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Answer:

To prove that every bounded sequence contains a convergent subsequence, we will use the Bolzano-Weierstrass theorem.

Bolzano-Weierstrass theorem: Every bounded sequence in Euclidean space has a convergent subsequence.

Proof: Let (a_n) be a bounded sequence in Euclidean space, which means that there exists some M > 0 such that |a_n| <= M for all n. We will use the bisection method to construct a subsequence that converges to some limit L.

Step 1: Divide the interval [-M, M] into two subintervals, [-M, 0] and [0, M]. Since (a_n) is bounded, at least one of these intervals contains infinitely many terms of the sequence. Let's call the interval that contains infinitely many terms I_1.

Step 2: Divide I_1 into two subintervals of equal length, and again, at least one of these subintervals must contain infinitely many terms of the sequence. Let's call the interval that contains infinitely many terms I_2.

Step 3: Repeat this process for each subinterval, dividing it into two subintervals of equal length and choosing the one that contains infinitely many terms of the sequence. We obtain a sequence of nested intervals I_1 ⊃ I_2 ⊃ I_3 ⊃ ... that contain infinitely many terms of the sequence.

Step 4: Since the length of each interval is decreasing and they are nested, the intersection of all the intervals is a single point L. By construction, every term of the sequence (a_n) belongs to one of the intervals I_k, which means that there exists a subsequence (a_nk) that converges to L.

Therefore, every bounded sequence in Euclidean space has a convergent subsequence, as required.

Step-by-step explanation:

We can use the binοmial distributiοn tο mοdel this situatiοn, where each questiοn is a Bernοulli trial with a prοbability οf success (getting the cοrrect answer) οf 0.5.

A prοbability is a numerical representatiοn οf the likelihοοd οr chance that a specific event will take place. Bοth prοpοrtiοns ranging frοm 0 tο 1 and percentages ranging frοm 0% tο 100% can be used tο describe prοbabilities.

Let X be the number οf False answers in the cοmpleted exam paper.

Prοbability that all answers are False:

[tex]P(X = 4) = (0.5)^4 = 0.0625[/tex]

Prοbability that exactly three οr fοur answers are False:

P(X = 3 οr X = 4) = P(X = 3) + P(X = 4)

[tex]= 4C3(0.5)^3(0.5)^1 + (0.5)^4[/tex]

= 0.25 + 0.0625

= 0.3125

Prοbability that at least οne answer is True (which is the cοmplement οf the event that all answers are False):

P(at least οne True) = 1 - P(all False) = 1 - 0.0625 = 0.9375

Prοbability that the last answer is True:

This prοbability can be calculated using cοnditiοnal prοbability, where we find the prοbability that the last answer is True given that we knοw that at least οne answer is True:

P(last answer is True | at least οne True) = P(last answer is True and at least οne True) / P(at least οne True)

P(last answer is True and at least οne True) = P(last answer is True) = 0.5

P(at least οne True) = 0.9375 (calculated in part 3)

Sο, P(last answer is True | at least οne True) = 0.5 / 0.9375 = 0.5333 (rοunded tο fοur decimal places).

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The linear model is Population = 92 + 2.35x and the exponential model is Population= 92(1.0187)ˣ.

An exponential model implies that the change in y is proportionate to its present value, whereas a linear model assumes that the change in y is constant for each unit change in the explanatory variable (x). To put it another way, an exponential model assumes a varying rate of change, whereas a linear model implies a constant rate of change.

Let us suppose number of years = x.

Given that, the U.S. population in 1910 was 92 million people. In 1990, the population was 280 million.

From 1910 to 1990 is a period of 80 years, the increase in population over this period is 280 million - 92 million = 188 million.

Therefore, the average annual increase in population is 188 million / 80 years = 2.35 million.

Thus, the linear model is:

Population = 92 + 2.35x

The exponential model is:

The population grew from 92 million in 1910 to 280 million in 1990, a factor of 280/92 = 3.0435.

The period of growth is 80 years, so the annual growth rate can be calculated as:

rate = (3.0435)^(1/80) - 1 = 0.0187 or 1.87%

Population= 92(1.0187)ˣ

Hence, the linear model is Population = 92 + 2.35x and the exponential model is Population= 92(1.0187)ˣ.

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1. The prοbability that yοu draw a red marble 7/24

2. The prοbability οf chοοsing exactly οne οf each cοlοr is 63/253

3. The prοbability that yοu draw 5 green marbles in a rοw is 0.0057

Prοbability is a branch οf mathematics in which the chances οf experiments οccurring are calculated. It is by means οf a prοbability, fοr example, that we can knοw frοm the chance οf getting heads οr tails in the launch οf a cοin tο the chance οf errοr in research.

1. The prοbability οf drawing a red marble is given by the number οf red marbles divided by the tοtal number οf marbles in the jar:

P(red) = 7/(7+9+8) = 7/24

2. The prοbability οf chοοsing exactly οne οf each cοlοr, if yοu pick 3 marbles frοm the jar, can be fοund using the hypergeοmetric distributiοn. The tοtal number οf ways tο chοοse 3 marbles frοm 24 is:

C(24,3) = 2024

The number οf ways tο chοοse οne οf each cοlοr is:

C(7,1) * C(9,1) * C(8,1) = 798 = 504

Therefοre, the prοbability οf chοοsing exactly οne οf each cοlοr is:

P(1 οf each cοlοr) = 504/2024 = 63/253

3. The prοbability οf drawing 5 green marbles in a rοw withοut replacement can be fοund as fοllοws. Therefοre, the prοbability can be calculated as:

P(5 green in a rοw) = (9/24) * (8/23) * (7/22) * (6/21) * (5/20) = 0.0057 (rοunded tο fοur decimal places)

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The logarithmic form of the equation a^3 = y is log_a(y) = 3

A logarithmic equation is an equation that involves the logarithm of one or more variables.

From the question, we have the following parameters that can be used in our computation:

a^3 = y

Take the logarithm of both sides of the equation

So, we have

3log(a) = log(y)

Divide both side by log(a)

So, we have

3 = log_a(y)

Rewrte as

log_a(y) = 3

Hence, the equation is (b) 3 = log_a(y)

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The fourth expression 3⁻⁷/3⁻⁵ is equivalent to  3⁻².

What exactly are expressions?

In mathematics, an expression is a combination of numbers, variables, operators, and functions that are combined in a particular way to represent a mathematical value or relationship. Expressions can be simple or complex, and they can involve any number of mathematical operations.

Expressions can be used to represent many mathematical concepts:

Now,

To find the equivalent expression of 3⁻², we can simplify each of the given choices using the rules of exponents:

1. 3⁻²/3³ = 1/3²⁺³ = 1/3⁵

2. 3⁻⁷/3⁵ = 3⁻⁷⁻⁵ = 3⁻¹²

3. 3³/3⁻¹² = 3³⁺¹² = 3¹⁵

4. 3⁻⁷/3⁻⁵ = 3⁻⁷⁺⁵ = 3⁻²

Out of the given choices, the fourth choice 3⁻⁷/3⁻⁵ is equivalent to 3⁻².

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It will take Lush Gardens Co. 3 years and 3 months to settle the loan if it makes payments of $1,600 at the end of every month.

First, we need to find the amount of the loan.

Amount of loan = Total cost of the truck - Down payment

Amount of loan = $54,000 - $5,940 = $48,060

Next, we need to find the monthly interest rate. Since the interest is compounded semi-annually, we first need to find the semi-annual interest rate:

Semi-annual interest rate = 4.46% / 2 = 2.23%

Then, we can find the monthly interest rate using the formula:

Plugging in the numbers:

Monthly interest rate = (1 + 0.0223)^(1/12) - 1 = 0.001845

Now we can use the formula for the loan payment:

Loan payment = Monthly interest rate * Loan amount / (1 - (1 + Monthly interest rate)^(-n))

Where n is the number of months.

Plugging in the numbers:

$1,600 = 0.001845 * $48,060 / (1 - (1 + 0.001845)^(-n))

Solving for n using a financial calculator :

n = 38.98 months

Rounding up to the next payment period (which is 39 months), we get:

n = 3 years, 3 months

Therefore, it will take Lush Gardens Co. 3 years and 3 months to settle the loan if it makes payments of $1,600 at the end of every month.

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The height of the can should be approximately 7 inches to have a surface area as close to 100 square inches as possible.

What is cylinder?

A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases of the same size and shape, and a curved lateral surface connecting the bases.

The formula for the surface area of a cylinder is:

where r is the radius of the base, h is the height, and π is approximately 3.14.

In this case, the radius is given as 2 inches, and we want to find the height that will give us a surface area as close to 100 square inches as possible. So we need to solve for h in the equation:

100 = 2π(2)² + 2π(2)h

Simplifying this equation:

100 = 8π + 4πh

92 = 4πh

h = 23/π

h ≈ 7.32

Rounding this answer to the nearest inch, we get:

h ≈ 7 inches

Therefore, the height of the can should be approximately 7 inches to have a surface area as close to 100 square inches as possible.

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Answer:

Step-by-step explanation:

o find a linear function that can be used to predict the number of nations participating x years after 1988, we can use linear regression analysis. Using a spreadsheet program or calculator with linear regression capabilities, we obtain:

y = 1.9632x + 51.3276

where y is the predicted number of nations participating and x is the number of years after 1988.

To predict those years in which more than 100 nations will participate, we can substitute y = 100 into the linear function and solve for x:

100 = 1.9632x + 51.3276

48.6724 = 1.9632x

x ≈ 24.8

Therefore, more than 100 nations will participate in the Winter Games approximately 24.8 years after 1988, or in the year 2013.

Starting with the angle of -270°, adding 360° gives us 90° (which is a coterminal angle with -270°).

If we rotate 180° counter-clockwise from 90°, we end up at the angle of 270°. Therefore, the new measurement of the coterminal angle is 270°.

I hope this helps :)

Answer:

1. the radius is increasing at a rate of approximately 0.265 cm/s when the radius is 3 cm

2. the radius is increasing at a rate of approximately 0.159 cm/s when the circumference is 1 cm

Step-by-step explanation:

We can use the formulas for the area and circumference of a circle to solve these problems:

a. To find how fast the radius is changing when the radius is 3 cm, we can use the formula for the area of a circle:

A = πr^2

Taking the derivative of both sides with respect to time, t, we get:

dA/dt = 2πr dr/dt

where dr/dt is the rate of change of the radius.

We know that dA/dt = 5 cm²/s, and when the radius is 3 cm, we can plug in these values to solve for dr/dt:

5 = 2π(3) dr/dt

dr/dt = 5/(6π) cm/s ≈ 0.265 cm/s

Therefore, the radius is increasing at a rate of approximately 0.265 cm/s when the radius is 3 cm.

b. To find how fast the radius is changing when the circumference is 1 cm, we can use the formula for the circumference of a circle:

C = 2πr

Taking the derivative of both sides with respect to time, t, we get:

dC/dt = 2π dr/dt

where dr/dt is the rate of change of the radius.

We know that when the circumference is 1 cm, C = 1 cm, so we can plug in these values to solve for dr/dt:

1 = 2π dr/dt

dr/dt = 1/(2π) cm/s ≈ 0.159 cm/s

Therefore, the radius is increasing at a rate of approximately 0.159 cm/s when the circumference is 1 cm.

The unknown length in the rectangle is ? = 8cm

We we want to find the unknown length.

First, if we look just at the pink and yellow rectangles we can see that these have a combined area of:

A = 12cm² + 18cm²

A = 30cm²

And we know that one side measures 6cm, then the length of AD is:

AD*6cm = 30cm²

AD = 30cm²/6cm

AD = 5cm

Now, the yellow and purple rectangles have a combined area of:

A = 18cm² + 22cm² = 40cm²

Then we can write:

?*AD = 40cm²

?*5cm = 40cm²

? = 40cm²/5cm

? = 8cm

That is the unknown lenght.

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The algebraic expressions that are equivalent to 7x-2 are -2 + 7x, 14x/2 - 4/2 and 5x + 2x - 2

The expression in the question is given as

7x - 2

The above expression is linear

Here are some algebraic expressions that are equivalent to 7x-2:

14x/2 - 4/2

When evaluated, we have

14x/2 - 4/2 = 7x - 2

5x + 2x - 2:

We can break up 7x into 5x and 2x, then combine 5x and 2x to get 7x.

So, we have

5x + 2x - 2 = 7x - 2

Hence, the expressions are 14x/2 - 4/2 and 5x + 2x - 2

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Answer:

The answer would be 8.1% (8.108% if you need the exact percentage.)

Step-by-step explanation:

All you would need to do is add 7 + 18 + 12 which = 37, and divide 3/37, which in percentage that gives you 8.1%. (8.108% if you need exact percentage.)

Answer:

144ft^2

Step-by-step explanation:

Answer:

144^2 feet

Step-by-step explanation:

To find the area you would multiply length x width for your answer. I hope this helps!

Answer:

-16

Step-by-step explanation:

1. multiply the minus sign on both sides now we have 4x=-64

2. divide 4 on both sides

You're done! x=-16

Answer:

7+2=3x-2+2

9/3=3x/3

3=x

The minimum number of children in the colony is 47 given the ratio 6 : 4 : 3

Given that

Ratio = 6 : 4 : 3

We are given the ratio of male to female to children as 6:4:3.

Let the common ratio be x, so the actual number of males, females, and children respectively will be 6x, 4x, and 3x.

Total number of members in the colony = 6x + 4x + 3x = 13x.

We are given that the minimum number of members in the colony is 200.

So,

13x = 200

x = 200/13

The actual number of children in the colony 3x is

Children =  3 * (200/13) = 46.15

Children = 46.15

However, since we cannot have a fraction of a child, we round to the nearest whole number.

Therefore, the minimum number of children in the colony is 47.

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Answer:

160 miles

Step-by-step explanation:

To find the unit rate, we want to know how many miles can we go in i hour.

96 ÷ 1 [tex]\frac{1}{2}[/tex]

1 [tex]\frac{1}{2}[/tex] can be written

[tex]\frac{2}{2}[/tex] + [tex]\frac{1}{2}[/tex] = [tex]\frac{3}{2}[/tex]      1 is the same as [tex]\frac{2}{2}[/tex]

Now we have

96 ÷ [tex]\frac{3}{2}[/tex]  if we get a common denominator, in this case 2, we can just divide a cross.

96 is equal to [tex]\frac{192}{2}[/tex]   ([tex]\frac{96}{1}[/tex] x [tex]\frac{2}{2}[/tex])

Now we have

[tex]\frac{192}{2}[/tex] ÷ [tex]\frac{3}{2}[/tex] =  [tex]\frac{64}{1}[/tex] = 64

We now know the unit rate is 64 miles in one hour.

The distance in 2 [tex]\frac{1}{2}[/tex] hours would be

64 + 64 + [tex]\frac{1}{2}[/tex](64)

64 + 64 + 32

160

Helping in the name of Jesus.

The equation of the hyperbola x²/36 = 1. The equation of a horizontal line passing through (±6,0).

The resulting curve of a hyperbola consists of two separate branches that are mirror images of each other, and they are characterized by their asymptotes, which are straight lines that the curve approaches but never touches. The asymptotes intersect at the center of the hyperbola, which is also the point where the two branches are closest together.

Since the foci of the hyperbola are at (0,-6) and (0,6), and the transverse axis is vertical, the center of the hyperbola is at the midpoint of the foci, which is the point (0,0).

Also, since the asymptote of the hyperbola is the line y = -x, we know that the slopes of the asymptotes are ±1, and since the transverse axis is vertical, the slope of the transverse axis is 0. Therefore, we have a hyperbola with center (0,0), vertical transverse axis, and asymptotes y = x and y = -x.

The standard form of the equation for a hyperbola with center (0,0), transverse axis of length 2a, and asymptotes y = ±(a/x) is:

x²/a² - y²/b² = 1

where b² = a² - c², where c is the distance from the center to each focus.

In this case, since the distance between the foci is 12, we have:

c = 6

Also, since the transverse axis is vertical and the distance from the center to each vertex is a, we have:

a = distance from center to vertex = 6

Therefore, we have:

b² = a² - c² = 36 - 36 = 0

So the equation of the hyperbola is:

x²/36 - y²/0 = 1

Simplifying, we get:

x²/36 = 1

or

x² = 36

This is the equation of a horizontal line passing through (±6,0).

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Part 1

a.  In function notation, we have that R(3) > D(3)

b. In function notation, we have that R(0) - D(0) = 25

Part 2

The drone is 20 feet above the ground when the rocket hit the ground.

Part 3

a. R(t) = D(t) when the graphs intercept at t = 4.75 s

b.  It tell us that the drone and rocket are at the same height at that time.

A function notation is the representation of a statement as a mathematical equation using symbols.

Part 1

a. To write the statement at 3 seconds the toy rocket is higher than the drone in function notation.

Since

At t = 3, the toy rocket is higher than the drone.

So, in function notation, we have that R(3) > D(3)

b. To write the statement at the start the toy rocket is 25 feet above the drone.

At the start t = 0, and the toy rocket is 25 feet above the drone.

So, in function notation, we have that R(0) - D(0) = 25

Part 2.

To find the height of the drone when the rocket hit the ground,

Since

We find R(t) = 0 and find the time t, where it intercepts the height of the drone.

So, from the graph R(t) = 0 at t = 5. And at t = 5, D(t) = 20

So, the drone is 20 feet above the ground when the rocket hit the ground.

Part 3.

a. The value of t at which R(t) = D(t) is where the graphs intercept.

The graphs intercept at t = 4.75 s

b. Since R(t) = D(t) at t = 4.75 s, it tell us that the drone and rocket are at the same height at that time.

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Answer:

[tex]\boxed{\textsf{y = 3}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to solve for y.}[/tex]

[tex]\textsf{We should begin by \underline{simplifying} the equation.}[/tex]

[tex]\textsf{First, begin by dividing 750 from \underline{both sides} of the equation.}\\[/tex]

[tex]\Large\underline{\textsf{Divide:}}[/tex]

[tex]\mathtt{\frac{162000}{750} = \frac{750 \times 6^y}{750} .}[/tex]

[tex]\mathtt{216=6^y}[/tex]

[tex]\Large\underline{\textsf{Continue Dividing By 6:}}[/tex]

[tex]\mathtt{\frac{216}{6} = 36 (^{y-1})}[/tex]

[tex]\mathtt{\frac{36}{6} = 6 (^{y-2} )}[/tex]

[tex]\Large\underline{\textsf{Identify y:}}[/tex]

[tex]\mathtt{6^1 = 6. \ 6^{1+2} = 216. \ (6^3)}[/tex]

[tex]\boxed{\textsf{y = 3}}[/tex]

The simplification by expressing fractional exponents instead of radicals is a^1/2. b^1/2 or  (ab)^(1/2)

The expression in radical form was been provided as √ab and this is been expected from use to express it ionform of fractional exponent form.

√ab

note that;

√a = a^1/2

√ab = √a * √b

By utilixzing the properties of square root function, then we can have the expressions as ;

√ab = √a √b

√ab = a^1/2 * b^1/2

√ab =a^1/2. b^1/2

=a^1/2. b^1/2 or  (ab)^(1/2)

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value from least to greatest is

A<B<C<D

A fraction is a numerical quantity that represents a part of a whole or a ratio of two quantities. It is expressed in the form of one number, called the numerator, divided by another number, called the denominator, and is typically written as a/b, where "a" is the numerator and "b" is the denominator.

Part a)

Let 5.7 (bar on 7) be x

x=5.777777... .....(1)

Multiply the equation by 10,

10x=57.77777... ....(2)

(2)-(1),

9x=52

x=52/9=5.777777778

Partb)

Let 5.75 (bar on 75) be x

x=5.75757575... .....(1)

Multiply the equation by 100,

100x=575.75757575.. ....(2)

(2)-(1),

99x=569.99

x=190/33=5.757575758

Part c)

Let 5.75 (bar on 5) be x

x=5.7555555... .....(1)

Multiply the equation by 10,

10x=57.55555555... ....(2)

Multiply the equation by 10,

100x=575.5555.. ....(3)

(3)-(2),

90x=52

x=259/45=5.755555556

Hence, value from least to greatest is

A<B<C<D

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The answer of the given question based on the ordering from the least to greatest  the answer is the correct answer is (A) DBCA.

The decimal system is a number system that uses the base 10, meaning that there are 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) that can be used to represent any quantity. This system is also called the base-10 numeral system, denary system, or Hindu-Arabic numeral system.

In the decimal system, each digit represents a different power of 10. For example, in the number 357, the 7 represents the units, 5 represents the tens, and 3 represents the hundreds.

This system is widely used in everyday life, in which numbers are expressed in decimal form, such as money, time, and measurements. It is also used in mathematics, science, and engineering as the standard way of expressing numbers.

The correct order from least to greatest will be :

A. 5.7

D. 5.75

B. 5.75

C. 5.75

Therefore, the correct answer is (A) DBCA.

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Answer:

(x, y) = (-1/2, 5/2)

Step-by-step explanation:

To solve this system of equations using substitution, we can use the first equation to express y in terms of x, and then substitute that expression into the second equation to get an equation in terms of x alone.

Starting with the first equation, we have:

y = x + 3

We can now substitute this expression for y into the second equation:

3x + 3y = 6

3x + 3(x + 3) = 6

Simplifying the right-hand side:

3x + 3x + 9 = 6

6x = -3

x = -1/2

Now that we have solved for x, we can substitute this value back into the first equation to find y:

y = x + 3

y = -1/2 + 3

y = 5/2

Therefore, the solution to the system of equations is (x, y) = (-1/2, 5/2).

To check our solution, we can substitute these values back into both equations and verify that they hold true:

y = x + 3

5/2 = -1/2 + 3

5/2 = 5/2

This equation holds true, so the first equation is satisfied by our solution.

3x + 3y = 6

3(-1/2) + 3(5/2) = 6

-3/2 + 15/2 = 6

6 = 6

This equation also holds true, so the second equation is satisfied by our solution as well.

Therefore, we can conclude that our solution is correct.

The equations have two variables, x and y, and we can use substitution to solve for one variable in terms of the other. By substituting the expression for y into the second equation and solving for x, we get the value of x as -1/2. We can then substitute this value back into the first equation to get the value of y, which is 5/2. This means that the solution to the system of equations is (x, y) = (-1/2, 5/2) and we can check that this solution satisfies both equations.

Hope this helps! Sorry if it doesn't. If you need more help, ask me! :]

Answer: The solution to a system of equations in two variables represents the values of the variables that satisfy both equations simultaneously.

To check which equations might make up the system with a solution of (5,-19), we can substitute x = 5 and y = -19 into each equation and see if they are both satisfied.

Substituting x = 5 and y = -19 into the first equation, we get:

y = 2x - 23

-19 = 2(5) - 23

-19 = -13

This is not true, so the first equation is not part of the system.

Substituting x = 5 and y = -19 into the second equation, we get:

y = x - 17

-19 = 5 - 17

-19 = -12

This is not true, so the second equation is not part of the system.

Substituting x = 5 and y = -19 into the third equation, we get:

y = -7x + 16

-19 = -7(5) + 16

-19 = -19

This is true, so the third equation is one of the equations in the system.

Substituting x = 5 and y = -19 into the fourth equation, we get:

y = -21 - 9x

-19 = -21 - 9(5)

-19 = -64

This is not true, so the fourth equation is not part of the system.

Substituting x = 5 and y = -19 into the fifth equation, we get:

y = -3x - 6

-19 = -3(5) - 6

-19 = -21

This is not true, so the fifth equation is not part of the system.

Therefore, one possible system of equations with a solution of (5,-19) is:

y = -7x + 16

We would need another equation to form a complete system with a unique solution.

Step-by-step explanation:

Answer:

4.5

Step-by-step explanation:

6+2=8

8/2=4

1.5x4=6

6-1.5=4.5

Answer:

i don’t know sorry

Step-by-step explanation:

nothing