(a) To show that {v1,…,vk,w1,…,wℓ} is an orthogonal set, we must show that the inner product of any two distinct vectors in this set is 0.
(b) To show that {v1,…,vk,w1,…,wℓ} is a basis of Rn, we must show that any vector in Rn can be written as a linear combination of these vectors.
(c) To show that dim V + dim V⊥ = n, we must show that the number of vectors in {v1,…,vk,w1,…,wℓ} is equal to n.
Let v and w be any two distinct vectors in the set {v1,…,vk,w1,…,wℓ}. By definition, since {v1,…,vk} is an orthogonal basis of a subspace V of Rn and {w1,…,wℓ} is an orthogonal basis of V⊥, we have that v and w are either in V or in V⊥, but not both.
Thus, if v is in V, then we have = 0, because w is in V⊥. Similarly, if w is in V⊥, then = 0, because v is in V. Therefore, = 0 for any two distinct vectors v and w in {v1,…,vk,w1,…,wℓ}, which implies that {v1,…,vk,w1,…,wℓ} is an orthogonal set.
(b) Let x be any vector in Rn. Since {v1,…,vk} is a basis of V, x can be written as a linear combination of the vectors in {v1,…,vk}, that is, x = a1v1 + a2v2 + ... + akvk.
Since {w1,…,wℓ} is a basis of V⊥, x can also be written as a linear combination of the vectors in {w1,…,wℓ}, that is, x = b1w1 + b2w2 + ... + bℓwℓ.
Combining these two equations, we can write x = (a1v1 + a2v2 + ... + akvk) + (b1w1 + b2w2 + ... + bℓwℓ). Thus, any vector in Rn can be written as a linear combination of the vectors in {v1,…,vk,w1,…,wℓ}, which implies that {v1,…,vk,w1,…,wℓ} is a basis of Rn.
By definition, {v1,…,vk} is an orthogonal basis of a subspace V of Rn, so dim V = k. Similarly, {w1,…,wℓ} is an orthogonal basis of V⊥, so dim V⊥ = ℓ. Thus, the number of vectors in {v1,…,vk,w1,…,wℓ} is k + ℓ = n, which implies that dim V + dim V⊥ = n.
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2
Alice went to a toy store. She paid for 2 jigsaw
puzzles and 4 packs of cards with 3 ten dollar bills.
The price for a pack of cards is $2. She received
$4 in change.
What was the total cost of the 2 jigsaw puzzles?
A $9
B $18
C $22
D $26
Answer:
Alice paid for the 4 packs of cards with 4 x $2 = $8.
She used 3 ten dollar bills, which is 3 x $10 = $30.
Therefore, the cost of the 2 jigsaw puzzles is:
Total cost of the purchase - Cost of the packs of cards
= ($30 + $4 change) - $8
= $26
So, the total cost of the 2 jigsaw puzzles was $26.
The answer is (D) $26.
Answer:
C. $22
Step-by-step explanation:
You have $30, then you multiply 4x2=8(Price of the Cards)
next subtract the $8 from $30 (30-8)
Your final answer is 22
An electric utility company charges $24 per month maintenance fee to customers generating their own solar power, but refunds them $0.08{ per kilowatt-hour of net electricity returned to the grid.
How many kilowatt-hours per month does one have to return to the grid to break even (do not pay nor receive any money)? Round your answer to the nearest kilowatt-hour. Do not include units in your answer
300 kilowatt-hours per month does one have to return to the grid to break even.
To find out how many kilowatt-hours per month one has to return to the grid to break even, we need to set up an equation and solve for x, where x is the number of kilowatt-hours returned to the grid.
The equation would be:
24 = 0.08x
To solve for x, we need to isolate the variable on one side of the equation. We can do this by dividing both sides of the equation by 0.08:
24/0.08 = x
x = 300
Therefore, one would have to return 300 kilowatt-hours per month to the grid to break even.
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Solve the following system of equations:
2x-6y+9z=-8
5x+y+2z=10
3x+y-8z = -28
(Please add an explanation for better understanding.)
I've been working on this for a week and I can't figure it out...
Answer:
This
Step-by-step explanation:
( x y z) = (-1,7,4) explanation is big
1. he marks of Calculus I in the final examination in a private college are normally distributed with a mean of 45 marks and a standard deviation of 10 marks. (a) If a student is chosen at random, find the probability that his/her mark is less than 52. 0.75$ (b) If the college has 220 students who sat for the examination, find the number of students whose marks are between 45 and 55. 75 (c) Find the percentage of students whose marks exceed 40. 69.15%
2. A survey on the study habits of 1000 HSM students shows that 550 use reference books, 750 have regular study times and all those who use reference books have regular study time. A HSM student is chosen at random; what is the probability that the student
a) only has regular study time?
b) either has a regular study time or uses
reference books?
c) neither studies regularly nor uses reference
books?
1. (a) Using a z-table, we can find the probability corresponding to this z-score, which is 0.758. Therefore, the probability that a student's mark is less than 52 is 0.758. (b) To find the number of students whose marks are between 45 and 55, we need to calculate the z-scores for 45 and 55 . The z-score for 45 is (45 - 45) / 10 = 0, and the z-score for 55 is (55 - 45) / 10 = 1. (c) The probability corresponding to this z-score is 0.3085. Therefore, the percentage of students whose marks exceed 40 is 1 - 0.3085 = 0.6915, or 69.15%. 2. (a) So, the probability that a student only has regular study time is 750/1000 - 550/1000 = 200/1000 = 0.2. (b) The probability that a student either has a regular study time or uses reference books is 750/1000 + 550/1000 - 550/1000 = 750/1000 = 0.75. (c) The probability that a student neither studies regularly nor uses reference books is 1 - 0.75 = 0.25.
Therefore, the probability that a student's mark is between 45 and 55 is 0.8413 - 0.3413 = 0.5. Since there are 220 students in the college, the number of students whose marks are between 45 and 55 is 0.5 * 220 = 110.
To find the probability that a student's mark is less than 52, we need to calculate the z-score for 52 using the formula z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation. So, z = (52 - 45) / 10 = 0.7.
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i need help on my dividend
Answer:
ok divide is easy
Step-by-step explanation:
dividing is not adding or subtracting
please help me out with anybody this if u can and answer it right i will give brainy
Question 12
Find the five-number summary of the data set:
135, 149, 156, 112, 134, 141, 154, 116, 134, 156
Minimum: 112
Quartile Q1: 129.5
Median: 138
Quartile Q3: 154.5
Maximum: 156
9. The sum of the squares of n standard gaussians is known as a Chi square and denoted by xh. In other words, if Z1, Z2, ..., , Zn are independent N(0, 1) then, x2 z +Z2 ...+22 where d over = means the two sides have the same distribution, i.e. they show the same values with the same probabilities. Use the CLT to find P(xż5 > 30). The closest value is, (A) 0.18 (B) 0.24 (C) 0.20 (D) 0.22 (E) 0.16
The sum of the squares of n standard gaussians is known as a Chi square and denoted by x2. In other words, if Z1, Z2, ..., Zn are independent N(0, 1) then x2 = Z1^2 + Z2^2 + ... + Zn^2. to use the Central Limit Theorem (CLT) to find P(x2 > 30). The closest value is (C) 0.20 The correct option is C) 0.20
The CLT states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed. In this case, the sum of the squares of n standard gaussians (x2) will be approximately normally distributed with mean n and variance 2n.
Therefore, we can standardize x2 to find the probability that it is greater than 30:
Z = (x2 - n) / sqrt(2n)
We want to find P(x2 > 30), which is equivalent to P(Z > (30 - n) / sqrt(2n)). Using the standard normal table, we can find the closest value to this probability. The closest value is (C) 0.20. Therefore, the answer is (C) 0.20.
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A water valve controls the amount of water flowing through the dabney dam. The probability for the water flow is uniform between 0 and 4 b/s (barrels per second), uniform between 4 and 9 b/s, and uniform between 9 and 14 b/s. The water flow cannot be 14 b/s or greater. The probability of being in the second range is half of the first range. The probability of being in the third range is a fifth of of being in the first range. In other words, the pdf looks like this:
The value of the constant "k", which make the given pdf valid is 6/17 .
Let x be the amount of water flowing through the Dabney dam;
The Probability Density Function(pdf) of x is given as
⇒ f(x) = { k , 0≤x≤2
(1/3)k , 2<x≤4
(1/6)k , 4<x<5
We know that for a valid p.d.f. [tex]\int\limits^{\infty}_{-\infty} {f(x)} \, dx[/tex] = 1 ;
Substituting the functions for the different intervals ,
We get;
⇒ [tex]\int\limits^{0}_{-\infty} {0} \, dx[/tex] + [tex]\int\limits^{2}_{0} {k} \, dx[/tex] + [tex]\int\limits^{4}_{2} {\frac{1}{3} k} \, dx[/tex] + [tex]\int\limits^{5}_{4} {\frac{1}{6} k} \, dx[/tex] + [tex]\int\limits^{\infty}_{5} {0} \, dx[/tex] = 1 ;
⇒ 0 + [kx]²₀ + [kx/3]⁴₂ + [kx/6]⁵₄ + 0 = 1 ;
⇒ 2k + (k/3)(4-2) + (k/6)(5-4) = 1 ;
⇒ 2k + 2k/3 + k/6 = 1 ;
⇒ (12k + 4k + k)/6 = 1;
⇒ 17k/6 = 1;
⇒ k = 6/17.
Therefore, the value k=6/17 will make the pdf valid.
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The given question is incomplete, the complete question is
A water valve controls the amount of water flowing through the Dabney dam. The probability for the water flow is uniform between 0 and 4 b/s (barrels per second), uniform between 4 and 9 b/s, and uniform between 9 and 14 b/s. The water flow cannot be 14 b/s or greater. The probability of being in the second range is half of the first range. The probability of being in the third range is a fifth of of being in the first range.
In other words, the pdf looks like this:
f(x) = { k , 0≤x≤2
(1/3)k , 2<x≤4
(1/6)k , 4<x<5.
Find the value of k , that will make the pdf valid.
btract polynomials Subtract the polynomials. ((3)/(2)c^(2)-(4)/(3)c+9)-((1)/(6)c^(2)+(1)/(9)c+2)
Subtracting the polynomial (1/6)c² + (1/9)c + 2 from (3/2)c² - (4/3)c + 9 will result to the polynomial (4/3)c² + (-13/9)c + 7.
To subtract the polynomials, we need to subtract the corresponding terms. That is, we subtract the coefficients of the c² terms, the coefficients of the c terms, and the constant terms. The result will be a new polynomial. Here is the solution:
((3)/(2)c² - (4)/(3)c + 9) - ((1)/(6)c² + (1)/(9)c + 2)
= (3/2)c² - (4/3)c + 9 - (1/6)c² - (1/9)c - 2
= (3/2 - 1/6)c² + (-4/3 - 1/9)c + (9 - 2)
= (8/6)c² + (-13/9)c + 7
= (4/3)c² + (-13/9)c + 7
So the answer is (4/3)c² + (-13/9)c + 7.
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Solve The Equation 6x^4 + 8x^2 = 26x^2
Answer: [tex]\sqrt{3}[/tex]
Step-by-step explanation:
take,
x^2 = Y
6Y^2 + 8Y = 26Y
6Y^2 + 8Y - 26Y = 0
6Y^2 - 18Y = 0
6Y (Y - 3) = 0
(Y - 3) = 0
Y = 3
x^2 = y
x^2 = 3
x = [tex]\sqrt{3}[/tex]
What is the remainder when f(x)=x^3+7x^2+9x-5 is divided by (x+4)
On dividing (x³ + 7x² + 9x - 5) by (x + 4), we get -
h(x) = x² + 3x - 3 + 7/(x + 4).
What is Division algorithm?Division algorithm states that -
Dividend = (Divisor x Quotient) + Remainder
Given is to find the remainder when -
(x³ + 7x² + 9x - 5) ÷ (x + 4)
We can write -
f(x) = (x³ + 7x² + 9x - 5)
g(x) = (x + 4)
So -
h(x) = f(x) ÷ g(x)
h(x) = (x³ + 7x² + 9x - 5) ÷ (x + 4)
h(x) = (x³ + 4x² + 3x² + 9x - 5)/(x + 4)
h(x) = x² + (3x² + 9x - 5)/(x + 4)
h(x) = x² + 3x + (-3x - 5)/(x + 4)
h(x) = x² + 3x - 3 + 7/(x + 4)
Therefore, on dividing (x³ + 7x² + 9x - 5) by (x + 4), we get -
h(x) = x² + 3x - 3 + 7/(x + 4).
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I need help pls pls pls pls help quickly.
Answer:
-7/8
Step-by-step explanation:
The two coordinates shown are (0,5) and (8,-2).
The slope formula is (y2-y1)/(x2-x1).
Using that, the slope of a line passing through both points is (-2-5)/(8-0).
That reduces to -7/8.
Answer:
-7/
Step-by-step explanation:
What is the solution to this equation?
9 ^x- 1 = 2
O A. 1
O B. 2
O C.
1/2
OD. -1/2
Which of the following shows a set of equivalent rational numbers?
A. 2/5 0.4 25%
B. 2/3 0.6 66%
C. 3/10 0.3 30%
D. 73/100 0.73 730%
Answer:
D is not equivalent because 730% is not a rational number.
4. An ant leaves the origin at time zero, travelling along the positive x-axis at a speed of two Poincare units of length per second.
(a) Find the (Euclidean) coordinates of the ant’s position after 2 seconds.
(b) How long will it take the ant to pass the point (0.999, 0)?
(a) After 2 seconds, the ant's position in the (Euclidean) coordinates is (4,0)
(b) To pass the point (0.999, 0), the ant needs 0.4995 seconds.
The ant is traveling along the positive x-axis at a speed of 2 Poincare units of length per second. This means that its position at any given time can be determined using the equation:
x = 2t
where x is the ant's position along the x-axis, and t is the time in seconds.
(a) To find the ant's position after 2 seconds, we can simply plug in t = 2 into the equation:
x = 2(2) = 4
So the ant's position after 2 seconds is (4, 0).
(b) To find how long it will take the ant to pass the point (0.999, 0), we can set x = 0.999 and solve for t:
0.999 = 2t
t = 0.999/2 = 0.4995
So it will take the ant 0.4995 seconds to pass the point (0.999, 0).
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If 1/6 of a gallon of milk is shared equally between three friends how much milk will each friend get
the friends receive one eighteenth each
e) Solve: f(x) = g(x) for x = [-240°; 240°]
Answer:
218.88
Step-by-step explanation:
A rocket is launched from the top of a 200-foot tall cliff with an initial velocity of 120 feet per second. The height, h, of the rock at time t seconds after it was launched can be modeled by the equation h = -16t2 + 120t + 200. Once the rocket reaches its maximum height, how long will it take to reach the ground? Round to the nearest tenth of a second.
By answering the above question, we may infer that The rocket will thus equation touch down 2.5 seconds after it was fired (to the nearest tenth of a second).
What is equation?A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.
Hence, we may solve for t by setting the equation for h equal to 0:
[tex]-16t^2 + 120t + 200 = 0\\2t^2 - 15t - 25 = 0\\(-b \sqrt(b2 - 4ac)) / 2a[/tex]
where a=2, b=15, and c=25
[tex]t = (15 \sqrt(625)) / 2(2) (-(-15) \sqrt((-15)2 - 4(2)(-25))) / 4\st = (15 + 25) / 4[/tex]
T thus equals 10/2 or -5/2. Time cannot be negative, thus we may ignore the negative number.
The rocket will thus touch down 2.5 seconds after it was fired (to the nearest tenth of a second).
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following exercises, perform the indicated operations. 7. (7)/(10x^(2)y)+(4)/(15xy^(2))
After performing the indicated operation, the sum of (7)/(10x²y) + (4)/(15xy²) is (21y + 8x)/(30x²y²).
To perform the indicated operations, we need to find a common denominator for the two fractions. The common denominator for 10x²y and 15xy² is 30x²y². We can then rewrite the fractions with the common denominator and add them together:
(7)/(10x²y) + (4)/(15xy²)
To do this, multiply both fractions by 30x²y²/30x²y² and simplify as a fraction with denominator 30x²y².
= (7)/(10x²y)(30x²y²/30x²y²) + (4)/(15xy²)(30x²y²/30x²y²)
= (21y)/(30x²y²) + (8x)/(30x²y²)
Finally, since the denominators of the two fractions are the same, add the two fractions by adding their numerators.
= (21y + 8x)/(30x²y²)
Therefore, the answer is (21y + 8x)/(30x²y²).
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How do you find the scale factors with fractions
To find the scale factors with fractions, you first need to understand what scale factors are. Scale factors are the ratios of corresponding lengths in two similar figures. They tell you how much larger or smaller one figure is compared to another.
To simplify fractions, you need to find the greatest common factor (GCF) of the numerator and denominator and divide them by it.
For example, if we have the fraction 6/12, we can simplify it by finding the GCF of 6 and 12, which is 6. Then, we divide both the numerator and denominator by 6:
6/12 = (6 ÷ 6) / (12 ÷ 6) = 1/2
So, 6/12 simplifies to 1/2.
If the fraction is already in its simplest form, then there is no need to simplify it further.
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The complete question i:
How do you find the scale factors with fractions? Give example.
ind all solutions of the system of equations algebraically. oordinate points. y=-3x^(2)+12x 3x+y=12
The given system of equations can be written as: y=-3x^2+12x 3x+y=12
To solve this system, we can use the substitution method. We will solve the first equation for y and substitute this value in the second equation to solve for x. From the first equation, we have y=-3x^2+12x. Substituting this in the second equation, we get 3x+(-3x^2+12x)=12. Simplifying, we get -3x^2+15x=12.
Now, we can solve this equation for x. To do this, we can divide both sides of the equation by -3 to get x^2+5x/3=4/3. Factoring the left side, we get (x+5/3)(x-4/3)=0. From here, we can find the two solutions for x: x=-5/3 and x=4/3. Substituting these values in the first equation, we get the two solutions for y: y=-17/3 and y=20/3. Therefore, the two solutions for the system of equations are (x=-5/3, y=-17/3) and (x=4/3, y=20/3).
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If the aspect ratio of a white board is 16:9 and the width is 64in, what is the height of the white board?
The height of the white board for the given aspect ratio is found as 36 in.
Explain about the aspect ratio?A proportional association among an image's height and width is known as an aspect ratio. It essentially describes the contour of a picture. Aspect ratios are expressed as a width-to-height formula, such as 3:2. A square image, for instance, has a 1:1 aspect ratio because its width and height are equal. Aspect ratio, in its most basic form, is the proportion between an image's width and height. That affects how readers compose your picture and how huge it will be, which is why it's so crucial.Aspect ratio of a white board = 16:9 .
Width = 64 in
Let the height be 'h'.
So, using proportions.
16/9 = 64/h
h = 64*9 /16
h = 36
Thus, the height of the white board for the given aspect ratio is found as 36 in.
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Choose two ratios that describe the collection of baseballs to gloves.
8:4
06:4
02:1
02:4
10
4 of 15 answered
need help?
->>
Two ratios that describe the collection of baseballs to gloves are
8:4 and 2:1 .
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. Ratios can be classified into two types. One is part to part ratio and the other is part to whole ratio.
Given,
Number of gloves = 16
Number of baseballs = 8
Ratio of gloves to baseballs
= 16/8
= 8/4 = 8:4
= 2/1 = 2:1
Hence, 8:4 and 2:1 are two ratios that describe the collection of baseballs to gloves.
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Complete question is:
There are 16 baseballs and 8 gloves,
Choose two ratios that describe the collection of baseballs to gloves.
8:4, 6:4, 2:1, 2:4, 10:4
a set of eight cards was labled m, u, t, i, p, l, y. what is the sample space for choosing one card
The sample space for choosing one card from the set of eight cards labeled m, u, t, i, p, l, y is 8.
What is sample space?An assortment or set of potential results from a random experiment make up a sample space. With the letter "S," the sample space is denoted. Events are a subset of the possible results of an experiment. Depending on the experiment, a sample space could have several different outcomes. Discrete sample spaces, often known as finite sample spaces, are those that have a finite number of outcomes.
The sample space for choosing 1 card is given as:
Sample space = 8C1 = 8! / (1!)(8 - 1)!
Sample space = 8.
Hence, the sample space for choosing one card from the set of eight cards labeled m, u, t, i, p, l, y is 8.
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HELP ME PLEASE!!!!!!!!!!!!!!!
Answer: 480
Step-by-step explanation: Since this is a triangle, we are using 1/2 * base * height. So since the height is 15, and the base is 64, that would equal 960. Then we are going to divide it by 2, which equals 480.
what is the 20th term of the sequence 1,3,9,27
Answer:
1162261467
Step-by-step explanation:
Note that this is a geometric sequence.
1,3,9,27
The pattern is to multiply the number by 3 to get to the next term.
a = 1
r = 3
n=20
arⁿ⁻¹ = nth term
You then substitute the values of a,r and n into the nth term:
arⁿ⁻¹=3¹⁹
3¹⁹=1162261467 which is the 20th term of the sequence
Hoped this helped
Answer:
To find the 20th term of the sequence 1, 3, 9, 27, we need to first identify the pattern of the sequence. It appears that each term of the sequence is obtained by multiplying the previous term by 3. This means that the sequence is a geometric sequence with first term (a) = 1 and common ratio (r) = 3.
Using the formula for the nth term of a geometric sequence:
an = a * r^(n-1)
where an is the nth term of the sequence, a is the first term, r is the common ratio, and n is the term number.
Substituting the values we have:
a = 1, r = 3, and n = 20
an = 1 * 3^(20-1) = 1 * 3^19 = 1162261467
Therefore, the 20th term of the sequence 1, 3, 9, 27 is 1162261467.
Step-by-step explanation:
brainliest pls
Un recipiente de aluminio tiene una capacidad de 6 litros a 28 grados celcius.
Determina el volumen del recipiente cuando este se caliento a 100 grados celcius
Answer: I don't speak Spanish but the answer might be 21.4285714286 liters
Step-by-step explanation:
"An aluminum container has a capacity of 6 liters at 28 degrees Celcius. Determine the volume of the container when it is heated to 100 degrees Celsius."
Help with my home work
Find the gradients of lines A and B.
B
-2
6
O
-21
Fet
A
4
72%
X
The gradient of line A is 4 and the gradient of line B is -2.
What is graph?Graph is a data structure that consists of nodes (or vertices) connected by edges. It is a way of representing data in a structured form and can be used to represent any type of real-world relationship. It can be used to represent geographical relationships between cities, social networks, and even computer networks. Graphs are often used in computer science, mathematics, and engineering to represent relationships between data points.
To calculate the gradient of a line, the formula is rise/run. Thus, the gradient of line A can be calculated as rise/run = 72/4 = 18. On the other hand, the gradient of line B can be calculated as rise/run = 6/-2 = -3
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find the equation and then find the ordered pair please!
Note that the pair of numbers that solves the system of equations:
y=11x+40; and
y= -7x+9
are, x = -31/18 ; and y = 379/18.
To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously. One way to do this is to set the expressions for y equal to each other, since both expressions represent the same value of y.
So we have:
11x + 40 = -7x + 9
Simplifying and solving for x, we get:
18x = -31
x = -31/18
Now that we have found the value of x, we can substitute it into either equation to find the corresponding value of y. Let's use the first equation:
y = 11(-31/18) + 40
y = -341/18 + 720/18
y = 379/18
Therefore, the pair of numbers that solves the system of equations is (-31/18, 379/18).
Learn more about system of equations at:
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