Answer:
7^3
Step-by-step explanation:
since seven is being multiplied by itself 3 times, it is being cubed. So it needs to be raised to the third power.
hope this helps!!! :3
Answer: (-7)(-7)(-7) is equal to -7^3 (just imagine that the three is on the top right outer corner of the 7)
Step-by-step explanation:
A power is when you multiply a number by itself a certain number of times, in this case, we are multiplying the number 7 by itself 3 times. Every time it is multiplied by its self, the power of that number goes up (if you are using a power of one for any number, it will stay the same). Here is an example:
-7 to the first power is just -7.
-7 to the second power, or squared is just (-7)(-7) or (-7) x (-7).
-7 to the third power, or cubed, is just (-7)(-7)(-7) or (-7) x (-7) x (-7).
Hope this helps!!!
Find the base area of a cone that
has a slant height of 4 inches and a
radius of 4 inches.
Answer:
AB≈50.27in²
Step-by-step explanation:
AB=πr2=π·42≈50.26548in²
Answer: 16π squared inches
Step-by-step explanation:
The base area of a cone is the area of a circle.
Formula for area of a circle = π[tex]r^{2}[/tex]
Since your r= 4 inches, you plug in 4 inches into the formula.
Area = π * [tex]4^{2}[/tex] = 16π
how to calculate standard deviation and variance with 7.2, 7.6, 8.5, 8.7, 9.0, 9.3, 9.4, 10.2, 10.9, 11.3
After determining the mean the variance is 1.5689 and standard deviation is 1.253.
The values are 7.2, 7.6, 8.5, 8.7, 9.0, 9.3, 9.4, 10.2, 10.9, 11.3.
Firstly we find the mean of the numbers.
Mean = Sum of the Numbers/Total numbers
Mean = (7.2 + 7.6 + 8.5 + 8.7 + 9.0 + 9.3 + 9.4 + 10.2 + 10.9 + 11.3)/10
Mean = 92.1/10
Mean = 9.21
Now we determine the variance.
The formula of variance is:
x x - Mean (x - Mean)^2
7.2 (7.2-9.21) = -2.01 4.0401
7.6 (7.6-9.21) = -1.61 2.5921
8.5 (8.5-9.21) = -0.71 0.5041
8.7 (8.7-9.21) = -0.51 0.2601
9.0 (9.0-9.21) = -0.21 0.0441
9.3 (9.3-9.21) = 0.09 0.0081
9.4 (9.4-9.21) = 0.19 0.0361
10.2 (10.2-9.21) = 0.99 0.9801
10.9 (10.9-9.21) = 1.69 2.8561
11.3 (11.3-9.21) = 2.09 4.3681
Total = 15.689
Variance = ∑(x - Mean)^2/10
Variance = 15.689/10
Variance = 1.5689
Standard Deviation = √Variance
Standard Deviation = √1.5689
Standard Deviation = 1.253
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when should you use the sum rule? what about the product rule? group of answer choices use the sum rule when determining the probability of independent events occurring together, the product rule when determining the probability of mutually exclusive events occurring use the product rule when determining the probability of independent events occurring together, the sum rule when determining the probability of mutually exclusive events occurring
The sum can be used when sum gives the situations in which the probability of a union of events can be calculated by summing probabilities together. and product rule If there are n(A) ways to do A and n(B) ways to do B, then the number of ways to do A and B is n(A) × n(B)
What is Probability?
Probability is an area of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.
Reason:
The probability rule of sum describes scenarios in which the probability of a union of events can be computed by adding probabilities. It is frequently used on mutually exclusive occurrences, which are events that cannot occur at the same moment.
If there are n(A) ways to perform A and n(B) ways to do B, then the total number of ways to accomplish A and B is n(A) n. (B). This is true if the number of possible ways to do A and B are independent; the number of options for doing B is the same regardless of the option you chose for A.
The probability product rule states that P(AnB) = P(A) * P (B)
This implies that the events must be distinct.
In contrast, for the probability sum rule.
P(A + B) = P(A) + P(B) (B)
This suggests that P(AnB) = 0, indicating that they are disjoint or mutually exclusive.
Box 2 states that the sum rule is used when events are independent, however this is not correct; it is employed when events are mutually exclusive.
Two events are said to be mutually exclusive in probability theory if they cannot occur at the same time or concurrently. In other words, discontinuous events are those that are mutually exclusive. If two occurrences are regarded discontinuous, the likelihood of both occurring at the same time is zero.
If A and B are the two events, then the probability of their disjoint is stated as:
P (A and B) = 0 for the probability of a disjoint (or mutually exclusive) event.
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The sum can be used when sum gives the situations in which the probability of a union of events can be calculated by summing probabilities together. and product rule If there are n(A) ways to do A and n(B) ways to do B, then the number of ways to do A and B is n(A) × n(B)
What is Probability?
Probability is an area of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.
Reason:
The probability rule of sum describes scenarios in which the probability of a union of events can be computed by adding probabilities. It is frequently used on mutually exclusive occurrences, which are events that cannot occur at the same moment.
If there are n(A) ways to perform A and n(B) ways to do B, then the total number of ways to accomplish A and B is n(A) n. (B). This is true if the number of possible ways to do A and B are independent; the number of options for doing B is the same regardless of the option you chose for A.
The probability product rule states that P(AnB) = P(A) * P (B)
This implies that the events must be distinct.
In contrast, for the probability sum rule.
P(A + B) = P(A) + P(B) (B)
This suggests that P(AnB) = 0, indicating that they are disjoint or mutually exclusive.
Box 2 states that the sum rule is used when events are independent, however this is not correct; it is employed when events are mutually exclusive.
Two events are said to be mutually exclusive in probability theory if they cannot occur at the same time or concurrently. In other words, discontinuous events are those that are mutually exclusive. If two occurrences are regarded as discontinuous, the likelihood of both occurring at the same time is zero.
If A and B are the two events, then the probability of their disjoint is stated as:
P (A and B) = 0 for the probability of a disjoint (or mutually exclusive) event.
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let a and b be two possible events when a random number generator is used once. suppose you are told that a and b are mutually exclusive. this implies that the probability that either A. A happens or B happens is equal to theproduct of the probabilities of A and B B. the probability that A and B both happen is equal to the sum ofthe probabilities of A and B C. the probability that either A happens or B happens is equal to thesum of the probabilities of A and B D. the probability that A happens given that B happens is equal tothe probability that A happens E. the probability that A and B both happen is equal to the productof the probabilities of A and B
The probability that either A happens or B happens is equal to the sum of the probabilities of A and B. Option C is the answer.
What is a probability?Probability is a measure of the likelihood of a particular event occurring, expressed as a number between 0 and 1. A probability of 0 means an event is impossible, while a probability of 1 means an event is certain. A probability of 0.5 means an event has a 50-50 chance of happening.
In the question given, the probability that either A happens or B happens is equal to the sum of the probabilities of A and B.
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How do you convert 68 Kilogram (kg) to Pound (lb)?
There are 149.91 pounds in 68 kilogram, or 68 kilogram weighs 149.91 pounds.
How much in kilogrammes are converted to pounds?The Price Per Kilogram (Pk) shall replace the Price Per Pound (Pp) (Kg)
Just divide the kg price by 2.2046 to get the lb price to convert cost per kilogramme to cost per pound.
In principle, two pounds are equal to one kilogramme (there is a longer decimal position, but I abbreviate it to 2.2046).
The weight of a kilogramme is roughly 2.2046226218 pounds. The technique can be used to convert kilogrammes to pounds and is relatively simple. The value of a kilogramme merely needs to be multiplied by 2.2046226218.
68 kg weights 149.91 pounds, or 68 kg is 149.91 pounds. Conversion of wages 149.91434 pounds (lb), 68 kilogrammes (kg)
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15. The Atlanta Braves had a run differential of +134 in the 2021 season. What does this mean?
O A. They scored a total of 134 runs
O B. They scored 134 more rurks than they allowed their opponents to score.
O C. They allowed their opponents to score a total of 134 runs.
OD. They allowed their opponents to score 134 more runs than they scored.
They allowed their opponents to score 134 more runs than they scored.
What is the differential equation?
A differential equation is an equation that contains at least one derivative of an unknown function, either a normal differential equation.
A differential equation is an equation that contains at least one derivative of an unknown function, either a normal differential equation or a partial differential equation.
A run differential of +134 for the Atlanta Braves in the 2021 season means that the team scored 134 more runs than their opponents over the course of the season.
A positive run differential indicates that a team has outscored their opponents, while a negative run differential means that a team has been outscored by their opponents.
Hence, They allowed their opponents to score 134 more runs than they scored.
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If AB = 9 and AB ∣∣∣∣ DC, what is the perimeter of ABCD?
Answer:
38
Step-by-step explanation:
Since the figure shown of ABCD has two pairs of parallel lines, it can be classified as a parallelogram.
Opposite sides are congruent in a parallelogram, so:
AB = CD = 9
BC = AD = 10
To find the perimeter you sum together all the sides. Therefore the perimeter of ABCD would be AB + BC + CD + AD. This is equal to:
9 + 10 + 9 + 10 = 38
Write these ratios in their simplest form
a 9:18
Answer:
1 : 2
Step-by-step explanation:
9 : 18
Both numbers are divisible by 9, thus:
9/9 : 18/2
[1 : 2]
What is irreflexive closure?
Answer: The irreflexive closure of a relation R R R cannot contain any elements of the form ( a , a ) ∉ R (a,a)\notin R (a,a)∈/R (by the definition of reflexive).
Step-by-step explanation:
Plans for a new park call for gardens directly across the sidewalk from each other to be congruent. This computer printout shows a rose garden.
If the vertices of a tulip garden are located at (x1,−y1), (x2,−y2), (x3,−y3), and (x4,−y4), will the tulip garden be congruent to the rose garden?
The lengths of all four sides of the tulip garden are equal to the lengths of the corresponding sides of the rose garden, then the two gardens will be congruent.
In order to determine if the tulip garden will be congruent to the rose garden, we must first calculate the lengths of the corresponding sides of each garden. The rose garden's vertices are labeled (x1, y1), (x2, y2), (x3, y3), and (x4, y4). The tulip garden's vertices are labeled (x1, −y1), (x2, −y2), (x3, −y3), and (x4, −y4). The length of the first side of the rose garden can be found using the distance formula, [tex]d = √((x2−x1)^2 + (y2−y1)^2)[/tex]. The corresponding length of the tulip garden can be found using the same distance formula, d = √((x2−x1)^2 + (y2−y1)^2) We can continue this process for the other three side of both gardens. If the lengths of all four sides of the tulip garden are equal to the lengths of the corresponding sides of the rose garden, then the two gardens will be congruent.
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Complete the sentence. Then, use the laws of exponents to write an equivalent expression for the given expression.
To divide powers that have the same base and different exponents, keep the ___ and ___ the exponents.
To divide powers that have the same base and different exponents, keep the base and subtract the exponents.
How to divide exponents with same baseWhen dividing two powers that have the same base but different exponents, you keep the base and subtract the exponents.
For example, if you want to divide a^m by a^n,
you can write it as a^m / a^n
= a^(m-n).
The new exponent is the result of subtracting the exponents of the two powers.
hence the division will result to a^(m-n).
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Find the balance if you deposit 5000 that earns 4% annual interest compounded yearly for 6 years
Answer:
The balance after 6 years, assuming the deposit of 5000 earns 4% interest compounded yearly, would be 7400. This is calculated by using the formula A=P(1+r/n)^nt, where A is the balance, P is the principal (5000), r is the interest rate (0.04), n is the number of times the interest is compounded per year (1), and t is the number of years (6).
Given p(x) = -37 +2/5(x²-14), what is the value of p(-9)?
The value of p(-9) is -10.2.
What is an Equation?An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side.
Given is an equation,
p(x) = -37 + (2/5) (x² - 14)
To find the value of p(-9), substitute -9 in the place of x.
p(-9) = -37 + (2/5) ((-9)² - 14)
= -37 + (2/5) (81 - 14)
= -37 + (2/5) × 67
= -37 + 26.8
= -10.2
Hence the value of the function when x = -9 is -10.2.
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in a survey, the of respondents are identified as for for for and for anything else. the average (mean) is calculated for respondents and the resu
Write an equation:
"Ehren worked 36 hours last week and earned a total of $500"
Please help me ;-; Thanks!
The correct equation to represent "Ehren worked 36 hours last week and earned a total of $500" is,
⇒ 36x = 500
Where, x represent the amount earn in per hour.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
We have to given that;
The expression is,
"Ehren worked 36 hours last week and earned a total of $500"
Let the amount earn in per hour = x
Hence, For Ehren worked 36 hours last week and earned a total of $500, the correct equation is,
⇒ 36x = 500
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The length of a cell phone is 1.21.2 inches and the width is 3.43.4 inches. The company making the cell phone wants to make a new version whose length will be 1.141.14 inches. Assuming the side lengths in the new phone are proportional to the old phone, what will be the width of the new phone?
Assuming the side lengths in the new phone are proportional to the old phone, the width of the new phone would be equal to 3.23 inches.
What is a proportion?In Mathematics, a proportion simply refers to an equation which is typically used to represent the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.
By applying direct proportion to the given information, we have the following mathematical expression:
1.2 inches = 3.4 inches
1.14 inches = w inches
Cross-multiplying, we have the following:
1.2w = 3.4 × 1.14
1.2w = 3.876
w = 3.876/1.2
w = 3.23 inches.
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Consider the problem of evaluating the function sin(x), in particular, the propagated data error, i. E. , the error in the function value due to a perturbation h in the argument x. (?) Estimate the absolute error in evaluating sin (x). (b) Estimate the relative error in evaluating sin(x). (c) Estimate the condition number for this problem. (d) For what values of the argument x is this problem highly sensitive?
The absolute error in evaluating sinx is [tex]cos x \triangle x[/tex].
(a)Consider the problem of evaluating the value of sin x.
The objective is to estimate the absolute error in sin x.
Assume r = sin x
The absolute error formula gives that:
[tex]\triangle r = \frac{\delta r}{\delta x} \triangle x[/tex]
[tex]= \frac{\delta sin x}{\delta x} \triangle x[/tex]
[tex]= cos x \triangle x[/tex]
(b) The objective is to estimate the relative error in sin x.
The relative error is given by:
[tex]\frac{\triangle r}{r} = \frac{cos x \triangle x}{sin x}[/tex]
[tex]= cot x \triangle x[/tex]
(c) The objective is to estimate the condition number for this problem.
The formula of a function r = f(x), its condition is given by:
[tex]\frac{x f'(x)}{f(x)}[/tex], here r = sin x
The condition number at point x is:
[tex]\frac{x\frac{d(sin x)}{dx} }{sin x}[/tex]
[tex]= \frac{x cos x}{sin x}[/tex]
= x cot x
(d) The objective is to find for what values of argument x is the problem highly sensitive.
Here, condition number is xcot x.
The cot function is highly sensitive at x=0.
Hence, the argument x is highly sensitive for x=0.
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What number must you multiply the second equation by so that the sum of the x-coefficients is zero?
7x−y=55
x+3y=33
1. -7
2. 1/3
3. -1/3
4. 7
-7 must you multiply the second equation by so that the sum of the x-coefficients is zero.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The system of equations are
7x−y=55..(1)
x+3y=33..(2)
We have to multiply with -7 second equation by so that the sum of the x-coefficients is zero
-7x-21y=-231..(3)
Add equation 1 and equation 2
7x−y-7x-21y=33-231
-22y=-198
Hence, -7 must you multiply the second equation by so that the sum of the x-coefficients is zero.
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how to find the empirical formula from percentages
The empirical formula of a compound is the simplest and lowest whole number ratio of the atoms of each element that makes up a compound.
To find the empirical formula from percentages, you need to first calculate the mass of each element present in the compound, based on the percentage of each element. Once you have the mass of each element, divide each mass by the atomic mass of that element to get the number of moles of each element. Then divide each mole by the smallest number of moles to get the mole ratio. This ratio is the empirical formula for the compound.
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solve the systen of linear equations by elimintion x-2y=-7 3x+2y=7
The solution to the equation is (0, 7/2)
How to solve the systen of linear equationsFrom the question, we have the following parameters that can be used in our computation:
x-2y=-7 3x+2y=7
Add the two equations to elimiate y
4x = 0
So, we have
x = 0
This means that
0 - 2y = -7
So, we have
y = 7/2
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what is the value of (x - y) (x - y) if xy = 3 and x2 y2 = 25?
Answer:
19
Step-by-step explanation:
step by step
ind the values of the trigonometric functions of t from the given information. csc(t) = 7, cos(t) < 0
So, the values of the trigonometric functions of t are sin(t) = 1/7, cos(t) = -√(1 - (1/7)^2) and tan(t) = -(1/7)/(-√(1 - (1/7)^2)).
Given that csc(t) = 7 and cos(t) < 0, we can find the values of the other trigonometric functions of t using the relationships between the trigonometric functions.
csc(t) = 1/sin(t), so sin(t) = 1/7
Since cos(t) < 0, t is in the second or third quadrant, where sin(t) is positive. Hence, t is in the third quadrant.
Therefore, the values of the trigonometric functions of t are:
sin(t) = 1/7
cos(t) = -√(1 - sin^2(t)) = -√(1 - (1/7)^2)
tan(t) = sin(t)/cos(t) = -(1/7)/(-√(1 - (1/7)^2))
Note that the values of the other trigonometric functions, such as cot(t), sec(t), and csc(t), can be found using the definitions of these functions and the values of sin(t) and cos(t) that we have found.
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So, the values of the trigonometric functions of t are [tex]Sin(t)=\frac{1}{7}[/tex], [tex]Cos(t)=-\sqrt{[1-(\frac{1}{7})^2] }[/tex] and [tex]tan(t)=\frac{\frac{1}{7} }{-\sqrt{1-(\frac{1}{7})^2 } }[/tex].
Given that csc(t) = 7 and cos(t) < 0, we can find the values of the other trigonometric functions of t using the relationships between the trigonometric functions.
[tex]Cosec(t)=7[/tex]
We known
⇒ [tex]Sin(t)=\frac{1}{Cosec(t)} =\frac{1}{7}[/tex]
Since cos(t) < 0, t is in the second or third quadrant, where sin(t) is positive. Hence, t is in the second quadrant.
Therefore, the values of the trigonometric functions of t are:
⇒ [tex]Sin(t)=\frac{1}{7}[/tex]
⇒ [tex]Cos(t)=-\sqrt{[1-(Sin^2t)]}[/tex]
⇒ [tex]Cos(t)=-\sqrt{[1-(\frac{1}{7})^2] }[/tex]
⇒ [tex]tan(t)=\frac{Sin(t)}{Cos(t)}[/tex]
⇒ [tex]tan(t)=\frac{\frac{1}{7} }{-\sqrt{1-(\frac{1}{7})^2 } }[/tex]
∴ Note that The definitions of the other trigonometric functions, along with the values of Sin(t) and Cos(t), can be used to determine the values of cot(t), Sec(t), and Cosec(t).
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consider the following data: −11,−5,−5,−11,13,−11,−5 step 2 of 3 : calculate the value of the sample standard deviation. round your answer to one decimal place.
The value of the sample variance rounded to one decimal place is 76.0.
What is the Mean?
The mean is a measure of the central tendency of a set of data, calculated as the sum of the values divided by the number of values in the set.
The sample variance of a set of data is a measure of the spread or dispersion of the data around the mean. To calculate the sample variance, you can follow these steps:
Calculate the mean of the data: First, find the sum of all the data points and divide by the number of data points (n).
For the given data, the mean is: (-11 + -5 + -5 + -11 + 13 + -11 + -5) / 7 = -7
Subtract the mean from each data point: Next, subtract the mean from each data point to get the deviations from the mean.
For the given data, the deviations are: (-11 - -7) = -4, (-5 - -7) = 2, (-5 - -7) = 2, (-11 - -7) = -4, (13 - -7) = 20, (-11 - -7) = -4, (-5 - -7) = 2
Square the deviations: Square each deviation to get rid of negative values.
For the given data, the squared deviations are: (-4)^2 = 16, (2)^2 = 4, (2)^2 = 4, (-4)^2 = 16, (20)^2 = 400, (-4)^2 = 16, (2)^2 = 4
Sum the squared deviations: Sum all the squared deviations.
For the given data, the sum of the squared deviations is: 16 + 4 + 4 + 16 + 400 + 16 + 4 = 456
Divide the sum by n - 1: Finally, divide the sum of the squared deviations by the number of data points minus 1 (n - 1). This gives us the sample variance.
For the given data, the sample variance is: 456 / (7 - 1) = 456 / 6 = 76
Hence, the value of the sample variance rounded to one decimal place is 76.0.
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Determine, for the typical algorithms that you use to perform calculations by hand, the running time to do the following: a. Add two N-digit integers. B. Multiply two N-digit integers. C. Divide two N-digit integers
The running time of the algorithms of addition, multiplication and division of two numbers are O(N), O(N^2), and O(N^2) respectively.
For the typical algorithms used to perform calculations by hand:
A. Adding two N-digit integers typically takes O(N) time. This is because each digit of the larger number needs to be added to the corresponding digit of the other number and the carry over needs to be taken into account.
B. Multiplying two N-digit integers typically takes O(N^2) time. This is because each digit of one number needs to be multiplied by each digit of the other number, and the intermediate results need to be summed.
C. Dividing two N-digit integers typically takes O(N^2) time. This is because the process of long division involves repeatedly dividing, multiplying, and subtracting the digits of the dividend and the divisor until the result is obtained.
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87 is 120% of what number?
Answer: 72.5
Step-by-step explanation:
120% = 120/100 = 1.2
1.2*x = 87
x = 87/1.2 = 72.5
For what values of h and k the system is consistent ?a)2h 2x1 – X2b)-6x1 + 3x2 k
For the system of equation " 2x₁ – x₂ = h ; -6x₁ + 3x₂ = k " , it will be consistent for all the values of h and k , which satisfy "k = -3h" .
The System Of Linear Equations is considered as consistent if it has at least one solution.
The system of equation is :
⇒ 2x₁ – x₂ = h .....equation(1) ;
⇒ -6x₁ + 3x₂ = k ....equation(2) ;
Now replacing the equation(2) by its sum with 3 times the first equation, the the system of equations become :
⇒ 2x₁ – x₂ = h .....equation(1) ;
⇒ 0 = k + 3h ;
Simplifying further ,
we have ;
⇒ k = -3h .
Therefore , the system will be consistent for values of "h" and "k" satisfying k = -3h .
The given question is incomplete , the complete question is
For what values of h and k the system Of Equations is consistent ?
2x₁ – x₂ = h ; -6x₁ + 3x₂ = k
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the lines containing the altitudes of a triangle are concurrent, and the point of concurrency is called the of the triangle.
The point of concurrency for the lines containing the altitudes of a triangle is called the orthocenter of the triangle.
The orthocenter of a triangle is the point where the perpendicular drawn from the vertices to the opposite sides of the triangle intersect each other.
The orthocenter for a triangle with an acute angle is located within the triangle.For the obtuse angle triangle, the orthocenter lies outside the triangle.The vertex of the right angle is where the orthocenter for a right triangle is located.The place where the altitudes connecting the triangle's vertices to its opposite sides intersect is known as the orthocenter.It is located inside the triangle in an acute triangle. For an obtuse triangle, it lies outside of the triangle. For a right-angled triangle, it lies on the vertex of the right angle.
The equivalent for all three perpendiculars is the product of the sections into which the orthocenter divides an altitude.Therefore, the point of concurrency for the lines containing the altitudes of a triangle is called the orthocenter of the triangle.
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Write the augmented matrix for each of the following systems of linear equations. a. x – 3y=5 b. x+2y=0 x+y=1 c. x - y + z=2 x - z=1 d. y + 2x=0 y+z=0
The augmented matrix for the given systems of linear equations are obtained. a. x – 3y=5 b. x+2y=0 x+y=1 c. x - y + z=2 x - z=1 d. y + 2x=0 y+z=0
Explain the term augmented matrix?Two matrices are combined to form enhanced matrices.
A augmented matrix is often one in which the matrices have been merged so that they are next to one another. A divider which expresses the "border" of each matrix connects the last column of the initial matrix to the initial column of a second matrix. Augmented matrices can be used to visualize systems of equations.Thus, augmented matrix for the given systems of linear equations are obtained as:
a. x – 3y = 5 : [1 - 3 : 5]
b. x+2y=0 x+y=1 : [1 2 : 0]
[1 1 : 1]
c. x - y + z=2 and x - z=1 : [1 -1 1 : 2]
[1 0 -1 : 1]
d. y + 2x=0 y+z=0 : [2 1 0 : 0]
[0 1 1 : 0]
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Solve the following system of equations for all three variables.
-3x - 5y + z = -8
2x + 8y - z = -4
x + 2y – z = 2
x,y,z
The solution to the system of equations is x = 0, y = 3, and z = 4.
What is algebraic equation ?
An algebraic equation or polynomial equation is an equation of the form P=0 where P is a polynomial with coefficients in some field, often the field of the rational numbers.
First, multiply the first equation by 2 and add it to the second equation to eliminate x:
-3x - 5y + z = -8
2 * (-3x - 5y + z) = 2 * -8
-6x - 10y + 2z = -16
2x + 8y - z = -4
-6x - 10y + 2z + 2x + 8y - z = -16 - 4
-4x - 2y + 3z = -20
Next, divide the third equation by 3 to obtain y:
x + 2y - z = 2
y = (2 - x + z) / 2
x + 2y - z = 2
x + 2 * (2 - x + z) / 2 - z = 2
x + 2 - x + z - z = 2
z = 4 - x,
y = (2 - x + 4 - x) / 2
y = (2 - 2x + 4) / 2
y = (6 - 2x) / 2
z = 4 - x
4 = 4 - x
x = 0
Plugging in the values of x and z into the equation for y:
y = (6 - 2x) / 2
y = (6 - 2 * 0) / 2
y = 6 / 2
y = 3
So, the solution to the system of equations is x = 0, y = 3, and z = 4.
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The solution to the system of equations for all three variables is x = -3, y = -5, and z = -15.
We can solve this system of equations using the elimination method or substitution method. Here, I will use the elimination method:
Starting with the first equation:
-3x - 5y + z = -8
Adding the second equation:
-3x - 5y + z + 2x + 8y - z = -8 + (-4)
-x + 3y = -12
Adding the third equation:
-x + 3y + x + 2y - z = -12 + 2
2y = -10
Solving for y:
y = -5
Substituting y back into the first equation:
-3x - 5(-5) + z = -8
-3x + 25 + z = -8
-3x + z = -33
Substituting y back into the second equation:
2x + 8(-5) - z = -4
2x - 40 - z = -4
2x - z = 36
Adding the two equations to eliminate x:
-3x + z + 2x - z = -33 + 36
-x = 3
Solving for x:
x = -3
Substituting x and y back into the third equation:
-3 + 2(-5) - z = 2
-3 - 10 - z = 2
-13 - z = 2
Solving for z:
z = -15
Therefore, the solution is x = -3, y = -5, and z = -15.
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What is the relationship between the ratios
Two ratios have proportional relationship.
What is ratio?The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.
A ratio is a mathematical expression written in the form of a:b, where a and b are any integers.
Given question,
What is the relationship between the ratios?
Ratio defines the relationship between the quantities of two or more objects. It is used to compare the quantities of the same kind. If two or more ratios are equal, then it is said to be in proportion.
Hence, two ratio have the proportional relationship.
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