Graph the following system of equations. y = 3x + 15 3x + 3y = 9 What is the solution to the system? There is no solution. There is one unique solution (−3, 6). There is one unique solution (0, 15). There are infinitely many solutions.

If we restrict the domain of the function to the portion of the graph with a positive slope, the domain of the inverse function will be the range of the original function for values of x greater than 4, and its range will be all real numbers greater than or equal to 4.

The given function f(x) = |x – 4| + 6 is a piecewise function that contains an absolute value. The absolute value function has a V-shaped graph, and the slope of the graph changes at the point where the absolute value function changes sign. In this case, that point is x=4.

If we restrict the domain of f(x) to the portion of the graph with a positive slope, we are essentially considering the piece of the graph to the right of x=4. This means that x is greater than 4, or x>4.

The domain of the inverse function, f⁻¹(x), will be the range of the original function f(x) for values of x greater than 4. This is because the inverse function reflects the original function over the line y=x. So, if we restrict the domain of f(x) to values greater than 4, the reflected section of the graph will be the range of f⁻¹(x).

The range of f(x) is all real numbers greater than or equal to 6 because the absolute value function always produces a positive or zero value and when x is greater than or equal to 4, we add 6 to that value. The range of f⁻¹(x) will be all real numbers greater than or equal to 4, as this is the domain of the reflected section of the graph.

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Given statement solution is :- It takes approximately 1.122 seconds for the ball to reach its maximum height.

To find the time it takes for the ball to reach its maximum height, we need to determine the vertex of the parabolic function given by the equation h(t) = [tex]-4.9t^2 + 11t + 7.[/tex]

The vertex of a parabola in the form y = [tex]ax^2 + bx[/tex] + c is given by the formula t = -b / (2a).

Comparing the equation h(t) = [tex]-4.9t^2 + 11t + 7[/tex] to the standard form, we have:

a = -4.9

b = 11

Using the formula for the vertex, we can calculate the time it takes to reach the maximum height:

t = -11 / (2 * -4.9)

t = -11 / -9.8

t ≈ 1.122

Therefore, it takes approximately 1.122 seconds for the ball to reach its maximum height.

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Answer:

Step-by-step explanation:

Answer:

D)

Step-by-step explanation:

ΔSTU is translated to ΔS'U'T'

    (i) moved right by 1 unit. Horizontal displacement right is denoted positive number. So, x +1.

   (ii) moved down by 7 unit. Vertical displacement down is denoted by negative number. So, y - 7

(x, y)  --> (x + 1, y -7)

Both EG and HF are expressed in terms of x and their Coefficients are equal (9 and 7), we can conclude that EG is congruent to HF. This means that segment EG and segment HF have the same length.

To prove that segment EG is congruent to segment HF, we need to show that they have the same length.

Given the information provided, we can use the fact that points E, F, G, and H lie on the same line segment in that order. Let's assume that the distances from E to F, F to G, and G to H are denoted as EF, FG, and GH, respectively.

According to the given information, the ratio of EF to FG to GH is equal to 4/5/2. This implies that EF is 4 parts, FG is 5 parts, and GH is 2 parts of the total length of the line segment.

Let's denote the length of EG as x. Since EG is the combination of EF and FG, we can express it as EF + FG. From the given information, EF is 4 parts and FG is 5 parts. Therefore, we can write:

EG = EF + FG = 4x + 5x = 9x.

Similarly, we can denote the length of HF as y. Since HF is the combination of FG and GH, we can express it as FG + GH. From the given information, FG is 5 parts and GH is 2 parts. Therefore, we can write:

HF = FG + GH = 5x + 2x = 7x.

Now, we have expressed the lengths of EG and HF in terms of x. To prove that EG is congruent to HF, we need to show that they have the same length. This can be done by comparing their expressions:

EG = 9x

HF = 7x

Since both EG and HF are expressed in terms of x and their coefficients are equal (9 and 7), we can conclude that EG is congruent to HF. This means that segment EG and segment HF have the same length.

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Answer:

x² + y² = 74

Step-by-step explanation:

given

(x - y) = 8 ( square both sides )

(x - y)² = 8² ← expand left side using FOIL

x² - 2xy + y² = 64 ← substitute xy = 5

x² - 2(5) + y² = 64

x² - 10 + y² = 64 ( add 10 to both sides )

x² + y² = 74

The weight of one bucket of lobster is 18 pounds per bucket. Option b

To determine the weight of one bucket of lobster, we can use the given information about the rate of 450 pounds over 25 buckets.

The rate is defined as the amount of weight (in pounds) divided by the number of buckets. In this case, the rate is 450 pounds over 25 buckets.

To find the weight of one bucket of lobster, we can divide the total weight by the number of buckets:

Weight of one bucket = Total weight / Number of buckets

Weight of one bucket = 450 pounds / 25 buckets

Weight of one bucket = 18 pounds per bucket

Therefore, the weight of one bucket of lobster is 18 pounds per bucket.

The correct answer is option b) 18 pound per bucket.

It's important to note that this calculation assumes a constant weight per bucket throughout the given scenario. In reality, the weight of one bucket of lobster may vary depending on factors such as the size and type of lobsters being weighed. Additionally, it's possible that the weight per bucket may change over time or across different batches of lobsters.

Option B

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Answer:

5

Step-by-step explanation:

its the middle of the first half

The number line that represents the solution set for the inequality 3(8 – 4x) < 6(x – 5) is a number line from negative 5 to 5 in increments of 1, with an open circle at 3 and a bold line starting at 3 and pointing to the right.

To determine which number line represents the solution set for the inequality 3(8 – 4x) < 6(x – 5), we need to solve the inequality and analyze its solution.

First, let's simplify the inequality:

3(8 – 4x) < 6(x – 5)

24 - 12x < 6x - 30

Next, let's bring all the terms involving x to one side of the inequality:

-12x - 6x < -30 - 24

-18x < -54

Now, divide both sides of the inequality by -18.

Since we are dividing by a negative number, we need to reverse the direction of the inequality:

x > -54/-18

x > 3

The inequality solution is x > 3, which means x is greater than 3. In interval notation, this can be represented as (3, +∞),

where the parentheses indicate that 3 is not included in the solution.

Now let's analyze the given options:

A number line from negative 5 to 5 in increments of 1.

An open circle is at 3 and a bold line starts at 3 and is pointing to the left: This option represents values less than 3, but we need values greater than 3.

A number line from negative 5 to 5 in increments of 1.

An open circle is at 3 and a bold line starts at 3 and is pointing to the right:

This option represents values greater than 3, which matches the solution set.

A number line from negative 5 to 5 in increments of 1.

An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the left: This option does not match the given inequality.

A number line from negative 5 to 5 in increments of 1.

An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the right:

This option does not match the given inequality.

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The vertex of the graphed function f(x) = [x - 4] + 3 is (4, 3).

The function f(x) = [x - 4] + 3 is in the form of an absolute value function, where the expression inside the absolute value brackets represents the input value shifted horizontally by 4 units to the right.

The constant term of 3 represents a vertical shift upwards by 3 units.

To find the vertex of this function, we need to determine the x-coordinate that corresponds to the minimum value of the absolute value function.

In this case, since the absolute value function is within brackets and not being multiplied or divided by a coefficient, the minimum value occurs at the point where the expression inside the brackets equals zero.

Therefore, setting x - 4 = 0, we find x = 4.

Substituting this x-value into the function, we find f(4) = [4 - 4] + 3 = 3. So the vertex of the graphed function is located at the point (4, 3), where x = 4 and y = 3. This means that the graph is shifted 4 units to the right and 3 units upward from the origin.

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The total number of frogs at the pond can be represented by the expression 2x + 1, where 'x' represents the number of lily pads with frogs on them.

Let's analyze the given information step by step to determine the number of frogs at the pond.

If lily pads with frogs on them have two frogs each:

Let's assume the number of lily pads with frogs is 'x'. According to the information, these lily pads with frogs would contribute a total of 2x frogs.

There is one lily pad with no frogs on it:

This means there is an additional lily pad without any frogs. Let's consider this as one extra lily pad without frogs, contributing 0 frogs.

If each lily pad has exactly one frog on it:

Now, let's consider the case where each lily pad has one frog. This implies that the number of frogs would be equal to the number of lily pads, which is 'x'.

There is a frog with no lily pad:

According to the information, there is a frog without a lily pad. We can consider this as one extra frog without a lily pad.

Combining the information from steps 1 and 3, we have the total number of frogs from the lily pads as 2x, and from step 4, we have one additional frog without a lily pad.

Now, let's consider the total number of frogs by summing up the frogs from the lily pads and the additional frog:

Total number of frogs = 2x + 1

Since the number of frogs from the lily pads is equal to the number of lily pads (x), we can rewrite the equation as:

Total number of frogs = 2(x) + 1

= 2x + 1

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The answers are given below:

a) A ∪ B = {1, 2, 3, 4, 6}

b) A ∩ B = {2}

c) Bc = {1, 3, 5}

Probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.

Given that a number cube with faces labelled 1 to 6 is rolled once. The number rolled will be recorded as the outcome.

A number cube with faces labelled 1 to 6.

Therefore the set of all possible outcomes is {1, 2, 3, 4, 5, 6}.

Event A

The number rolled is even.

Therefore, the set of outcomes for event A is {2, 4, 6}.

Event B

The number rolled is less than 4.

Therefore, the set of outcomes for event B is {1, 2, 3}.

Part (a)

Event "A or B" means the outcomes in A or B or both.

[tex]\Rightarrow \text{A} \cup \text{B} = \{1,2,3,4,6\}[/tex]

Part (b)

Event "A and B" means the outcomes in both A and B.

[tex]\Rightarrow \text{A} \cap \text{B} = \{2\}[/tex]

Part (c)

The complement of event B means everything that is not in B.

[tex]\Rightarrow \text{B}\text{c} = \{1,3,5\}[/tex]

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Answer:

(3/13) = -1 ≡ 3^6 (mod 13)

Step-by-step explanation:

We can use Euler's criterion theorem to prove or disprove the statement.

Euler's criterion states that for an odd prime p and integer a not divisible by p,

a^((p-1)/2) ≡ (a/p) (mod p)

where (a/p) is the Legendre symbol. If (a/p) = 1, then a is a quadratic residue modulo p, and if (a/p) = -1, then a is a quadratic non-residue modulo p.

In this case, we have p = 13 and a = 3. Therefore, we need to evaluate:

3^6 ≡ (3/13) (mod 13)

To evaluate (3/13), we need to use the properties of the Legendre symbol. First, we note that:

(3/13) ≡ (-1/13) (3/13)

since (-1/13) = (-1)^((13-1)/2) = 1 by Euler's criterion. Next, we use the quadratic reciprocity law to simplify (-1/13):

(-1/13) = (-1)^((13-1)/2) * (13-1)/2 = 1

Therefore, we have:

(3/13) ≡ (-1/13) (3/13) ≡ 1 * (3/13) ≡ (3/13)

So we need to evaluate (3/13) to determine if it is equal to 1 or -1.

To do this, we can use the quadratic reciprocity law again:

(3/13) = (13/3) * (-1/3)

Since 13 ≡ 1 (mod 3), we have:

(13/3) = (1/3) = 1

Since 3 ≡ 3 (mod 4), we have:

(-1/3) = -1

Therefore, we have:

(3/13) = (13/3) * (-1/3) = 1 * (-1) = -1

So (3/13) = -1.

Therefore, the statement is true, and we have:

(3/13) = -1 ≡ 3^6 (mod 13)

The original goal of the Kickstarter project was $23,217.49.

Here's the calculation:

Original goal = $318,544 / (1372% / 100)

= $318,544 / 1.372

= $23,217.49

The project was able to raise more than 13 times its original goal, which is a great success. It shows that there is a high demand for the product and that the project team was able to effectively market it to potential backers.

Answer:

C)  $2,443.71

Step-by-step explanation:

To calculate the amount of interest earned on the savings account, use the compound interest formula.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ I=P\left(1+\dfrac{r}{n}\right)^{nt}-P$\\\\where:\\\\ \phantom{ww}$\bullet$ $I =$ interest accrued.\\ \phantom{ww}$\bullet$ $P =$ principal amount. \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form). \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year. \\ \phantom{ww}$\bullet$ $t =$ time (in years). \\ \end{minipage}}[/tex]

Given values:

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

[tex]I=4570.65\left(1+\dfrac{0.039}{12}\right)^{12\cdot 11}-4570.65[/tex]

[tex]I=4570.65\left(1+0.00325\right)^{132}-4570.65[/tex]

[tex]I=4570.65\left(1.00325\right)^{132}-4570.65[/tex]

[tex]I=4570.65\left(1.534653130...\right)-4570.65[/tex]

[tex]I=7014.362330...-4570.65[/tex]

[tex]I=2443.712330...[/tex]

[tex]I=\$2,443.71\[ \sf (nearest\;cent)[/tex]

Therefore, the amount of interest the account has earned is $2,443.71, rounded to the nearest cent.

Answer:

$2,443.71

Step-by-step explanation:

To calculate the interest earned, we can use the formula for compound interest:

[tex]\rm\implies A = P(1 + \frac{r}{n})^{(nt)}[/tex]

where:

Given:

Substitute the given values into the above formula:

[tex]\begin{aligned}\rm\implies A& =\rm 4570.65(1 + \frac{0.039}{12})^{(12 \cdot 11)}\\& \approx \rm{\$9,014.36}\end{aligned}[/tex]

To find the interest earned, we subtract the initial deposit from the final amount:

[tex]\begin{aligned}\rm\implies Interest& =\rm A - P\\& \approx \$9,014.36 - \$4,570.65\\& \approx \boxed{\rm{\$2,443.71}}\end{aligned}[/tex]

[tex]\therefore[/tex] The account has earned $2,443.71 in interest.

If we lined up all the people in the world (about 7.4 billion people) and assume that each person occupied 12 inches of the line, the number of times around Earth that they would reach would be: 59.1 times

To determine the number of times that all the people on the earth will move around the planet the dimensions above would be gotten by first harmonizing the dimensions.

1 mile = 63360 inches

So, 25,000 miles = 1.58 * 10⁹ in

Now we multiply the number of people on the earth and divide this number u the circumference in inches.

7.4 * 10⁹ * 12

= 88.8 * 10⁹

88.8 * 10⁹/1.58 * 10⁹ in

= 59.1

So all the people on the earth will move around its circumference, 59.1 times.

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Answer: the answer is c i just finished the test

Step-by-step explanation:

bTo determine the percent of the original price you end up paying after the discounts, we can calculate the final price as a percentage of the original price.

Let's assume the original price of an item is $100.

The first discount reduces the price by 50%, so the price after the first discount is 50% of the original price, which is $100 * 0.5 = $50.

The second discount of 45% is applied to the price after the first discount. The price after the second discount is 45% of $50, which is $50 * 0.45 = $22.50.

Therefore, the final price you end up paying after both discounts is $22.50.\

To find the percent of the original price you end up paying, we divide the final price by the original price and multiply by 100:

Percent of original price = (final price / original price) * 100

In this case, the final price is $22.50 and the original price is $100:

Percent of original price = ($22.50 / $100) * 100

                       = 0.225 * 100

                       = 22.5%

Therefore, you end up paying 22.5% of the original price after the discounts.

It's important to note that the percent of the original price you end up paying will depend on the original price of the item. The calculations above were based on the assumption that the original price was $100. If the original price is different, the resulting percentage will vary accordingly.

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Carys is reviewing her tax bill for the previous year to ensure accuracy and compliance with tax regulations.

Carys checking a tax bill for the last year involves reviewing and verifying the information related to her tax obligations and payments for that specific year. This process typically includes several steps:

Gathering relevant documents: Carys would collect all the necessary documents, such as income statements (W-2 or 1099 forms), expense receipts, investment statements, and any other relevant financial documents.

Reviewing income sources: Carys would examine all the sources of income she received during the year, including employment income, self-employment income, rental income, investment income, etc. She would ensure that all income sources are accurately reported and accounted for.

Deductions and credits: Carys would assess if she is eligible for any deductions or tax credits that can reduce her taxable income or provide tax benefits. These may include deductions for expenses like mortgage interest, medical expenses, education expenses, and credits such as the Child Tax Credit or the Earned Income Tax Credit.

Calculating taxable income: Carys would determine her taxable income by subtracting eligible deductions and adjustments from her total income. This step helps in determining the portion of income that is subject to taxation.

Applying tax rates: Carys would review the applicable tax brackets and rates to calculate her income tax liability based on her taxable income. Different income levels are subject to different tax rates, and Carys would ensure she is applying the correct rates.

Checking for errors: Carys would carefully review all calculations and entries on her tax return to identify any errors or discrepancies. This includes verifying that all income, deductions, and credits are accurately reported and that the final tax liability is correctly calculated.

Filing the tax return: Once Carys is satisfied with the accuracy of her tax bill, she would file her tax return with the appropriate tax authority, such as the Internal Revenue Service (IRS) in the United States. This step involves submitting the required forms and supporting documents to report her income and claim any applicable deductions or credits.

By going through these steps, Carys can ensure that her tax bill for the last year is accurate, complete, and in compliance with the tax regulations and laws of her jurisdiction.

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Answer:

4/15

Step-by-step explanation:

2/5 drive

1/3 bus

and rest walk

fraction of those who walk is 1-(2/5+1/3)

2/5+1/3=(6+5)/15=11/15

15/15-11/15=4/15

The sequence of transformations of the function are

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

f(x) = -3(x - 2)² + 1

The parent function is

y = x²

The transformation of the function from the parent function is as follows

Next, we plot the graph of the function

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The bank reconciliation shows an adjusted bank balance of Br.5,301.42 and an adjusted ledger balance of Br.4,173.83.

To prepare the bank reconciliation and journal entry for ABC Company based on the given information, we need to compare the transactions recorded in ABC's ledger with the transactions shown in the bank statement. Here's the bank reconciliation:

1. Bank Statement Balance:

Cash on deposit at July 31: Br.4,964.47

2. Adjusted Bank Balance:

Bank Statement Balance: Br.4,964.47

3. Deposit not on the statement: + Br.410.90

(Adjusted Bank Balance: Br.5,375.37)

Erroneously charged check: + Br.79

(Adjusted Bank Balance: Br.5,454.37)

Outstanding checks:

Check 801: - Br.100.00

Check 888: - Br.10.25

Check 890: - Br.294.50

Check 891: - Br.205.00

(Adjusted Bank Balance: Br.4,844.62)

4. Incorrectly charged check: + Br.90

(Adjusted Bank Balance: Br.4,934.62)

5. Collection of interest-bearing note: + Br.500

(Adjusted Bank Balance: Br.5,434.62)

6. Interest earned: + Br.24.75

(Adjusted Bank Balance: Br.5,459.37)

7. Incorrectly recorded check: - Br.90

(Adjusted Bank Balance: Br.5,369.37)

8. Collection fee: - Br.5.00

(Adjusted Bank Balance: Br.5,364.37)

9. NSF check: - Br.50.25

(Adjusted Bank Balance: Br.5,314.12)

10. Service charge: - Br.12.70

(Adjusted Bank Balance: Br.5,301.42)

Adjusted Ledger Balance:

Bank Balance in ABC's Ledger: Br.4,173.83

Reconciled Bank Balance: Br.5,301.42

Reconciled Ledger Balance: Br.4,173.83

Journal Entry:

To record the bank reconciliation, we make the following journal entry:

Debit: Cash (Ledger Balance adjustment) - Br.5,301.42

Credit: Cash (Bank Statement Balance adjustment) - Br.4,964.47

Credit: Miscellaneous Income/Expense - Br.336.95 (the difference between the two balances)

This journal entry ensures that the ledger balance matches the reconciled bank balance by adjusting for the discrepancies identified during the reconciliation process.

The journal entry reflects the necessary adjustments to reconcile the two balances, with a debit to Cash (Ledger Balance adjustment), a credit to Cash (Bank Statement Balance adjustment), and a credit to Miscellaneous Income/Expense to account for the difference.

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The length of side a is approximately B. 24.3 (to the nearest tenth).

The correct answer is B. 24.3

To find the length of side a in triangle ABC, we can use the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle.

The Law of Sines can be expressed as:

a/sin(A) = c/sin(C)

where a is the length of side a, A is the measure of angle A, c is the length of side c, and C is the measure of angle C.

Angle A (m∠A) = 45°

Side c (c) = 17

Angle B (m∠B) = 25°

We need to find the length of side a.

Using the Law of Sines:

a/sin(45°) = 17/sin(25°)

To find a, we isolate it by multiplying both sides of the equation by sin(45°):

a = 17 * (sin(45°) / sin(25°))

Using a calculator to evaluate the expression:

a ≈ 24.3

Therefore, the length of side a is approximately 24.3 (to the nearest tenth).

The correct answer is:

B. 24.3

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Each friend had originally 13 action figures before buying 11 more.

Given that there are 5 friends and each of them has x action figures in his or her collection. When they buy 11 more action figures, the total number of action figures they will have is 120.

a) We need to find an equation that models the problem. Let x be the original number of action figures that each friend had.

Therefore, the total number of action figures that each friend will have after purchasing 11 more is x + 11.

The total number of action figures will be 5(x + 11) = 5x + 55.

Now, according to the problem,5x + 55 = 120

This is our equation that models the problem.

b) We have to solve the equation 5x + 55 = 120 to find the original number of action figures, x, that each friend had before buying 11 more action figures.

5x + 55 = 120

5x = 120 - 55 (subtract 55 from both sides)

5x = 65x = 13

Therefore, each friend had originally 13 action figures before buying 11 more.

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Answer:

105% = 105/100 = 21/20 or 1 & 1/20

The conclusion remains the same as mentioned earlier. Danielle's monthly payment for the entertainment center with the new plasma television is $115. She will make equal monthly payments of $115 for 1 year to pay off the remaining amount owed.

To find Danielle's monthly payment, we need to subtract the down payment from the total cost of the entertainment center with the new plasma television.

The total cost of the entertainment center with the new plasma television = $1680

Down payment = $300

The remaining amount to be paid off is $1680 - $300 = $1380.

Since Danielle pays off the remaining amount in equal monthly payments for 1 year, we need to calculate the number of months in a year. There are 12 months in a year.

To find Danielle's monthly payment, we divide the remaining amount by the number of months:

Monthly payment = Remaining amount / Number of months

= $1380 / 12

= $115

Therefore, Danielle's monthly payment is $115. She will make equal monthly payments of $115 for 1 year to pay off the rest of the amount owed for the entertainment center with the new plasma television.

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Answer:

(12, 11, 10)

(2, 3, 4)

(5, 7, 13)

(1, 1, 1)

(7, 10, 3)

(4, 6, 7)

(x, y, x + y) for any positive values of x and y

(x, y, x - y) if x is greater than y and x - y is greater than 0

(x, x, 2) for any positive value of x

Step-by-step explanation:

The given equation is:

4a + 3 = 7a - 2

To solve for a, we can start by simplifying both sides of the equation. First, we can combine the constants on the right side:

4a + 3 = 7a - 2

4a + 5 = 7a

Next, we can isolate the variable terms on one side of the equation and the constant terms on the other side. Let's subtract 4a from both sides:

4a + 5 = 7a

5 = 3a

Finally, we can solve for a by dividing both sides by 3:

5 = 3a

5/3 = a

Therefore, the solution is:

a = 5/3

We can check this solution by substituting it back into the original equation:

4a + 3 = 7a - 2

4(5/3) + 3 = 7(5/3) - 2

20/3 + 3 = 35/3 - 2

29/3 = 29/3

Since both sides of the equation are equal when we substitute a = 5/3, we can confirm that this is the correct solution.

C 3) 1) 2) B. triangle and a (X) if it does not form a triangle. Which of the following could be the lengths of the sides of a triangle. Put a () if it is forms 12, 11, 10 2, 3, 4 3, 2, 1 4) 5) 6) 7) 7, 13, 7 8) 13, 12, 5 F 9) 3, 7, 10 10) 4, 6, 7 11) x, y, x + y 1 12) x, y, x-y 13) 1, 1, 2 nd all possible value of x.​

To determine whether a set of lengths could form the sides of a triangle, we need to check if the sum of the two shorter sides is greater than the longest side. If it is, then the lengths can form a triangle; otherwise, they cannot.

Using this criterion, we can determine which sets of lengths form triangles:

(12, 11, 10) - (O) forms a triangle, since 10 + 11 > 12.

(2, 3, 4) - (O) forms a triangle, since 2 + 3 > 4.

(3, 2, 1) - (X) does not form a triangle, since 1 + 2 is not greater than 3.

(5, 7, 13) - (O) forms a triangle, since 5 + 7 > 13.

(1, 1, 1) - (O) forms a triangle, since all sides are equal.

(8, 8, 16) - (X) does not form a triangle, since 8 + 8 is not greater than 16.

(13, 12, 5) - (O) forms a triangle, since 5 + 12 > 13.

(7, 10, 3) - (X) does not form a triangle, since 3 + 7 is not greater than 10.

(4, 6, 7) - (O) forms a triangle, since 4 + 6 > 7.

(x, y, x + y) - (O) forms a triangle for any positive values of x and y, since x + y is always greater than x and y individually.

(x, y, x - y) - (O) forms a triangle if x is greater than y and x - y is greater than 0.

(1, 1, 2) - (X) does not form a triangle, since 1 + 1 is not greater than 2.

(x, x, 2) - (O) forms a triangle for any positive value of x, since x + x > 2.

Therefore, the sets of lengths that can form triangles are:

(12, 11, 10)

(2, 3, 4)

(5, 7, 13)

(1, 1, 1)

(7, 10, 3)

(4, 6, 7)

(x, y, x + y) for any positive values of x and y

(x, y, x - y) if x is greater than y and x - y is greater than 0

(x, x, 2) for any positive value of x

The different Algebraic Properties that were used are:

1) Associative Property

2) Commutative Property

3) Inverse Property

4) Identity Property

There are different Algebraic Properties such as:

Commutative Property

Associative Property

Inverse Property

Identity Property

1) A + (B + C) = (A + B) + C  

The algebraic property used here is referred to as: associative  property

2) A + B = B + A  

The algebraic property used here is referred to as: commutative  property

3) A + (−A) = 0  

The algebraic property used here is referred to as: inverse property

4) A + 0 = A  

The algebraic property used here is referred to as: identity  property

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The expression to determine the perimeter of the triangle in terms of the shortest side (s) is 3s + 15.

To determine the perimeter of the triangle, we need to express the lengths of all three sides in terms of the shortest side (s).

The longest side is three more than twice the length of the shortest side. This can be expressed as 2s + 3.

The third side is twelve feet long.

Now, let's denote the lengths of the sides as follows:

Shortest side = s

Second side = 2s + 3

Third side = 12

To find the perimeter, we sum up the lengths of all three sides:

Perimeter = s + (2s + 3) + 12

Simplifying the expression:

Perimeter = s + 2s + 3 + 12

Perimeter = 3s + 15

Therefore, the expression to determine the perimeter of the triangle in terms of the shortest side (s) is 3s + 15.

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Answer:

d = [tex]\sqrt{89}[/tex]

Step-by-step explanation:

to calculate the distance d use the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = (- 4, - 1 ) and (x₂, y₂ ) = (1, - 9 )

d = [tex]\sqrt{(1-(-4))^2+(- 9-(-1))^2}[/tex]

  = [tex]\sqrt{(1+4)^2+(-9+1)^2}[/tex]

  = [tex]\sqrt{5^2+(-8)^2}[/tex]

  = [tex]\sqrt{25+64}[/tex]

  = [tex]\sqrt{89}[/tex]

The elements that compose an ellipse are:

1. Center: The center is the point in the middle of the ellipse. It is denoted as (h, k), where h represents the x-coordinate and k represents the y-coordinate of the center.

2. Major Axis: The major axis is the longer of the two axes of the ellipse. It passes through the center and is perpendicular to the minor axis.

3. Minor Axis: The minor axis is the shorter of the two axes of the ellipse. It also passes through the center and is perpendicular to the major axis.

4. Vertices: The vertices are the points on the ellipse where the major axis intersects the ellipse. In this case, the vertices are V1(-1, -8) and V2(11, -8).

5. Co-vertices: The co-vertices are the points on the ellipse where the minor axis intersects the ellipse.

6. Foci: The foci are two points inside the ellipse that help define its shape. The sum of the distances from any point on the ellipse to the two foci is constant. In the given problem, the foci are F1(0, 3) and F2(0, -3).

7. Eccentricity: The eccentricity of an ellipse is a measure of how "squished" or elongated the ellipse is. It is denoted by the letter e and is calculated by dividing the distance between the center and one of the foci by the distance between the center and a vertex.

To find the equation of the ellipse, you can use the standard form equation:

((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1

Where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes respectively.

To find the length of the major axis, you can simply double the length of the semi-major axis.

To find the length of the minor axis, you can double the length of the semi-minor axis.

To find the length of each side of the rectangle, you can use the formula 2 * b.

To graph the ellipse, plot the center, vertices, co-vertices, and foci on a coordinate plane and sketch the shape of the ellipse using these points.

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