A (-3, 4),  B (1, 3), C (3, 6) and D (1, 6).

As EFGH is congruent to ABCD.

And

A reflection of ABCD across the y-axis would bring the following transformation:

A(-3, 4), B (1, 3), C(3, 6) and D(1, 6) ⇒ A' (3, 4), B'(-1, 3), C'(-3, 6) and D'(-1, 6)

As EFGH ≅ ABCD

So,

Here are the matching vertices when a reflection across the y-axis is made:

E(3,4), F(-1,3), G(-3, 6), and H(-1,6) ⇒ a reflection across the y-axis

Analyzing "a reflection across the x-axis"

As the given ABCD Quadrilateral has vertices as A (-3, 4),  B (1, 3), C (3, 6) and D (1, 6).

As EFGH is congruent to ABCD.

And

A reflection of ABCD across the x-axis would bring the following transformation:

A(-3, 4), B (1, 3), C(3, 6) and D(1, 6) ⇒ A' (-3, -4), B'(1, -3), C'(3, -6) and D'(1, -6)

As EFGH ≅ ABCD

So,

Here are the matching vertices when a reflection across the x-axis is made:

E(-3,-4), F(1,-3), G(3,-6), and H(1,-6) ⇒ a reflection across the x-axis

Analyzing "a translation 5 units down"

As the given ABCD Quadrilateral has vertices as A (-3, 4),  B (1, 3), C (3, 6) and D (1, 6).

As EFGH is congruent to ABCD.

And

A translation of ABCD 5 units down brings the following transformation:

A (-3, 4),  B (1, 3), C (3, 6) and D(1, 6) ⇒  A' (-3, -1),  B'(1, -2), C'(3, 1) and D'(1, 1)

As EFGH ≅ ABCD

So,

Here are the matching vertices when a translation 5 units down is made:

E(-3,-1), F(1,-2), G(3, 1), and H(1,1) ⇒ a translation 5 units down

Here is summary of matched Pairs:

E(4, 4), F(8, 3), G(10, 6), and H(8, 6) ⇒ a translation 7 units right

E(3,4), F(-1,3), G(-3, 6), and H(-1,6) ⇒ a reflection across the y-axis

E(-3,-4), F(1,-3), G(3,-6), and H(1,-6) ⇒ a reflection across the x-axis

E(-3,-1), F(1,-2), G(3, 1), and H(1,1) ⇒ a translation 5 units down

Keywords: reflection, translation, transformation

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Answer:

[tex]\sqrt{85}[/tex]

Step-by-step explanation:

Given

x - [tex]\frac{1}{x}[/tex] = 9 ← square both sides

(x - [tex]\frac{1}{x}[/tex])² = 9²

x² - 2 + [tex]\frac{1}{x^2}[/tex] = 81 ( add 2 to both sides )

x² + [tex]\frac{1}{x^2}[/tex] = 83

Now

(x + [tex]\frac{1}{x}[/tex])² = x² + [tex]\frac{1}{x^2}[/tex] + 2, thus

x² + [tex]\frac{1}{x^2}[/tex] = 83 + 2 = 85

(x + [tex]\frac{1}{x}[/tex] )²= 85 ( take the square root of both sides)

x + [tex]\frac{1}{x}[/tex] = [tex]\sqrt{85}[/tex]

The missing question is find x , y and the measure of each angle

The values of x and y are x = 28 and y = 30

The measures of ∠ABC is 26°, ∠CBD is 64° and ∠DBE is 116°

Step-by-step explanation:

Let us revise the meaning of complementary angles and supplementary angles

∵ ∠ABC = x

∵ ∠CBD = 2y + 4

∵ ∠ABC and ∠CBD are complementary

- That means their sum is 90°, add their values and equate

   the sum by 90

∴ x + (2y + 4) = 90

∴ x + 2y + 4 = 90

- Subtract 4 from both sides

∴ x + 2y = 86 ⇒ (1)

∵ ∠DBE = (3y + x)

∵ ∠CBD and ∠DBE are supplementary

- That means their sum is 180°, add their values and equate

   the sum by 180

∵ ∠CBD = (2y + 4)

∴ (2y + 4) + (3y + x) = 180

- Add like terms

∴ x + 5y + 4 = 180

- Subtract 4 from both sides

∴ x + 5y = 176 ⇒ (2)

Now we have a system of equations to solve them

Subtract equation (1) from equation (2) to eliminate x

∵ 3y = 90

- Divide both sides by 3

∴ y = 30

- Substitute the value of y in equation (1) to find x

∵ x + 2(30) = 86

∴ x + 60 = 86

- Subtract 60 from both sides

∴ x = 26

∵ ∠ABC = x

∴ The measure of angle ABC is 26°

∵ ∠CBD = 2y + 4

∴ ∠CBD = 2(30) + 4 = 60 + 4 = 64°

∴ The measure of angle CBD is 64°

∵ ∠DBE = 3y + x

∴ ∠DBE = 3(30) + 26 = 90 + 26 = 116°

∴ The measure of angle DBE is 116°

The values of x and y are x = 28 and y = 30

The measures of ∠ABC is 26°, ∠CBD is 64° and ∠DBE is 116°

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The question involves using the properties of complementary and supplementary angles to form equations and solve for the variables x and y.

This question is about solving for variables using information about complementary and supplementary angles. A pair of angles are complementary if the sum of their measures is 90 degrees, and supplementary if the sum of their measures is 180 degrees.

In this case, ∠ABC= x and ∠CBD=(2y+4) are complementary, so their sum is 90 degrees. This gives us the equation: x + 2y + 4 = 90.

Similarly, ∠CBD and ∠DBE=(3y+x) are supplementary, meaning their sum is 180 degrees. This gives us the second equation: 2y + 4 + 3y + x = 180.

By solving these two equations, we can determine the values of x and y.

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