How do you use a system of equations to find the solution algebraically?

If you factor 3x from the expression, you have

[tex]3x^3+9x^2-12x=3x(x^2+3x-4)[/tex]

So, we have

[tex]3x(x^2+3x-4)=0 \iff 3x=0\lor x^2+3x-4=0[/tex]

We easily have

[tex]3x=0\iff x=0[/tex]

So, one solution is x=0.

The other solutions depend on the quadratic equation:

[tex]x^2+3x-4=0 \iff x=-4 \lor x=1[/tex]

So, the solutions are [tex]x=-4,\ 0,\ 1[/tex]

Answer:  B) 115

Step-by-step explanation:

S₅ means the sum of the first 5 numbers in the sequence.

3 + 13 + 23 + 33 + 43 = 115

To determine if pentagon DEFGH has 5 congruent angles, Peter needs to consider if it is a regular pentagon. Regular pentagons have 5 congruent sides and angles. In such pentagons, each angle measures 108°.

Peter can determine if the pentagon has 5 congruent angles by checking if it is a regular pentagon. A regular pentagon is defined as having all its sides and angles equal. If the pentagon DEFGH is regular, it will have 5 congruent sides and 5 congruent angles.

The Pythagorean theorem would apply if DEFGH were a triangle, where D and L are sides with hypotenuse s. For pentagons, the theorem is not applicable. It's important to remember that the angles of any pentagon add up to 540°. So, in a regular pentagon, each angle measures 108° because 540°/5=108°.

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Answer: A

Hope this helps you out!

A common ratio between 0 and 1

This function starts at 30 where x=0, and then gains 15 units each time x is increased by 1.

So, we deduce that renting the storage has a fixed price of 30, and then you have to pay 15 each month.

It will add 15 dollar for each month.

Answer:

  Hari is 20; Harry is 28

Step-by-step explanation:

The ratio in 4 years is equivalent to the ratio 6:8, which has each of the original ratio unit numbers increased by 1. That means each of those ratio units stands for 4 years, and the present ages are ...

  Hari: 5·4 = 20

  Harry: 7·4 = 28

_____

Conventional method of solution

If you like, you can write equations for the ages of Hari (x) and Harry (y):

  x/y = 5/7

  (x+4)/(y+4) = 3/4

These can be solved a variety of ways. For some methods, it may be useful to write them in standard form:

These have solution (x, y) = (20, 28).

[tex]\bf \qquad \qquad \textit{sum of a finite arithmetic sequence} \\\\ S_n=\cfrac{n}{2}(a_1+a_n)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\[-0.5em] \hrulefill\\ n=12\\ S_{12}=156 \end{cases}\implies 156=\cfrac{12}{2}(a_1+a_{12}) \\\\\\ 156=6(a_1+a_{12})\implies \cfrac{156}{6}=a_1+a_{12}\implies 26=a_1+a_{12}[/tex]

Final answer:

The sum of the first and twelfth terms of an arithmetic progression, whose first 12 terms sum to 156, is 26. This is determined by using the arithmetic progression sum formula and understanding that the sum of equidistant terms from the beginning and end of the series is constant.

Explanation:

To determine the sum of the first and twelfth terms of an arithmetic progression (AP) given the sum of the first 12 terms, we can use the properties of AP. The sum of an arithmetic progression can be expressed using the formula Sn = n/2(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.

For this question, the sum of the first 12 terms (S12) is given as 156, and we want to find the sum of the first and twelfth terms. By properties of AP, the sum of equidistant terms from the beginning and end (first and last terms in this case) is the same. So the sum of the first and twelfth terms is equal to the sum of the second and eleventh terms, and so on. This sum is consistent and equals a + a + 11d, which simplifies to 2a + 11d.

Since S12 = 12/2(2a + 11d) and we know S12 = 156, we can simplify the formula to get 156 = 6(2a + 11d). However, to solve for 2a + 11d directly, we do not need the value of d; we only need the fact that the sum of the first and last terms will be constant and equal to the sum of 2a + 11d. Therefore, the sum of the first and twelfth terms is 156/6, which equals 26.

Answer:

I think it would be true

The provided statement "When researchers incorrectly interpret the responses to a survey question, this is poor analysis" is true.

The practice of assessing client insights is known as survey analysis. It could be customer satisfaction scores or other customer experience measures.

We have a statement:

When researchers incorrectly interpret the responses to a survey question,

this is a poor analysis.

The above statement is true because when we have survey data that is incorrectly interpreted, it will create a problem in the final phase or implementation phase.

Thus, the provided statement "When researchers incorrectly interpret the responses to a survey question, this is poor analysis" is true.

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Percent error = |(Measured - Actual) / Actual| × 100. For Measured 68, Actual 60: [tex]\( \frac{|68 - 60|}{60} \times 100 ≈ 13.3\% \).[/tex]

To find the percent error, you can use the formula:

[tex]\[ \text{Percent Error} = \left| \frac{\text{Measured Value} - \text{Actual Value}}{\text{Actual Value}} \right| \times 100 \]\\[/tex]

Given:

- Measured Value = 68

- Actual Value = 60

Substitute these values into the formula:

[tex]\[ \text{Percent Error} = \left| \frac{68 - 60}{60} \right| \times 100 \]\[ \text{Percent Error} = \left| \frac{8}{60} \right| \times 100 \]\[ \text{Percent Error} = \left| 0.1333... \right| \times 100 \]Now, rounding to the nearest tenth:\[ \text{Percent Error} ≈ 13.3\% \]\\[/tex]

So, the percent error is about 13.3%. This indicates that the measured value is approximately 13.3% higher than the actual value.

The percent error, rounded to the nearest tenth, is about 13.3%.

To find the percent error, you can use the following formula:

[tex]\[\text{Percent Error} = \frac{{|\text{Estimated Value} - \text{Actual Value}|}}{{\text{Actual Value}}} \times 100.\][/tex]

Here, the estimated value is 68 marbles, and the actual value is 60 marbles.

Calculate the absolute error :

  [tex]\[ |\text{68 - 60}| = 8. \][/tex]

Calculate the relative error :

[tex]\[ \frac{{8}}{{60}}. \][/tex]

Convert to percentage :

[tex]\[ \frac{{8}}{{60}} \times 100 = 13.333. \][/tex]

Round to the nearest tenth :

[tex]\[ \approx 13.3.[/tex]

Thus, the percent error, rounded to the nearest tenth, is about 13.3%.

Question :

You estimate that a jar contains 68 marbles. The actual number of marbles is 60. Find the percent error. Round your answer to the nearest tenth. The percent error is about %.

Answer:

d

Step-by-step explanation:

13 red, 1 king of spade, 1 ace of club

Answer:

B

Step-by-step explanation:

Given the 2 equations

x³ + y = 0 → (1)

11x² - y = 0 → (2)

Add (1) and (2) term by term to eliminate the term in y

x³ + 11x² = 0 ← factor out x² from each term

x²(x + 11) = 0

Equate each factor to zero and solve for x

x² = 0 ⇒ x = 0

x + 11 = 0 ⇒ x = - 11

Substitute these values into (1) for corresponding values of y

(1) y = - x³

x = 0 ⇒ y = 0 ⇒ (0, 0) is a solution

x = - 11 ⇒ y = - (- 11)³ = 1331 ⇒ (- 11, 1331) is a solution

Answer:

x = 48.37

Step-by-step explanation:

x^2 = 58^2 - 32^2

x^2 = 3364 - 1024

x^2 = 2340

x = 48.37

Answer:

48.37

Step-by-step explanation:

a. On Graph B, at 0 minutes, the height of the graph will be 12 miles.

b. Then, the graph will decrease linearly until 15 minutes, when Adalyn reaches the school.

Certainly, let's delve a bit deeper into each part of Graph B in relation to Adalyn's journey from home to school and back:

a. Initial Point at 0 Minutes:

On Graph B, the vertical axis represents the distance from the school, while the horizontal axis represents time. At the start of her journey (0 minutes), Adalyn is at her home, which is 12 miles from the school. This is the farthest point from the school she will be during her trip. Therefore, the height of Graph B at 0 minutes must be the full distance to the school, which is 12 miles. This is the starting point for the graph on the vertical axis.

b. Graph Behavior from 0 to 15 Minutes:

As time progresses from 0 to 15 minutes, Adalyn is driving towards the school, getting closer every minute. On the graph, this is represented by a decreasing line, since the vertical distance from any point on the line to the horizontal axis represents her distance from the school, which is getting smaller. The rate of decrease is constant because we're assuming she's driving at a steady pace.

At exactly 15 minutes, Adalyn reaches the school. Since she is now at the school, her distance from it is 0 miles. On Graph B, this is represented by the graph touching the horizontal axis. At this point, the height of the graph is 0 miles, indicating that there is no distance between Adalyn and the school.

After 15 minutes, the graph would stay at 0 miles (the line would run along the horizontal axis) until she begins her journey back home. Once she starts the return trip, the graph would begin to increase linearly from 0, reflecting her increasing distance from the school as she drives back home.

If we were to continue completing Graph B based on Graph A, after the 15-minute mark, we'd see a flat line at 0 miles until 60 minutes when Adalyn starts her journey back home, at which point the line would ascend back to 12 miles by 75 minutes, mirroring the initial descent but in the opposite direction.

The word which completes the statement are,

12 miles,

Steady,

Decrease

Complete each statement about Graph B.

On graph B, at 0 minutes, the height of the graph will be 12 miles.

Then, the graph will remain steady until 15 minutes, the height of graph B will be decreased.

Therefore, graph B is the reverse of graph A which is the distance from the school to her home.

Answer:

Through trial and error:

7 + (16 - 8) / 2 + (2 * 25 / 5)

The probability with replacement is [tex]= 1 \div 16[/tex]

The probability without replacement is [tex]= 5 \div 16[/tex]

(a)

With replacement

Since cards should be replaced, 5 lions or 5 legs

So, the probability is

[tex]= 5\div 20 \times 5\div 20\\\\= 1 \div 16[/tex]

(b)

Without replacement

Since cards should not be put back, for the second draw only 19 cards are left

So, the probability is

[tex]= 5\div 20 \times 5\div 19\\\\= 5 \div 16[/tex]

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To calculate probability with and without replacements, we adjust the number of total outcomes depending on whether the card is replaced or not after each draw. For this specific scenario, when there is no replacement, the probability of drawing a red bird and then a lion is (5/20)*(5/19), while with replacement it is (5/20)*(5/20).

Your question involves the concepts of probability and specifically the difference between sampling with and without replacement. Let's assume, for the sake of this example, that there are 5 red bird cards and 5 lion cards in the deck of 20 cards.

1. Sampling without replacement: After the first card (a red bird) is taken out, it is not added back into the deck. So the deck now only has 19 cards. So, the probability of drawing a red bird card first is 5/20 or 1/4. Then, the probability of drawing a lion card second is 5/19.

2. Sampling with replacement: After the first card (a red bird) is pulled out, it is replaced back into the deck. So the deck continues to have 20 cards. So, the probability of drawing a red bird first is still 5/20 or 1/4 and the probability of drawing a lion card second is still 5/20 or 1/4.

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Answer:

Step-by-step explanation:

Start with the distance squared.

The distance from the center to the point on the circle is found from

Formula

d^2 = r^2 = (x2 - x1)^2 + (y2 - y1)^2

Givens

x2 = 2

x1 = - 5

y2 = -8

y1 = - 8

Center= (-5,-8)

Solution

r^2 = (2 - - 5)^2 + (-8 - - 8)^2

r^2 = 49 + 0

r^2 = 49

equation

(x + 5)^2 + (y + 8)^ = 49

Answer:

Area: pir^2

r=4

4pi^2

if pi=3.14

then the area is 50.24ft^2

Circumference: pi x d

d=8

8pi

if pi=3.14

then the circumference is 25.12ft.

ANSWER

Circumference=8πft

Area=16π ft²

EXPLANATION

The circumference of a circle calculated using the formula:

C=πd

The diameter of the circle is 8ft.

The circumference is

C=8π ft.

The area of a circle is given by:

A = πr²

Where r=8/2=4 ft is the radius of the circle.

Therefore the area is

A = π×4²

A=16π ft²

Answer:

She had more correct answers on the first quiz

Explanation:

1- For the first quiz:

We know that the quiz had a total of 15 questions and that Kelly had [tex]\frac{4}{5}[/tex] of those correct

This means that:

Number of correct answers = [tex]\frac{4}{5}*15 = 12[/tex] answers

2- For the second quiz:

We know that the quiz had a total of 12 questions and that Kelly had [tex]\frac{5}{6}[/tex] of those correct

This means that:

Number of correct answers = [tex]\frac{5}{6}*12=10[/tex] answers

3- Comparing the number of correct answers:

From the above calculations, we know that she had 12 correct answers on the first quiz and 10 on the second

This means that she had more correct answers on the first quiz

Hope this helps :)

Kelly got more questions correct on the first quiz with 12 correct answers compared to 10 correct answers on the second quiz.

To figure out on which quiz Kelly got more questions correct, we need to calculate the number of correct answers for each quiz.

First Quiz (Decimals): Kelly answered 4/5 of 15 questions correctly.

To find the number of correct answers, multiply:

(4/5) × 15 = 12

So, Kelly got 12 questions correct in the first quiz.

Second Quiz (Fractions): Kelly answered 5/6 of 12 questions correctly.

To find the number of correct answers, multiply:

(5/6) × 12 = 10

So, Kelly got 10 questions correct in the second quiz.

Therefore, Kelly got more questions correct on the first quiz with 12 correct answers compared to 10 on the second quiz.

Answer:

50 inches cubed

Step-by-step explanation:

To solve the volume of a rectangular pyramid, the equation l×w×h divided by 3 can be used. Plug in the numbers given to you in the question- the length of the base is 3, width of base is 5, and height of pyramid is 10 so the equation becomes 3×5×10 divided by 3. When solved, you get 50.

Vinyl is = 15 in. * 33 in. = 495 in.²

Vinyl is needed 495 inches² to cover both of triangular sides

We have to given that,

Gymnastics incline mat is shaped like a wedge two sides of the matter shaped like right triangles.

Now, We know that,

Area of triangle is,

⇒ A = 1/2 x  base x height

Here, base = 33 inches

Height = 15 inches

Since, Vinyl needed to cover both of the triangular sides.

Hence, Area is,

A = 2 (Area of triangle)

A = 2 (1/2 x  base x height)

A = base x height

A = 15  x 33

A = 495 inches²

Thus, Vinyl is needed 495 inches² to cover both of triangular sides.

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Answer:

B

Step-by-step explanation:

The function is cubic (the highest power is 3), so it can't be even.  It's either odd or neither.

We can either graph the function, or we can test that it's symmetrical about the origin by seeing if f(x) = -f(-x).

f(x) = -2x³ + 3x²

f(-x) = -2(-x)³ + 3(-x)²

f(-x) = 2x³ + 3x²

f(x) ≠ -f(-x), so the function is neither even nor odd.

The answer is B because if you add the negative sign it will change

Answer:

Step-by-step explanation:

The factors of each term are ...

So, the greatest common factor is 3a. Factoring that out gives ...

  3a(16a^2 -25) . . . . . . matches answer 2

The factor in parentheses is the difference of squares, so it can be factored. You have memorized the form for the difference of squares ...

  p^2 -q^2 = (p -q)(p +q)

so you know the factoring of this with p=4a and q=5 will be ...

  3a(4a -5)(4a +5) . . . . . . matches answer 4

__

Any pair of factors can be multiplied by -1 without changing the value of the expression. So, two more answers are equivalent:

   (-3a)(-(16a^2 -25) = -3a(25 -16a^2) . . . . . . matches answer 5

  (-3a)(-(4a -5)(4a +5) = -3a(5 -4a)(5 +4a) . . . . . . matches answer 6

Answer:

2 4 5 6

Step-by-step explanation:

The highest common factor is 3a That means that 48a^3 must be divided by 3a which means 48a^3 / 3a = 16a^2. Notice what happened. 48/3 = 16. a^3/a = a^2.

Now you have to pull out 3a from 75a. That wasn't done to Answer 1. So answer one is incorrect.

75a/3a = 25

So far what you answer looks like is

3a(16a^2 - 25) which is answer 2

===========================================

Answer 3 is wrong because one of the factor has to be - 5. Neither one is.

===========================================

Answer 4 is correct

16a^2 - 25 factors into (4a - 5)(4a + 5)

So what is written reflects those factors + the 3a

==============================================

Now we come to the brutal ones.

If you take out a minus sign from the brackets, it has the effect of turning the two terms inside the brackets around.

-3a(25 - 16a^2) is what the above sentence means. so 5 is correct

==============================================

The two terms inside the brackets still factor

-3a ( 5 - 4a)(5 + 4a) It is just that they are turned around.

6 is correct.

===============================================  

Answer:

4:15, 4 to 15, 4/15

Answer:

a. <-28, -8>

Step-by-step explanation:

The given vector is

u = <-7, -2>.

To find 4u, we multiply the given vector by the scalar 4.

4u =4 <-7, -2>.

4u =<-7\times4, -2\times4>.

We multiply out to get;

4u =<-28, -8>.

The correct choice is a. <-28, -8>

The answer is <-28 or >8

[tex]\bf \qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}}\qquad \qquad y=\cfrac{k}{x}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{\textit{\underline{v}olume varies inversely with \underline{p}ressure}}{V=\cfrac{k}{p}}~\hfill[/tex]

Answer:

120

Step-by-step explanation:

5x4x3x2x1 =120

The number of ways to arrange the trophies is 120. The correct answer is option A.

The arrangement of the different things or numbers in a number of ways is called the combination.

Given that:-

The number of the ways will be calculated as:-

N  =  5!

N  =  5  x 4  x 3  x 2 x 1

N  =  120 ways

Therefore the number of ways to arrange the trophies is 120.

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let's recall that on the II Quadrant x/cosine is negative whilst y/sine is positive,

also let's recall that the hypotenuse is simply the radius distance and thus is never negative.

[tex]\bf cot(\theta )=\cfrac{\stackrel{adjacent}{-2}}{\stackrel{opposite}{1}}\impliedby \textit{let's find the hypotenuse} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{(-2)^2+1^2}\implies c=\sqrt{5} \\\\[-0.35em] ~\dotfill\\\\ csc(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{5}}}{\stackrel{opposite}{1}}\implies csc(\theta )=\sqrt{5}[/tex]

The exact value of csc theta when the value of cot theta is  -2 and the terminal side of theta lies in quadrant II (2) is √(5).

The terminal side of an angle is the rotated side of the initial side around a point to form an angle. This rotation can be clockwise or counter clock wise.

The exact value of csc theta has to be found out. The value of cot theta is,

cot θ = -2

cot θ =-2/1.

By the property of right angle triangle, the ratio of adjacent side to the opposite side is equal to the cot theta. Thus,

Adjacent side= -2

Opposite side= 1

The value of hypotenuse side is equal to the square root of the sum of the square of adjacent side and opposite side. Thus,

Hypotenuse side=√((-2)²+1²)

Hypotenuse side=√(5)

By the property of right angle triangle, the ratio of hypotenuse side to the opposite side is equal to the coses theta. Thus,

coses θ =√(5)/1

coses θ =√(5)

Hence, the exact value of csc theta when the value of cot theta is  -2 and the terminal side of theta lies in quadrant II (2) is √(5).

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Calculate the cost per square foot for each size tile.

1 square foot = 144 square inches.

Small tile = 4 x 4 = 16 square inches.

16 / 144 = 1/9 square foot.

Cost per square foot = 3.50 / 1/9 = $31.50 per square foot.

Large tile = 6 x 12 = 72 square inches.

72 / 144 = 1/2 square foot.

Cost per square foot = 4.50 / 1/2 = $9 per square foot.

The minimum area of the mosaic is 3 feet x 5 feet = 15 square feet.

The large tiles are cheaper per square foot.

Total tiles needed = 15 sq. ft.  / 1/2 sq. ft = 30 tiles.

30 tiles x 4.50 each = $135

Answer:

x=6,-4

Step-by-step explanation:

Use the formula (b/2)^2 in order to create a new term.  Solve for x by using this term to complete the square.

x=6,-4