The value of (1 + i)^8 is 16. De Moivre's describes how the power of a complex number can be calculated using the properties of its modulus and argument.
De Moivre's theorem states that for any complex number z = r * (cos(θ) + i * sin(θ)) and any integer n, we have:
zⁿ = rⁿ * (cos(n * θ) + i * sin(n * θ))
To use this theorem, we need to first express the complex number (1 + i) in terms of its magnitude and angle. We can do this by using the Pythagorean theorem to find the magnitude of the complex number, which is the square root of the sum of the squares of the real and imaginary parts.
In this case, the magnitude is the square root of 1^2 + 1^2, or √2.
The angle of the complex number can be found using the inverse tangent function. In this case, the angle is 45 degrees, since the tangent of 45 degrees is 1. Now that we have the magnitude and angle of the complex number, we can use De Moivre's theorem to find the value of (1 + i)^8. Substituting the values we found into the formula above, we get:
(1 + i)^8 = √2^8 * (cos(8 * 45) + i * sin(8 * 45))
= 2^4 * (cos(360) + i * sin(360))
= 16 * (1 + i * 0)
= 16
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queuing system a has a utilization of 80 percent, and queuing system b has a utilization of 90 percent. both have a single server. say the utilization of both systems increases by 5 percent; that is, a increases from 80 percent to 85 percent while b increases from 90 percent to 95 percent. which system is likely to experience the bigger change in the average time in queue?
For queuing system B, the average time in queue ids going to change significantly as a result of increased utilization.
This is because queuing system B is initially heavily loaded and has less capacity to handle more requests. As a result, an increase in utilization can cause a significant increase in average queue latency on system B compared to system A.
Average Queue Time is a measure of the time a customer or request spends being processed by a server. It is affected by several factors such as server load, customer or request arrival rate, and server service rate. As the load on the queuing system increases, it means that the server is being used more frequently and is no longer available to process incoming requests. This will increase the average time in the queue as more requests accumulate and have to wait to be processed. The greater the increase in occupancy, the greater the impact on average latency.
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Find the 21st term in the arithmetic sequence.
3, 7, 11, 15, 19, ...
The 21st term of the given arithmetic sequence is 83. The nth term of an arithmetic sequence is applied to find the required value where n = 21.
What is the nth term of an arithmetic series?The nth term of an arithmetic sequence is calculated by the formula
aₙ = a + (n - 1) · d
Here the first term is 'a' and the common difference is 'd'.
Calculation:The given sequence is an arithmetic sequence.
3, 7, 11, 15, 19, ....
So, the first term in the sequence is a = 3 and the common difference between the terms of the given sequence is d = 7 - 3 = 4.
Thus, the required 21st term in the sequence is
a₂₁ = 3 + (21 - 1) × 4
⇒ a₂₁ = 3 + 20 × 4
⇒ a₂₁ = 3 + 80
∴ a₂₁ = 83
So, the 21st term in the given arithmetic sequence is 83.
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Suppose IQ scores were obtained for 20 randomly selected sets of twins. The 20 pairs of measurements yield x=98.13, y=100.6, r=0.904, P-value=0.000, ŷ=14.68+0.88x, where x represents the IQ score of the twin born first. Find the best predicted value of ModifyingAbove y with caret given that the twin born first has an IQ of 104? Use a significance level of 0.05.
The predicted value of the twin born first has an IQ of 104 is 106.2..
We have given that,
sample size n = 10
x = 98.13
y = 100.6
r = 0.904
p-value = 0.000
y'= 14.68 + 0.88x
significance level = 0.05
and we have to find predicted value of the twin born first has an IQ of 104.
We know that x represents the IQ score of the twin born first.
Therefore value of x is 104.
Hence the predicted value P will be,
P = 14.68 + 0.88(104)
= 14.68 + 91.52
P = 106.2
Therefore the best predicted value of the twin born first has an IQ of 104 is 106.2.
The positive predicted value is the probability that a test gives result for a true statistic. The negative predictive value is the probability that a test gives a false result for a false statistic.
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Solve Proportion
y/32=16/10
Answer:
y = 51.2
Step-by-step explanation:
y/32=16/10
First, we time 32 on both sides to get y alone
y = 32(16/10)
y = 32(1.6)
y = 51.2
Let's check:
51.2/32 = 16/10
1.6 = 1.6
The answer y = 51.2 is correct!
If you can, please give me a Brainliest; thank you, and have a good day!
Please help, giving thanks and brainliest:) (Please give an explanation if possible and please dont give a fake answer)
Please answer all four questions (they are simple and go with the diagram)
(1) The measure of the angle CFD = 48°.
(2) The measure of the angle AFE = 48°.
(3) Pair of supplementary angles: Angle BFC and angle CFD.
(4) Pair of adjacent angles: Angle AFB and angle BFD.
What are the properties of angles?
The angle properties of lines are:
Vertically opposite angles are equal, for example, a = d, b = c.
Adjacent angles add to 180, for example, a + b = 180, a + c = 180. Corresponding angles are equal, for example, a = e, b = f, c = g, d= h.
(2) In the given figure, m∠BFC = 42°
By using the property of linear pair, we can write
∠BFC + ∠CFD = 180
42 + ∠CFD = 180
∠CFD = 58.
(2) Angle CFD and angle AFE are opposite angles and opposite angles are same in measurement so we can write
m∠AFE = 58°.
(3) Pair of supplementary angles: Angle BFC and angle CFD.
(4) Pair of adjacent angles: Angle AFB and angle BFD.
Hence,
(1) The measure of the angle CFD = 48°.
(2) The measure of the angle AFE = 48°.
(3) Pair of supplementary angles: Angle BFC and angle CFD.
(4) Pair of adjacent angles: Angle AFB and angle BFD.
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If f(x) = (x ^ 2 - x + 1)/(x - 1) find f(- 1/2)
Answer:
[tex] \frac{7}{6} [/tex]
peterson's solution is a valid solution to the critical section problem on modern multi-core systems. true false
Therefore, Peterson's solution is not a valid solution to the critical section problem on modern multi-core system so it is false.
What is multi core system?A multicore processors is an electronic component that has two or more processor cores attached for higher performance and lower power consumption. These processors also enable more efficient execution of numerous activities, such as with parallelization and multithreading.
Here,
It is because of the following disadvantages:
Although Peterson's technique is effective for two processes, it is the best plan for the important part in user mode.
Additionally, this solution involves busy waiting, wasting CPU time. So that “SPIN LOCK” problem can emerge. And any busy waiting solution may encounter this issue.
Therefore, Peterson's solution is not a valid solution to the critical section problem on modern multi-core system so it is false.
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define the level of significance. multiple choice it is the probability of a type ii error. it is the probability of a type i error. it is a z-value of 1.96. it is the beta error.
By information provided in the question, we can say that It is the probability of a Type I error.
what is probability?The proportion of favorable outcomes to all possible outcomes for an event is known as the probability. The quantity of positive outcomes in an experiment with 'n' outcomes is represented by x. The following equation can be used to determine the likelihood of an event:
Probability (event) = positive outcome / overall outcome = x/n
It refers to the likelihood of a Type I mistake.
The likelihood that a Type I error will occur, or the likelihood that a null hypothesis will be rejected when it is true, determines the degree of significance.
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Write the sum using summation notation. There may be multiple representations. Use i as the index of
summation.
1/3 - 1/9 + 1/27 - 1/81 + 1/243 - 1/729
The summation notation of the sum is given as follows:
[tex]\sum_{i = 1}^{6} \frac{(-1)^{i-1}}{3^i}[/tex]
How to write the summation notation?The summation notation of the sum is written considering the patterns of the terms that compose the terms.
First we look at the fact that the sum has six terms, and the first term is considered as having index 1, hence:
[tex]\sum_{i = 1}^{6}[/tex]
Then we consider that the signal is alternating, with the terms with odd indices being positive and those with even indices being negative, hence the numerator is of:
[tex](-1)^{i - 1}[/tex]
(any of -1 or +1 would work here).
Then we considering that the denominator is composed by the powers of 3, hence it is given by:
[tex]3^i[/tex]
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Name Yala. Al Write down each of the following expressions in the form of a power of one number a) 5³x5⁴ b) (-7)² ×(-7)³ c) 3⁶x3³ d) 27⁰x27² e) (-2)² ×(-2)³ × (-2)⁴ f) 3⁴x9²x(-3)⁶ g) (-4)⁶ × (-4)⁵ (-4)³ h) (-8)x(-8)x2² i) (0.25)×(0.25)⁴ × (0.25)⁰ J) 7³x7²x7
Answer:
A. [tex] {5}^{7} [/tex]
B. [tex] ( { - 7})^{5} [/tex]
C. [tex] {3}^{9} [/tex]
D. [tex] {27}^{2} [/tex]
E. [tex] ({ - 2})^{9} [/tex]
F. [tex] {3}^{14} [/tex]
G. [tex] {( - 4)}^{14} [/tex]
H. [tex] {2}^{8} [/tex]
I. [tex] {(0.25)}^{5} [/tex]
J. [tex] {7}^{6} [/tex]
Step-by-step explanation:
A.
[tex] {5}^{3} \times {5}^{4} \\ = {5}^{3 + 4} \\ = {5}^{7} [/tex]
B.
[tex]( { - 7})^{2} \times ( { - 7})^{3} \\ = ( { - 7})^{2 + 3} \\ = ( { - 7})^{5} [/tex]
C.
[tex] {3}^{6} \times {3}^{3} \\ = {3}^{6 + 3} \\ = {3}^{9} [/tex]
D.
[tex] {27}^{0} \times {27}^{2} \\ = {27}^{0 + 2} \\ = {27}^{2} [/tex]
E.
[tex]( { - 2})^{2} \times ( { - 2})^{3} \times ( { - 2})^{4} \\ = ( { - 2})^{2 + 3 + 4} \\ = ({ - 2})^{9} [/tex]
F.
[tex] {3}^{4} \times {9}^{2} \times ( { - 3})^{6} \\ = {3}^{4} \times {3}^{2 \times 2} \times {3}^{6} \\ = {3}^{4} \times {3}^{4} \times {3}^{6} \\ = {3}^{4 + 4 + 6} \\ = {3}^{14} [/tex]
G.
[tex] {( - 4)}^{6} \times {( - 4)}^{5} \times {( - 4)}^{3} \\ = {( - 4)}^{6 + 5 + 3} \\ = {( - 4)}^{14} [/tex]
H.
[tex] ( - 8) \times ( - 8) \times {2}^{2} \\ = 8 \times 8 \times {2}^{2} \\ = {2}^{3} \times {2}^{3} \times {2}^{2} \\ = {2}^{3 + 3 + 2} \\ = {2}^{8} [/tex]
I.
[tex] 0.25 \times {(0.25)}^{4} \times {(0.25)}^{0} \\ = {(0.25)}^{1} \times {(0.25)}^{4} \times {(0.25)}^{0} \\ = \times {(0.25)}^{1+4+0} \\ = \times {(0.25)}^{5} [/tex]
J.
[tex] {7}^{3} \times {7}^{2} \times 7 \\ = {7}^{3} \times {7}^{2} \times {7}^{1} \\ = {7}^{3+2+1} \\ = {7}^{6} [/tex]
20
Use the box and whisker plot to answer the following question.
73
81 84
70
→
65 70 75
80
Low temperatures (°F)
85
87
王
90
What is the interquartile range IQR of the data set? (IQR = Q3 - Q1)
The interquartile range IQR for the given data set is IQR=10
What is block and whisker plot?A box and whisker plot, often known as a box plot, shows a data set's five-number summary. The minimum, first quartile, median, third quartile, and maximum make up the five-number summary. A box is drawn from the first quartile to the third quartile in a box plot. At the median, a vertical line passes through the box.
How do you calculate box and whisker plot?Finding the median, lower quartile, and upper quartile of a given set of data (Q2, Q1, and Q3) is the first stage in creating a box-and-whisker figure. Now that you know how to calculate the interquartile range (IQR). The distance between the upper and lower quartiles is known as the interquartile range.
Lower data set:
70 73 81 84
Q1=73+81/2=77
Middle data set
65 70 75 80
Q2= 70+75/2=72.5
Upper data set
85 87 90
Q3=87
IQR=Q3-Q1=87-77=10
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How do you write 145,000,000,000 in scientific notation?
The scientific notation is [tex]1.45*10^{11}[/tex]
Scientific notation is a way to present extremely large or extremely small numbers in a more understandable way. We are aware that full numbers can go on forever, but we are unable to write such enormous figures on paper. Additionally, a simpler method of representation was required for the numbers that appear at the millions place after the decimal. This makes it challenging to represent a small number of integers in their enlarged form. We, therefore, employ scientific notations.
We need to move the decimal point 11 places to the left therefore the exponent for the 10s term will be positive:
[tex]145,000,000,000=145,000,000,000.0=1.45*10^{11}[/tex]
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-4-0.2x+11=-0.8x+10
please help meeeeeee i need the answerrrr
The solution of the expression - 4 - 0.2x + 11 = - 0.8x + 10 is 28.33.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, A numerical expression and we have to solve for x or find the value of x.
- 4 - 0.2x + 11 = - 0.8x + 10.
- 0.2x - 7 = - 0.8x + 10.
- 0.2x + 0.8x = 10 + 7.
0.6x = 17.
x = 17/0.6.
x = 28.33.
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Fifteen less than twice a number is the same as the number plus two. What is the number?
The number is 17.
The first part is "Fifteen less than twice a number".
Let us say that n represents the unknown number.
Twice a number is the same as two times the number or 2n, and fifteen less than 2n is 2n−15.
The next part "is the same as the" part of the word problem indicating that the two parts are equal. So we use an equal sign to represent it⇒2n−15 =
The third part of the problem is "the number plus two". This is the same as n+2
The final equation is 2n-15=n+2
To solve for n, we would add 15 to both sides: 2n=n+17
Then subtract 1n from both sides: n=17
The number is 17.
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Use I=PRTI=PRT to find the total amount in the savings account. Type in "money form".
$4500.00 at 8% for 3 years
The total amount earned after 3 years at 8% rate is $5580
What is interest?Interest is the price you pay to borrow money or the cost you charge to lend money.
Given that, $4500 is invested at 8% for 3 year,
SI = PxRxT/100
= 4500x8x3/100
= 1080
Amount = interest + principal
Amount = 1080 + 4500 = 5580
Hence, The total amount earned after 3 years at 8% rate is $5580
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An oil company fills 1 over 12 of a tank in 1 over 4 hour with gasoline. At this rate, which expression can be used to determine how long it will take for the tank to fill completely? (1 point)
Group of answer choices
1 over 4⋅ 1 over 12
1 over 12⋅ 4
1 over 4⋅ 12
12 ⋅ 4
The expression that can be used to determine how long it will take for the tank to fill completely is C. 1 over 4 . 12.
How to illustrate the rate?Ratio demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B). This indicates that you're dividing information A by B. For instance, the ratio will be 5/10 if A is 5 and B is 10.
In this case, the oil company fills 1 over 12 of a tank in 1 over 4 hour with gasoline.
The rate will be:
= 1/4 ÷ 1/12
= 1/4 × 12
= 3
The correct option is C
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Which column of elements is the most reactive?
The alkali metals, which are located in Group 1A of the periodic table, are the most reactive metals.
The blue column on the left of the periodic table contains the alkali metals, which are the most reactive metals. As we move down column (group) one, which is depicted in blue, their responsiveness improves.
What are Alkali metals ?
Alkali metals are those that react with water to produce strong bases.
It is composed of six elements collectively referred to as the lithium family. These metals have one electron in their outermost shell, making them extremely reactive elements of the periodic table.
A few instances of alkali metal
lithium (Li), sodium (Na), rubidium (Rb), potassium (K), francium (Fr) and Caesium (Cs).
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Line segment su is dilated to create s'u' using point q as the center of dilation. the scale factor of the dilation is
The scale factor of the dilation is 2 which can be shown in the details given below.
In geometry, a line segment is bounded with the aid of using awesome factors on a line. Or we are able to say a line phase is a part of the road that connects factors. A line has no endpoints and extends infinitely in each the course however a line phase has constant or particular endpoints.
QS : QS'
= 4 : 8
= QU : QU'
= 5 : 10
= 1 : 2
Then the scale factor is of the dilation is 2 (magnification).
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Complete question
Line segment SU is dilated to create S'U' using point Q as the center of dilation.
The scale factor of the dilation is:
I need to simplify the exponent for my math test asap.
Calculate the distance between the points Q=(1,0) and B= (-2.-3) in the coordinate plane.
Round your answer to the nearest hundredth.
Answer: QB≈4.24
Step-by-step explanation:
Q(1,0) B(-2,-3) QB=?
[tex]\boxed{L=\sqrt{(x_2-x_1)+(y_2-y_1)^2} }\\\\x_1=1\ \ \ \ x_2=-2\ \ \ \ y_1=0\ \ \ \ y_2=-3[/tex]
Hence,
[tex]QB=\sqrt{(-2-1)^2+(-3-0)^2} \\\\QB=\sqrt{(-3)^2+(-3)^2} \\\\QB=\sqrt{9+9} \\\\QB=\sqrt{18} \\\\QB=\sqrt{9(2)} \\\\QB=\sqrt{3^2(2)} \\\\QB=3\sqrt{2} \\\\QB\approx4.24[/tex]
like terms onto each other to simplify fully.
6+3x-6+4+5y
The simplified form of the given expression is 3x+5y+4. Here only the constants are simplified since there are no like terms in the expression.
What are like terms?An expression is the combination of terms with any of the arithmetic operations in between them. Here the terms are formed by a coefficient and a variable.In an expression, if two or more terms with the same variable with the same power are said to be like terms otherwise they are said to be unlike terms.The like terms undergo addition and subtraction operations.But the unlike terms can't be added or subtracted.Calculation:The given expression which is to be simplified is 6+3x-6+4+5y.
Here to simplify the given expression, we need to add or subtract the like terms.
In the given expression, the variables x and y.
They are unlike terms. So, we cannot add or subtract them. But there are three constants. Then, we can add or subtract them.
Thus, the simplified expression for the given expression is
6+3x-6+4+5y = 3x+5y+4
Here 6 and -6 are added so that we get 0.
Hence the answer is 3x+5y+4.
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e altitude of a triangle is increasing at a rate of 1 cm!min while the area of the triangle is increasing at a rate of 2 cm !min. at what rate is the base of the triangle changing when the altitude is 10 cm and the area i
The base of the triangle is decreasing at the rate of 1.6 cm/min.
Given
dh/dt = 1 cm/min
dA/dt = 2 cm2/min
When A = 100 cm2, h = 10 cm
We know that
A = 1/2 × b × h
b = 20 cm
Let us differentiate both sides
dA/dt = 1/2 [b dh/dt + h db/dt]
Substituting the values
2 = 1/2 [20 × 1 + 10 × db/dt]
By further simplification
4 = 20 + 10 db/dt
db/dt = -1.6 cm/min
Therefore, the base of the triangle is decreasing at the rate of 1.6 cm/min.
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there
Write an equivalent expression for 6a + a + 8b + b.
A
Answer:
7a+9b
Step-by-step explanation:
8y−5<3 = ??
please help me im so confused
Answer:
It looks like you have an inequality equation with one variable, but the equation is incomplete. In order to solve the inequality, we need to have a complete equation with a left-hand side and a right-hand side.
For example, if we wanted to solve the inequality 8y - 5 < 3, we could complete the equation by adding 5 to both sides:
8y - 5 + 5 < 3 + 5
8y < 8
Then, we could divide both sides of the inequality by 8 to solve for y:
8y/8 < 8/8
y < 1
Therefore, the solution to the inequality 8y - 5 < 3 is y < 1.
Would a square or a circle contain more area if they have enough brick to make a 10-kilometer fence/wall? Please show your calculations.
Length of wall = 10km
side of square= 10km
area of square= 10²= 100km²
radius of circle = 10/2 = 5km
area of circle = 3.14 × 5²
area of circle = 3.14 × 25
area of circla = 78.5 km²
area of square >>> area of circle
Please help!!
Find the y-intercept
Answer:
-2,0
very simple it is so easy
find an equation of the tangent plane to the given parametric surface at the specified point. x = u v, y = 8u2, z = u − v; (2, 8, 0)
The equation for the normal line B gives us :
L: (x,y,z) = (0,0,7) + t(1, 1, 1), t ∈ R
Find the solution ?
Full question is:
Find equations of the tangent plane and the normal line to the given surface at the specified point. x + y + z = 7e(xyz) at a specified point (0, 0, 7)
If we have a level surface, then it will give us
f(x,y,z) = x+y+z = 7e(xyz). where f is the function of the x, y, and z coordinates.
Now let us calculate the ∇ gradient of f at point (0,0,7):
∇ = (fx,fy,fz) = (1−7yz,1−7xz,1−7xy)
= (1, 1, 1)
We get the equation for the tangent plane A:
A: 1(x−0) + 1(y−0) + 1(z−7)=0
This can also be written as:
x+y+z = 7 ------------------------------------------------------------------(a)
The equation for the normal line B gives us :
L: (x,y,z) = (0,0,7) + t(1, 1, 1), t ∈ R --------------------------------(b)
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Which of the following is a factor of (x+y)³ - (x³+y³) ?
choose the correct answer
x²+xy²+2xy
x²+y²-xy
xy²
3xy
Answer:
3 x y (y + x)
Step-by-step explanation:
Factor the following:
(y + x)^3 - (y^3 + x^3)
Factor the sum of two cubes. y^3 + x^3 = (y + x) (y^2 - x y + x^2):
(y + x)^3 - ((y + x) (y^2 - x y + x^2))
Factor y + x out of (y + x)^3 - (y + x) (y^2 - x y + x^2), resulting in (y + x) ((y + x)^(3 - 1) - (y^2 - x y + x^2)):
(y + x) ((y + x)^(3 - 1) - (y^2 - x y + x^2))
3 - 1 = 2:
(y + x) ((y + x)^2 - (y^2 - x y + x^2))
(y + x) (y + x) = (x) (x) + (x) (y) + (y) (x) + (y) (y) = x^2 + x y + x y + y^2 = y^2 + 2 x y + x^2:
(y + x) ((y^2 + 2 x y + x^2) - (y^2 - x y + x^2))
-(y^2 - x y + x^2) = -y^2 + x y - x^2:
(y + x) (x^2 + 2 x y + y^2 + (-y^2 + x y - x^2))
Grouping like terms, y^2 - y^2 + 2 x y + x y + x^2 - x^2 = (2 x y + x y) + (y^2 - y^2) + (x^2 - x^2):
((2 x y + x y) + (y^2 - y^2) + (x^2 - x^2)) (y + x)
x y 2 + x y = x y×3:
(3 x y + (y^2 - y^2) + (x^2 - x^2)) (y + x)
y^2 - y^2 = 0:
(3 x y + (x^2 - x^2)) (y + x)
x^2 - x^2 = 0:
Answer: 3 x y (y + x)
a linear ___ is a mathematical statement that two linear expressions, or a linear expression and a constant, are not equal.
A linear equation is used to prove the mathematical result that two linear expressions, or a linear expression and a constant, are not equivalent.
What is a linear equation?The mathematical assertion that two linear expressions, or a linear expression and a constant, are not equal is made via a linear equation.
The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.
A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included.
The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."
A mathematical equation must take the form y=mx+b in order to be classified as a linear function (in which m is the slope and b is the y-intercept).
Therefore, a linear equation is used to prove the mathematical result that two linear expressions, or a linear expression and a constant, are not equivalent.
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what number multiplied by 3 and subtracted by 2 equals 40
The number multiplied by 3 and subtracted by 2 equals 40 is 14.
Linear Equation:The Equation that is used to find an unknown value is known as Linear Equation. To find an unknown value or quantity, first form a linear equation according to the conditions that are given the problem, here we use variables like x, y, etc. to represent the unknown value.
Here we have
A Number is multiplied by 3 and subtracted by 2 equals 40
Let the number be x
Given that 'x' is multiplied by 3 (3x); subtracted by 2 (3x-2) equals to 40
=> 3x - 2 = 40
Add 2 on both sides
=> 3x - 2 + 2 = 40 + 2
=> 3x = 42
Divide by 3 into both sides
=> 3x/3 = 42/3
=> x = 14
Therefore,
The number multiplied by 3 and subtracted by 2 equals 40 is 14.
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