a.) A U B U C is the event of A n B, A U B^ n C, A U B U C.
b.) The sets of outcomes that correspond to the event D = are (T, T, H) (T, H, H) (H, T, H) (T, H, T) (H, H, T)
(a) A ∩ B is the event that the first two tosses show heads, i.e., the outcome is (H, H, T) or (H, H, H).
A U (B ∩ C) is the event that at least one of the first two tosses shows heads and the third toss shows heads, i.e., the outcome is (H, H, H), (H, T, H), or (T, H, H).
A U B U C is the event that at least one of the three tosses shows heads, i.e., the outcome is (H, T, T), (T, H, T), (T, T, H), (H, H, T), (H, T, H), (T, H, H), or (H, H, H).
(b) The sets of outcomes that correspond to the event D = {"3rd toss shows heads for the first time"} are:
(T, T, H)
(T, H, H)
(H, T, H)
(T, H, T)
(H, H, T)
These outcomes correspond to the event D because they show the first occurrence of heads on the third toss. The outcome (H, H, H) does not correspond to the event D because it shows three consecutive heads, not the first occurrence of heads on the third toss.
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find the orthogonal complement w⊥ of w and give a basis for w⊥.w = xyz: x = 12t, y = − 12t, z = 6t
The orthogonal complement w⊥ of w has a basis given by {v1, v2} = {(1, 0, 0), (0, 1, 2)}.
How to find the orthogonal complement w⊥ of w?To find the orthogonal complement w⊥ of w, we need to find the set of all vectors that are orthogonal (perpendicular) to w.
Given w = (x, y, z) = (12t, -12t, 6t), we can find a vector v = (a, b, c) that is orthogonal to w by taking their dot product equal to zero:
w · v = 0
Substituting the values of w and v:
(12t, -12t, 6t) · (a, b, c) = 0
(12t)(a) + (-12t)(b) + (6t)(c) = 0
12at - 12bt + 6ct = 0
Now, we can solve this equation to find the values of a, b, and c that satisfy the orthogonal condition for all values of t.
12at - 12bt + 6ct = 0
Factor out t:
t(12a - 12b + 6c) = 0
For this equation to hold true for all values of t, the expression inside the parentheses must equal zero:
12a - 12b + 6c = 0
Divide by 6:
2a - 2b + c = 0
This equation represents a plane in three-dimensional space. To find a basis for w⊥, we can express this equation in the form of a linear combination of vectors. Let's solve for c:
c = 2b - 2a
Now, we can express the basis vectors for w⊥ in terms of a and b:
v = (a, b, 2b - 2a)
We can choose any values for a and b to get different vectors in the orthogonal complement w⊥. For example, we can set a = 1 and b = 0:
v1 = (1, 0, 0)
Or we can set a = 0 and b = 1:
v2 = (0, 1, 2)
These two vectors, v1 and v2, form a basis for w⊥.
Therefore, the orthogonal complement w⊥ of w has a basis given by {v1, v2} = {(1, 0, 0), (0, 1, 2)}.
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For \( \bar{A}=x \bar{a} x+y \bar{a} y+z \bar{a} z \) and \( \bar{B}=2 x \bar{a} x+3 y \bar{a} y+3 z \bar{a} z \). Find the followingat \( (2,2,1) \). a) \( \bar{C}=\bar{A} \times \bar{B} \) b) Find \
a. At point (2, 2, 1) the vector \(\bar{C} = - 2\bar{a}y+4\bar{a}z\)
b. At (2, 2, 1) the value of D = 23
Given that,
For \(\bar{A}=x \bar{a} x+y \bar{a} y+z \bar{a} z \)\) and \(\( \bar{B}=2 x \bar{a} x+3 y \bar{a} y+3 z \bar{a} z \)\).
Here, A and B are vectors
We know that,
a. At (2, 2, 1) we have to find \(\bar{C}=\bar{A} \times \bar{B}\).
C is a vector by using matrix,
\(\bar{C}=\left[\begin{array}{ccc}\bar{a}x&\bar{a}y&\bar{a}z\\x&y&z\\2x&3y&3z\end{array}\right]\)
Now, determine the matrix,
\(\bar{C} = \bar{a}x(3yz - 3yz) - \bar{a}y(3xz - 2xz)+\bar{a}z(3xy - 3xy)\)
\(\bar{C} = - \bar{a}y(xz)+\bar{a}z(xy)\)
At point (2,2,1) taking x = 2 , y = 2 and z = 1
\(\bar{C} = - \bar{a}y(2\times 1)+\bar{a}z(2\times 2)\)
\(\bar{C} = - 2\bar{a}y+4\bar{a}z\)
b. At (2, 2, 1) we have to find \(D=\bar{A} .\bar{B}\)
\(D=\bar{A} .\bar{B}\)
\(D = (x \bar{a} x+y \bar{a} y+z \bar{a} z )(2 x \bar{a} x+3 y \bar{a} y+3 z \bar{a} z)\)
D = 2x² + 3y² + 3z²
At point (2,2,1) taking x = 2 , y = 2 and z = 1
D = 2(2)² + 3(2)² + 3(1)²
D = 23.
Therefore, At (2, 2, 1) D = 23
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The question is incomplete the complete question is -
For \(\bar{A}=x \bar{a} x+y \bar{a} y+z \bar{a} z \)\) and \(\( \bar{B}=2 x \bar{a} x+3 y \bar{a} y+3 z \bar{a} z \)\).
Find the following at (2,2,1)
a. \(\bar{C}=\bar{A} \times \bar{B}\)
b. \(D=\bar{A} .\bar{B}\)
find the following measures of dispersion for the data below: 17 14 1 10 0 9 1 14 11 5 7 12 3 6 15 8 13 6 19 12 18 12 20 12 0 sorted data (list data with commas separating values)
Dispersion is a measure of how spread out a set of data values is. For the given data, I'll calculate three common measures of dispersion: range, variance, and standard deviation.
Sorted data: 0, 0, 1, 1, 3, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 13, 14, 14, 15, 17, 18, 19, 20
1. Range: The range is the difference between the highest and lowest values in the dataset. In this case, the range is 20 (highest value) - 0 (lowest value) = 20.
2. Variance: Variance is the average of the squared differences between each data point and the mean of the dataset. First, calculate the mean (sum of values / number of values) which is 233/25 = 9.32. Next, find the squared differences between each data point and the mean, and then average these squared differences. The variance for this dataset is approximately 34.93.
3. Standard Deviation: The standard deviation is the square root of the variance. In this case, the standard deviation is approximately √34.93 ≈ 5.91.
In summary, the range for the given data is 20, the variance is approximately 34.93, and the standard deviation is approximately 5.91. These measures of dispersion provide insights into the variability of the data.
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What is the value of the sin3x/6x; as x approaches to zero
Step-by-step explanation:
When x = 0,
sin3x / 6x = 0 / 0, which is an indeterminate form.
Hence we use L'Hospital's rule:
d/dx (sin3x) = 3cos3x
d/dx (6x) = 6
Now we have 3cos3x / 6 or 0.5cos3x.
When x = 0, 0.5cos3x = 0.5(1) = 0.5.
Hence the limit is 0.5.
Using limits, it is found that the value of the function is of 0.5.
The limit given is:
\(\lim_{x \rightarrow 0} \frac{\sin{(3x)}}{6x}\)
The following property is used:
\(\lim_{x \rightarrow 0} \frac{\sin{(ax)}}{ax} = 1\)
Hence, multiplying the numerator and denominator by 0.5:
\(\lim_{x \rightarrow 0} \frac{\sin{(3x)}}{6x} = 0.5\lim_{x \rightarrow 0} \frac{\sin{(3x)}}{3x} = 0.5(1) = 0.5\)
The value of the function is of 0.5.
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What is 22 1/4 in fraction and divided by 890
The fraction 22 1/4 divided by 890 gives 1/40 or 0.025.
What are Mixed Fractions?Mixed fractions are type of fractions which consist of a whole number part and a fractional part.
Mixed fractions are of the form a b/c where a is the whole number part and b/c is the fractional part.
Given is a mixed fraction 22 1/4.
This has to be first converted to improper fraction.
22 1/4 = (22 × 4 + 1) / 4 = 89/4
Now we have to divide the fraction 89/4 by 890.
89/4 ÷ 890 = 89/4 × 1/890
(since in the division of fractions, the second term becomes the reciprocal and the operation changes to multiplication).
89/4 ÷ 890 = 89/4 × 1/890
= 1/40
= 0.025
Hence the quotient when 22 1/4 divided by 890 is 0.025.
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a river in one part of Rocky Mountains has a flow rate of 250 cubic meters of water per second. The flow of the water can be modeled mathematically with the function, w(t)=250 t, where w(t) represents the volume of the water in cubic meters, at time t, in seconds.
The volume of the water in cubic meters, in five seconds will be 1250 cubic meters
Flow rate = 250 cubic meters of water per second
Model for the water flow = 250t
m(t) = volume of the water in cubic meters
where t = time t, in seconds
Let's assume that we've to calculate the volume of the water for five seconds. This will be:
m(t) = 250t
m(t) = 250 × 5
m(t) = 1250 cubic meters of water.
Therefore, the volume of the water in cubic meters, in five seconds will be 1250 cubic meters.
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find all real values of $p$ such that \[2(x 4)(x-2p)\]has a minimum value of $-18$ over all real values of $x$.
All the real values of p such that 2(x - 4)(x - 2p) has a minimum value of -18 over all real values of x is p = 1 and p = -5.
Given that:
2(x + 4)(x - 2p)
It is required to find the values of p such that this expression has a minimum value of -18.
Expand the given expression.
2(x² + 4x - 2px - 8p)
2(x² + (4 - 2p)x - 8p)
2x² + (8 - 4p)x - 16p
The minimum value of the function can be found by first, setting the derivative equal to 0.
Differentiating,
4x + (8 - 4p) = 0
4x = 4p - 8
x = p - 2
Substitute x = p - 2 to the function 2(x + 4)(x - 2p).
When the value of the function is -18,
2(p - 2 + 4)(p - 2 - 2p) = -18
2(p + 2)(-p - 2) = -18
(p + 2)(-p - 2) = -9
(p + 2)² = 9
p + 2 = ±3
When p + 2 = 3, p = 1.
When p + 2 = -3, p = -5.
Hence the p values are 1 and -5.
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mother ate pies in the morning, she ate one pie, while she gave half of all those left to Little Red Riding Hood who gave it to her grandmother. On the way, Little Red Riding Hood ate 2 pies and gave a third of the rest to the wolf. Little Red Riding Hood brought 8 pies to her grandmother. How many pies did mother bake?
The number of pies that Red Riding Hood's mother made is 25.
How many pies did the mother bake?Let x represent the number of pies mother baked in the morning.
Pies left after mother ate 1 = x - 1
The number of pies that Red Riding Hood takes to her grandmother: 1/2(x - 1)
0.5(x - 1)
0.5x - 0.5
Pies left after Red Riding Hood ate 2: [(0.5x - 0.5) - 2]
Pie after she gave the wolf 1/3 ; (1 - 1/3) × [(0.5x - 0.5) - 2]
2/3 × [(0.5x - 0.5) - 2] = 8
In order to determine the value of x, take the following steps:
Multiply both sides of the equation by 3/2.
[(0.5x - 0.5) - 2] = 12
Add 2 to both sides of the equation: 0.5x - 0.5 = 12 + 2
0.5x - 0.5 = 12
Add 0.5 to both sides of the equation
0.5x = 12 + 0.5
0.5x = 12.5
Divide both sides by 0.5
x = 25
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What is the value of t in this equation?
(-12)-5 =jt
Answer:
t = 60
Step-by-step explanation:
Using the rule of exponents
\((a^m)^{n}\) = \(a^{mn}\) , then
\((j^{-12}) ^{-5}\)
= \(j^{(-12(-5))}\)
= \(j^{60}\)
Then t = 60
If a = -1, b = 2 and c = -3 determine the value of a² – 2b³+ c²
Answer:
-26
Step-by-step explanation:
-1^2-2(2)^3 -3^2
-1-2(8)-1(9)
-1-16-9
-26
Solve for x.
X=?
4x+4 4x -32
Step-by-step explanation:
4x-4x=-32-4
X=28
Is the answer
Answer:
No solution
Step-by-step explanation:
As the slopes are same and y-intercepts are different, we can conclude this system of equations has no solution.
Kendra made a scaled copy of the following square. She used a scale factor less than 111. What could be the side length of the scaled copy of the square.
Answer:
B,C,E
Step-by-step explanation:
khan academy
Side length of the scaled copy of square is smaller than original.
What is scale factor?A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).
Given,
Scale factor of scaled copy of square is less than 1.
Scale factor basically defines the ratio between the sides of the constructed square to that of the original square.
if scale factor is less than 1 then the sides of the constructed square is smaller than that of the original square.
Hence, side length of constructed square is smaller than the original square.
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what is the slope of the line that goes through the points (4,25) and (-4,-21)?
What is 5.6666 as a fraction
Jakim deposited $10,000 in an account that earned simple interest annually.
- He did not make additional deposits nor withdrawals.
-At the end of 7 years, the balance was $12,100.
What is the interest rate on this account?
Answer:
Step-by-step explanation:
Simple interest is a method for calculating the percentage of interest paid on a sum over a predetermined time period at a predetermined rate. The annual simple interest rate on the principal amount of 10000 at the end of 7 years would be 3%.
What exactly is simple interest?In simple interest, the principal amount doesn't change. Simple interest is a clear-cut and simple method for figuring out how much money has earned interest. It is a way to figure out how much interest will be charged.
Simple interest can be computed by multiplying the principal amount by the rate multiplied by the time period. In banks and other financial institutions, simple interest is frequently used to complete a variety of calculations. The total amount of cash that was spent on interest during the course of the loan is described.
Simple Interest = ( Principal × Rate × Time ) / 100
First, we need the Simple Interest (S.I)
Total = Principal + Simple Interest
T = P + S.I
$10,000 = $12,100. + S.I
S.I = $12,100 - $10,000
S.I = 2100
Simple Interest = (Principal × Rate × Time) / 100
S.I = (P × R × T) / 100
2100 = (10000 × R × 7) / 100
2100 = 70000R / 100
2100 = 700R
R = 2100 / 700
R = 3
Thus, The annual interest rate is 3%
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factorise the following expressions:
a) 14w + 21
b) 27x+18
Step-by-step explanation:
a) 14w + 21
- Factor out 7 from the expression
7 (2w + 3)
b) 27x + 18
- Factor out 9 from the expression
9 (3x + 2)
Hi!
___________________________________________________________
\(\rightsquigarrow\circ\boldsymbol{Answer}\circ\leftharpoonup\)
#1 7(2w+3)
#2 9(3x+2)
\(\rightsquigarrow\circ\boldsymbol{Explanation}\circ\leftharpoonup\)
We need to factor these two expressions.
\(\twoheadrightarrow\sf 14w+21\) and \(\twoheadrightarrow\sf 27x+18\)
For now let's focus on the first expression.
Note that both terms, 14w and 21, have something in common.
In fact, they have 7 in common. So we factor it out by dividing 14w and 21 by 7.
\(\twoheadrightarrow\sf14w\div7=2x\\\twoheadrightarrow\sf21\div7=3\)
\(\bullet\) Put the terms together, with a plus sign in between since they are positive
\(\twoheadrightarrow\sf 2w+3\)
\(\bullet\) Put parentheses around 2w+3
\(\twoheadrightarrow\sf (2w+3)\)
\(\bullet\) Put7 outside the parentheses
\(\twoheadrightarrow\sf7(2w+3)\)
\(\bullet\) Great job! We factored the expression
__________________________________________________________
This problem can be solved the same way!
This time, both terms have a 9 in common.
\(\twoheadrightarrow\sf 27x+18\)
So we divide both terms by 9.
\(\twoheadrightarrow\sf 27x\div9=3x\\\twoheadrightarrow\sf18\div9=2\)
\(\bullet\) Put the terms together, with a plus sign in between since they're positive
\(\twoheadrightarrow\sf 3x+2\)
\(\bullet\) Next, put parentheses around 3x+2 and put 9 outside the parentheses
\(\twoheadrightarrow\sf 9(3x+2)\)
\(\bullet\) Great job! We factored both expressions
--
Hope that this helped! Best wishes.
\(\textsl{Reach far. Aim high. Dream big.}\\\boldsymbol{-Greetings!-}\)
--
I need help with this question
The length of the legs of the right triangle are 2.83 units.
How to find the side of a right triangle?A right tangle triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.
Therefore, the legs of the triangle can be found using trigonometric ratios.
Hence,
sin 45 = opposite / hypotenuse
sin 45 = a / 4
cross multiply
a = 4 × 0.70710678118
a = 2.83 units
Therefore,
cos 45 = b / 4
cross multiply
b = 0.70710678118 × 4
b = 2.82842712475
b = 2.83 units
Therefore, the legs are 2,83 units
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if you know this please help
Answer:
its c hope this helps:)
Step-by-step explanation:
pls help there are 5 pics with multiple questions inside of them pls thank you
Answer:
1. LHJ
2. AB
3. Acute
4. Complement 30, supplement 120
5. AOD
6. 61
7. acute
8. 36
9. Octogon
Step-by-step explanation:
SIMPLIFY USING THE DISTRIBUTIVE PROPERTY OF RATIONAL NUMBERS
1/5+ (2/5+3/5)
Answer:
6/5
Step-by-step explanation:
1/5 + (2/5 + 3/5)
solve for the bracket first
1/5 +(2+3/5)
1/5 + (5/5)
1/5 + 1
1+1*5/5
6/5
another method
1/5 + (2/5 +3/5)
open brackets)
1/5 + 2/5 + 3/5
since their denominator are same u can add their numerator
1+2+3/5
6/5
Jason has applied to be a camp counselor for the summer. The job pays $9 per hour. The equation to re
job is y = 9x, where x is the number of hours he works and y is the total amount he earns,
Mia has applied to be a lifeguard for the summer. The lifeguard job is three days a week with hours and
the table below.
Hours worked
Amount paid
Tuesday
6
$52.50
Thursday
8
$70,00
Saturday
5
$43.75
Which statement best describes the hourly rates?
O The two jobs pay the same hourly rate.
The comparison cannot be made with the information given.
O Jason's camp counselor job pays a higher hourly rate than Mia's lifeguard job.
Jason's camp counselor job pays a lower hourly rate than Mia's lifeguard job.
Answer:
The correct statement is (3).
Step-by-step explanation:
Jason's hourly pay rate as a camp counselor for the summer is:
J = $9 / hour.
The table representing Mia's pay as a lifeguard for the summer is as follows:
Days Hours worked Amount paid
Tuesday 6 $52.50
Thursday 8 $70,00
Saturday 5 $43.75
Compute Mia's hourly rate for Tuesday: \(\frac{\$52.50}{6}=\$8.75\)
Compute Mia's hourly rate for Thursday : \(\frac{\$70.00}{8}=\$8.75\)
Compute Mia's hourly rate for Saturday: \(\frac{\$43.75}{5}=\$8.75\)
Thus, Mia's hourly rate as a lifeguard for the summer is,
M = $8.75 / hour
This implies that Jason's camp counselor job pays a higher hourly rate than Mia's lifeguard job.
Thus, the correct statement is (3).
Answer:
The answer is C because if u multiply 6 by 9 its 54 that's how much Jason gets for 6 hours while on the other hand Mia for six hours gets 52 dollars.
Step-by-step explanation:
can u guys make this a brainlyest answer
help i do nut understand
Answer:
500$ is the answer....
Answer:
$60 and $560
Step-by-step explanation:
Prt= Interest
(500)(0.04)(3)= $60
A = P(1 + rt)
A = 500 (1 + 0.04(3))
A= $560
When 293 college students are randomly selected and surveyed, it is found that 114 own a car. The upper limit for the 90% confidence interval for the percentage of all college students who own a car is __
O 31.6% O 46.3% O 44.5% O43.6%
When 293 college students are randomly selected and surveyed, it is found that 114 own a car. The upper limit for the 90% confidence interval for the percentage of all college students who own a car is 46.3%.
The given problem is an example of confidence intervals in statistics. The formula for finding the confidence interval in percentage is given below:
Confidence Interval in Percentage = P ± Z Score x √ (PQ/n)
where
P = Sample proportion
Q = 1 - P
n = Sample Size
Z Score is obtained from the Z-Table where the confidence level is given.
To obtain the upper limit, the following formula should be used:
Upper Limit of the Confidence Interval = P + Z Score x √ (PQ/n)
Substitute the given values of the question into the formula:
P = 114/293 = 0.3891
Q = 1 - 0.3891 = 0.6109
n = 293
The upper limit for the 90% confidence interval is given by the Z-Table at 1.645:
Upper Limit of the Confidence Interval = 0.3891 + 1.645 x √ [(0.3891 x 0.6109)/293]
≈ 0.3891 + 1.645 x 0.0467
≈ 0.3891 + 0.07701
≈ 0.4661 = 46.3%
Therefore, the answer is 46.3%.
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Suppose $726.56 is deposited at the end of every six months into an account earning 6.45% compounded semi-annually. If the balance in the account four years after the last deposit is to be $31 300.00, how many deposits are needed? (This question asks for 'n')
We need approximately 10 deposits to reach a balance of $31,300 four years after the last deposit which is compounded semi-annually.
To solve this problem, we can use the formula for the future value of an annuity:
\(FV = P * ((1 + r)^n - 1) / r\)
Where:
FV is the future value of the annuity
P is the periodic payment or deposit amount
r is the interest rate per period
n is the number of periods
In this case, the deposit amount is $726.56, the interest rate is 6.45% compounded semi-annually, and the future value is $31,300. We need to find the number of deposits (n).
We can rearrange the formula and solve for n:
n = log((FV * r) / (P * r + FV)) / log(1 + r)
Substituting the given values:
n = log((31,300 * 0.03225) / (726.56 * 0.03225 + 31,300)) / log(1 + 0.03225)
Using a calculator or software, we find that n ≈ 9.989.
Therefore, we need approximately 10 deposits to reach a balance of $31,300 four years after the last deposit.
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A dance studio charges an application fee and a monthly rate for dance classes. This is represented by the equation, C=50x+75. In this function, x represents the
Responses
number of dance classes taken each month
cost per dance class
application fee
total cost of dance classes each month
In linear equation, 50x is the total cost of dance classes each month.
What is a linear equation example?
Ax+By=C is the typical form for linear equations involving two variables. A linear equation in standard form is, for instance, 2x+3y=5.Finding both intercepts of an equation in this format is rather simple (x and y).C = 50x + 75, where 75 is the application fee. When they take out the application fee, it will become:
C = 50x +75 (-75)
C = 50x
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Find the equation of the line through the point (−7,2) and parallel to y=2/5x−1/2
Answer:
y = 2/5x + 24/5
Step-by-step explanation:
A line that is parallel to y = (2/5)x - 1/2 will have the same slope as this line, which is 2/5. Therefore, the equation of the line we are looking for will be of the form:
y = (2/5)x + b
Where b is the y-intercept of the line. To find the value of b, we can use the fact that the line passes through the point (-7, 2). Substituting these coordinates into the equation, we get:
2 = (2/5)(-7) + b
Simplifying this equation, we get:
2 = -14/5 + b
b = 2 + 14/5
b = 24/5
So, the equation is y = 2/5x + 24/5
which graph shows the solution to the system of linear equations?
y=-1/3x+1
y=-2x-3
The equivalent graph of the function are shown below.
Equation of a lineThe equation of a line in slope-intercept form is expressed as y = mx + b where:
m is the slope
b is the intercept
Given the equations below
y=-1/3x+1
y=-2x-3
The slope of both equations are -1/3 and -2 respectively showing a negative slope. This means that both lines are oriented downward the plane as shown below;
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QUESTION 9 of 10: You enter a bid. Your cost on the product you make is $112 per unit. If you bid at $138 per unit, what percent is the price
you bid over your cost? (Round to the nearest percent)
a) 19%
b) 23%
C) 81%
Submit
From the information given, the percentage increase in the price bid over the cost will be 23%.
Cost on the product = $112.
Amount bid = $138
Therefore, the Percentage increase will be:
= (138 - 112) / 112 × 100
= 26/112 × 100
= 23%
Therefore, from the calculation above, it can be seen that the percentage increase in the price bid over the cost will be 23%.
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Answer:
the answer is A. y=-2x+3
i hope this helps
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Answer:
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Step-by-step explanation: