Which uses the GCF to generate an expression equivalent to 8/3−
2/3x?

Step-by-step explanation:

Using perfect square trinomial rules,

[tex](a + b) {}^{2} = {a}^{2} + 2ab + {b}^{2} [/tex]

Here, a is sin x

b is cos x

So

[tex]( \sin(x) + \cos(x) ) {}^{2} = \sin {}^{2} (x) + 2 \sin(x) \cos(x) + \cos {}^{2} (x) [/tex]

We know

[tex] \sin {}^{2} (x) + \cos {}^{2} (x) = 1[/tex]

So we know have

[tex]1 + 2 \sin(x) \cos(x) [/tex]

[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]

So our final answer is

[tex]1 + \sin(2x) [/tex]

Based on the box-and-whisker plot, the numbers that are considered possible outliers include the following: B. 31, 87, 90.

In Mathematics, an outlier is also referred to as anomalous data and it can be defined as a numerical value that is either unusually too small or large (big) in comparison with the overall pattern of the numerical values contained in a data set.

In Mathematics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

By critically observing the box-and-whisker plots or box plot, we can logically deduce that 31, 87, and 90 are possible outliers.

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Answer: B

Step-by-step explanation:

(7x^3-10x^2- 5)-(2x^3+7x^2-9)

= 5x^3 -3x^2-14

The correct answer is option a. 1 - e^-2.

To find the probability of a random variable X with exponential distribution, we use the following formula:P[X > x] = e^(-λx)Where λ is the rate parameter and x is the value we are trying to find the probability of.

In this case, we are given that the mean of the distribution is 1, so we can use this information to find the rate parameter:λ = 1/mean = 1/1 = 1Now, we can plug in the values for λ and x into the formula to find the probability:P[X > 2] = e^(-1*2) = e^-2 = 0.1353Therefore, the correct answer is option a. 1 - e^-2.

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In respοnse tο the query, we can state that While persistence might be equatiοn crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.

A. Flexibility fοsters inventiοn and creativity.

In a math equatiοn, twο assertiοns are cοnnected by the equals sign (=), which denοtes equivalence. A mathematical assertiοn used in algebraic equatiοns establishes the equivalence οf twο mathematical statements. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign creates a space between the values 3x + 5 and 14.

Tο cοmprehend the relatiοnship between the twο sentences written οn οppοsing sides οf a letter, utilise a mathematical fοrmula. The lοgο and the specific prοgramme typically cοrrespοnd. An illustratiοn wοuld be 2x - 4 = 2.  

Peοple and οrganizatiοns whο are flexible are better equipped tο adjust tο shifting cοnditiοns, explοre new avenues, and adapt. Its adaptability encοurages experimentatiοn and taking chances, which can result in fresh, creative ideas.

Fοr reaching οbjectives and cοmpleting wοrk quickly, realistic expectatiοns and well-οrganized preparatiοn are crucial, but they may nοt always fοster creativity and inventiοn. While persistence might be crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.

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The sοlutiοn οf the given prοblem οf expressiοn cοmes οut tο be examples abοve are equal tο 16t - 0.25.

Mathematical prοcesses like multiplying, dividing, adding, and nοw even deleting are required. If they were merged, the fοllοwing claim wοuld be made: An equatiοn, sοme statistics, and a mathematical fοrmula Values, cοmpοnents, and mathematical prοcesses like additiοns, subtractiοns, reversals, algebraic fοrmulas, and divisiοns make up a statement οf truth. Wοrds and phrases can be rated and analyzed.

Here,

We pοssess 16t – 0.25.

A) The fοrmula 42t/40.5 equals (22t)² / (20.5) = 24t / (20.5) = 16t / (20.5)4 = 16t / 4 = 4(16t).

Expressiοn A is nοt equal because 4(16t) is nοt the same as 16t - 0.25.

B) [tex](16)^t/2^4 = (2^4)^t / 2^4[/tex]

=> [tex]2^{4t} / 2^4 = 2^{(4t-4)} = 16^{(t-1)} (t-1).[/tex]

Expressiοn B is not equivalent because 16(t-1) is nοt equal to 16(t - 0.25).

C) [tex]16^t/2 = (2^4)^t / 2[/tex]

[tex]= > 2^{4t} / 2 = 2^{(4t-1)} = 16^t \times 0.5.[/tex]

Expression C is nοt equivalent because 16t * 0.5 does not equal 16t - 0.25.

Hence, As a result, nοne of the examples above are equal to 16t - 0.25.

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Answer:

Step-by-step explanation:

For the airplane, we have v = -453.01, 79.88. Similarly, for the wind, we have w = -75.18, -27.36.

The actual speed would be 530.79 mph.

The actual ground speed and direction of the airplane are 530.79 mph and 174.32 deg, respectively.

The component form of the velocity of the airplane can be found by using the formula:

v = r < cos theta, sin theta >, where r is the speed and theta is the bearing. For the airplane, we have v = 460 < cos 170, sin 170 > = <-453.01, 79.88>. Similarly, for the wind, we have w = 80 < cos 200, sin 200 > = <-75.18, -27.36>.

To find the actual ground speed and direction of the airplane, we need to add the velocity vector of the airplane and the wind vector. This gives us the velocity vector = v + w = <-453.01, 79.88> + <-75.18, -27.36> = <-528.19, 52.52>.

The actual speed can be found by taking the magnitude of the velocity vector, which is given by |v+w| = sqrt((-528.19)^2 + (52.52)^2) = 530.79 mph.

The actual direction can be found by using the formula theta = arctan(y/x), where x and y are the x and y components of the velocity vector. In this case, we have theta = arctan(52.52/-528.19) = -5.68 deg. However, since the velocity vector is in the third quadrant, we need to add 180 deg to get the actual direction, which is 180 + (-5.68) = 174.32 deg.

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Answer:

infinity

Step-by-step explanation:

There are "infinity" rational numbers between 0 and 5​.

What are rational numbers :

A rational number is one that has the form p/q, where p & q are both integers and q is not zero.

How to find rational numbers :

The denominators must be equal to get the rational numbers between two rational numbers with differing denominators.

Finding the LCM of the denominators or multiplying the denominators of one to both the numerator and denominator of the other are two options for equating the denominators.

Answer:

Step-by-step explanation:

Answer:

Let's start by using the formula for mixing two solutions to set up an equation. The formula is:

(concentration of solution 1) x (volume of solution 1) + (concentration of solution 2) x (volume of solution 2) = (concentration of resulting solution) x (total volume of resulting solution)

In this problem, solution 1 is the 25% sugar syrup, solution 2 is the 10% sugar syrup, and the resulting solution is 15% sugar syrup. We are also given that the total volume of the resulting solution is 120 grams. Let's use x to represent the volume of the 10% sugar syrup that is mixed with the 25% sugar syrup. Then we can write:

(0.25)(120-x) + (0.10)(x) = (0.15)(120)

Simplifying this equation, we get:

30 - 0.25x + 0.10x = 18

0.15x = 12

x = 80

Therefore, we need to mix 80 grams of the 10% sugar syrup with 40 grams of the 25% sugar syrup to get 120 grams of 15% sugar syrup.

To find the amount of pure sugar in the 120 grams of 15% syrup, we can use the fact that the concentration of pure sugar in the resulting solution is 15%. That means that 15% of the 120 grams is pure sugar. We can find this by multiplying 120 by 0.15:

120 x 0.15 = 18

So there are 18 grams of pure sugar in the 120 grams of 15% syrup.

The number of adults is 120, and the total money made from ticket sales is £1120.

The mathematical connection between two or more numbers is called a ratio, and it is represented as the product of the division of two values. Several formats, such as fractions, decimals, or percentages, can be used to express ratios. Mathematicians employ ratios in many different areas, including geometry, probability, and finance. The connection between the lengths of two or more sides of a form is described in geometry using ratios. Ratios, sometimes in the form of odds, are used in probability to indicate the possibility of an event occurring.

Let x be the number of adults.

Given that, ratio of children to adults is 2 : 3.

Thus,

2/3 = 80/x

Cross-multiplying gives:

2x = 240

x = 120

The number of adults is 120.

Each child ticket costs a quarter of the adult ticket, so the cost of a child ticket is:

1/4 * £8 = £2

The total money made from child tickets is:

£2 * 80 = £160

The total money made from adult tickets is:

£8 * 120 = £960

Total money made from ticket sales is:

£160 + £960 = £1120

Hence, the total money made from ticket sales is £1120.

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[tex]10^{32} \times 10^{36}[/tex] can be written as [tex]10^{68}[/tex] using a single exponent.

What is an exponents?

In mathematics, exponents are a way to indicate the repeated multiplication of a number or phrase.

Exponents are numbers that are superscripted above other numbers. In other words, it denotes that a certain level of power has been conferred upon the base. Index and power are other names for the exponent. If m is a positive number and n is its exponent, the expression Mn means that m has been multiplied by itself n times.

Exponents are required for a more comprehensible representation of numerical quantities. Repeated multiplication is simple to write down when using exponents. If both n and x are positive integers, the expression xn means that x has been multiplied by itself n times.

When multiplying two numbers with the same base, we can add their exponents. Therefore:

[tex]10^{32} \times 10^{36 }= 10^{(32+36) }= 10^{68}[/tex]

Hence, [tex]10^{32} \times 10^{36}[/tex]can be written as [tex]10^{68}[/tex] using a single exponent.

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The first step would be to multiply both sides by 5 when solving the equation (3(x+4))/(5)=6.

This will allow us to eliminate the denominator on the left side of the equation and simplify the equation.After multiplying both sides by 5, the equation becomes: 3(x+4) = 30

Next, we can distribute the 3 on the left side of the equation: 3x + 12 = 30

Then, we can subtract 12 from both sides of the equation: 3x = 18

Finally, we can divide both sides by 3 to solve for x: x = 6

Therefore, the solution to the equation is x=6.

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Using the unit tile the area of the three diagrams is - square tile = 1 unit², rectangular tile = x unit², and square tile = x² unit².

What is area?

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.

For the square tile with length and breadth = 1 unit, the area is simply the product of the length and breadth, which is 1 unit × 1 unit = 1 unit².

This is the same as the area of the unit tile, so we don't really need to use it to find the area of this tile.

For the rectangular tile with length = x units and breadth = 1 unit, we can use x unit tiles to cover the length, and 1 unit tiles to cover the breadth.

Therefore, the area of the rectangular tile is x unit × 1 unit = x unit².

For the square tile with length and breadth = x units, we can use x unit tiles to cover the length, and x unit tiles to cover the breadth.

Therefore, the area of the square tile is x unit × x unit = x² unit².

In general, for any tile with length = a units and breadth = b units, the area is given by the product of the length and breadth, which is a unit × b unit = ab unit².

Therefore, we can use the unit tile to find the exact area of any tile, as long as we know its dimensions.

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Answer:

-14/16

-35/40

Step-by-step explanation:

7/-8 times [tex]\frac{2}{2}[/tex] is -14/16

7/-8 times [tex]\frac{5}{5}[/tex] is -35/40

Answer:

7-8-87-87-87-7-767&7&7

Step-by-step explanation:

そう笑ここに何を書けばいいのかわからないので、ここで私は答えを知っているふりをしています

The value of the series 4 + 9 + 14 + ... + (5n-1) using the summation notation is (5n^2 + 3n)/2.

The value of the series 4 + 9 + 14 + ... + (5n-1) can be found using the summation notation as follows:

Identify the first term, common difference, and number of terms in the series. The first term is 4, the common difference is 5, and the number of terms is n.

Use the summation notation to write the series as ∑_{i=1}^{n} (5i-1).

Use the formula for the sum of an arithmetic series to find the value of the series. The formula is S_n = n/2 (a_1 + a_n), where S_n is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term.

Substitute the values of n, a_1, and a_n into the formula and simplify. S_n = n/2 (4 + (5n-1)) = n/2 (5n+3) = (5n^2 + 3n)/2.

The value of the series is (5n^2 + 3n)/2.

Therefore, the value of the series 4 + 9 + 14 + ... + (5n-1) using the summation notation is (5n^2 + 3n)/2.

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The number of cases which Judge Jerry hears every hour is 2 2/19

An element of a whole or, more broadly, any number of equal pieces, is represented by a fraction.

When used in conversational English, a fraction indicates the number of components of a particular size, as in one-half, eight-fifths, and three-quarters.

How to calculate:

GIven that he tries 5 cases in 2 3/8 hours

The unit rate of cases per hour is calculated by dividing the number of cases by the number of hours.

5/(2 3/8) = 5/(19/8) = 5 * 8/19 = 40/19 = 2 2/19

Therefore, the number of cases which Judge Jerry hears every hour is 2 2/19

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The zeros of the polynomial x4 + 2x3 + 22x2 + 50x - 75 = 0 are x = 3, x = -5, x = ±√(37).

To find all the zeros (real and complex) of the polynomial x^(4)+2x^(3)+22x^(2)+50x-75=0, we can use the Rational Root Theorem and synthetic division.

The Rational Root Theorem states that if p/q is a rational root of a polynomial equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

The factors of the constant term -75 are: ±1, ±3, ±5, ±15, ±25, ±75
The factors of the leading coefficient 1 are: ±1

Therefore, the possible rational roots of the polynomial are: ±1, ±3, ±5, ±15, ±25, ±75

We can use synthetic division to test each of these possible roots until we find one that is a root. Once we find a root, we can use synthetic division again to divide the polynomial by the factor (x - root) to get a smaller polynomial, and then repeat the process until we have found all the roots.

Using synthetic division, we find that 3 is a root of the polynomial. Dividing the polynomial by (x - 3) gives us a smaller polynomial: x^(3)+5x^(2)+37x+25=0

We can repeat the process with this smaller polynomial to find the remaining roots. Using synthetic division again, we find that -5 is a root of the smaller polynomial. Dividing the smaller polynomial by (x + 5) gives us an even smaller polynomial: x^(2)+37=0

This polynomial has no real roots, but it has two complex roots: x = ±√(-37) = ±√(37)i

So, the complete list of zeros (real and complex) of the original polynomial is: 3, -5, ±√(37).

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The polar coordinates corresponding to the rectangular coordinates (0, 5) are (5, π/2).

To find the distance from the origin to the point (r), we can use the Pythagorean theorem. The distance from the origin to a point (x, y) is given by the formula √(x² + y²).

In this case, since the x-coordinate is 0, we only need to find the distance from the origin to the y-coordinate. Therefore, r = √(0² + 5²) = 5.

To find the angle (θ) that the line from the origin to the point makes with the positive x-axis, we can use trigonometry.

However, this is undefined since we cannot divide by zero. Therefore, we need to use a special case. Since the x-coordinate is 0, the point lies on the y-axis.

Therefore, the angle θ is either π/2 or 3π/2. However, we are given the constraint 0 ≤ θ < 2π. Therefore, the angle θ = π/2 since it satisfies the constraint.

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Complete Question:

Convert the following rectangular coordinates into polar coordinates. Always choose 0 ≤ θ < 2 π . (a) ( 0 , 5 )

The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.

Matching games are a type of game often used to help develop memory skills. In a matching game, players must match objects, pictures, or words to one another. This can involve matching a picture to a word, or a word to a definition. Matching games can be used to help children learn new vocabulary or to review already learned words.

The card with 2⁻³ has a base of 2 and an exponent of -3. This means that the value of the card is 1/8, which is the same as 2⁻³.

The card with (12)⁻² has a base of 12 and an exponent of -2. This means that the value of the card is 1/144, which is the same as 12⁻².

The card with (22)⁻⁴ has a base of 22 and an exponent of -4. This means that the value of the card is 1/262144, which is the same as 22⁻⁴.

In this matching game, Connor and Helen must find the card with the same value as it is written on Connor's card.

The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.

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The base, which is equal to 1.3, is one of the values in the equation g(x)=18(1.3)x.

The monthly growth rate of the insect population is represented by this number. Since 1.3 is 1 + 30% represented as a decimal, the population is specifically growing by 30% each month. When the growth rate is compounded each month, this indicates that the insect population is expanding quickly over time.

If we enter x=3 into the function, for instance, we obtain g(3)=18(1.3)3=18(2.197)=39.546. This indicates that the anticipated bug population after three months is 39,546. Pest control is frequently required to preserve ecosystem balance since this exponential growth might, if unchecked, result in a very huge insect population.

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The given function, n(x)=x^(2)+10x+24, can be written in vertex form by completing the square. Vertex form is given by y=a(x-h)^2+k.

The vertex is at (h,k). To find h and k, first find the average of the x-values of the two roots:

h = ( -b +- sqrt(b^2 - 4ac) ) / 2a
 = ( -10 +- sqrt( 10^2 - 4(1)(24) ) ) / 2(1)
 = ( -10 +- sqrt(100 - 96) ) / 2
 = ( -10 +- sqrt(4) ) / 2
 = ( -10 +- 2 ) / 2
 = -6

Substituting h into the equation y=a(x-h)^2+k, we have:

k = y - a(x-h)^2
 = n(x) - a(x+6)^2
 = x^2 + 10x + 24 - a(x+6)^2
 = 24 - a(x+6)^2

We know that when x=-6, k=24, so

24 = a( -6+6 )^2
24 = 36a
a = 2/3

Therefore, the equation in vertex form is y = 2/3(x+6)^2 + 24.

The vertex is (h,k) = (-6, 24).

The x-intercepts (s) are the roots of the equation, so they can be found by setting the equation equal to 0 and solving for x.

0 = x^2 + 10x + 24
0 = (x+6)(x+4)

Therefore, the x-intercepts are x=-6 and x=-4.

The y-intercept is when x=0, so it is y = 24.

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When copying segments and angles, drawing a ray with one endpoint step is the same.

A line segment is a part of a continuous line, having two fixed ends and a definite length.

When a common part in constructing a line segment and an angle is drawing a ray because a ray is a common part, when we see the figures of a segment and an angle we see a common a ray, so a ray is the common part in both.

Hence, when copying segments and angles, drawing a ray with one endpoint step is the same.

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a) The sales 2 months after the end of the campaign will be $29,600. b)  it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.

The sales 2 months after the end of the campaign can be found by plugging x = 2 into the equation:
S = 80,000e-0.5x
S = 80,000e-0.5*2
S = 80,000e-1
S = 80,000 * 0.37
S = 29,600

Therefore, the sales 2 months after the end of the campaign will be $29,600.

We can solve this by setting the equation equal to 1000 and solving for x:
1000 = 80,000e-0.5x
1000/80,000 = e-0.5x
0.0125 = e-0.5x
ln(0.0125) = ln(e-0.5x)
-4.81 = -0.5x
x = 9.62

Therefore, it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.

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The distance between the middle school and the fire station is 4.5 miles

Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

The distance between the middle school and the fire station is is the hypotenuse

therefore;

c² = 2² + 4²

c² = 4 + 16

c² = 20

c = √20

c = 4.5 miles(nearest tenth)

therefore the distance between the middle school and fire station is 4.5 miles

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Joey walked 110 meters from  A to B and Lana started at point A and walked in a straight line to point B which is a distance of about 80.6 meters, obtained using Pythagorean Theorem, therefore;

Joey walked about 29.4 meters further than Lana.

Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is the sum of the squares of the lengths of the legs of the right triangle.

The distance Joey walked = 20 m + 40 m + 50 m = 110 m

The distance Lana walked can be found as follows;

The similar triangles formed by the path Lana walked and the path Joey walked, indicates that the ratio of the base lengths of the right triangles are;

50/20 = 5/2

The base length of the larger right triangle 5/(2 + 5) × 40 m = 200/7 m

Base length of the smaller right triangle = 2/(5 + 2) × 40 m = 80/7 m

The length of the path Lana walked, l, found using Pythagorean Theorem is therefore;

The distance further Joey walked = 110 m - 80.6 m = 29.4 m

Joey

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The trinomial in standard form is x^2 + 10x + 25.

To express (x+5)^(2) as a trinomial in standard form, we need to expand the expression using the distributive property.

First, we will distribute the first term, x, to each term inside the parentheses:

(x+5)(x+5) = x(x) + x(5) + 5(x) + 5(5)

Next, we will simplify the terms:

= x^2 + 5x + 5x + 25

Finally, we will combine like terms to get the trinomial in standard form:

= x^2 + 10x + 25

Therefore, the trinomial in standard form is x^2 + 10x + 25.

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Answer:15.75 min

Step-by-step explanation:

7/8)60= 52.5

3/10)52.5= 15.75

Answer:

When you multiply two numbers with the same sign (either both positive or both negative), the rule is that the result is always positive.

For example, if you multiply +2 and +3, the result is +6 because both numbers have the same positive sign. Likewise, if you multiply -4 and -5, the result is +20 because both numbers have the same negative sign.

This rule applies to any two numbers with the same sign, regardless of their values or whether they are whole numbers, fractions, or decimals.

Given vectors are v1 = |2| v2 = |3| v3 = |1| |2| |1| |h| |-4| |-4| |4| Let's consider the vectors are linearly dependent. If they are linearly dependent, then the rank of the matrix will be less than 3. So, Values of the parameter h are |2| 3 1 |2| 1 h |-4| |-4| 4

For the matrix to have a rank of less than 3, it must satisfy the following condition : |2| 3 1 |2| 1 h |-4| |-4| 4It can be rewritten as follows: |2| 3 1 |2| 1 h |-4| |-4| 4It can be simplified as follows: |2| 3 1 |2| 1 h |-4| |-4| 4. The matrix can be reduced to the following form: |2| 3 1 |2| 1 h |-4| |-4| 4, By subtracting R1 from R2, we get: |-1| 0 |h-1| |-6| |-h| |-4| |-4| 4The determinant of this matrix is: -4h + 4 + 8h = 4h + 4

Now, let's evaluate the matrix's rank. We can observe that the matrix's second row is equal to twice the matrix's first row. So, the rank of the matrix is 2.Since the rank of the matrix is 2, the given vectors are linearly dependent if h = -1.

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