Example 3: A triangle has angles x,y, and z. The measure of angle y is 10 degrees smaller than the measure of three times angle x. The measure of angle y is half the measure of angle z. Find the measure of all three angles in the triangle. (Using matrix to solve this word problems.)
Answers
The measure of angle x is 10/3 degrees, the measure of angle y is 0 degrees, and the measure of angle z is 0 degrees.
To solve this word problem using a matrix, we can set up a system of equations based on the given information. Let's assign variables to the angles in the triangle: x for angle x, y for angle y, and z for angle z.
From the given information, we can establish the following equations:
y = 3x - 10 (angle y is 10 degrees smaller than three times angle x)
y = (1/2)z (angle y is half the measure of angle z)
We can rewrite these equations in standard form:
3x - y = 10
y - (1/2)z = 0
To solve this system of equations using a matrix, we can represent the coefficients of the variables in a matrix and solve for the variables.
The matrix equation can be written as:
[3 -1 0] [x] [10]
[0 1 -1/2] [y] = [0]
Using matrix operations, we can find the values of x, y, and z.
By performing row operations on the augmented matrix, we can reduce it to row-echelon form or row-reduced echelon form. After the row operations, we obtain the following matrix:
[1 0 1/6] [x] [10/3]
[0 1 -1/2] [y] = [0]
From the row-reduced echelon form, we can determine the values of x, y, and z:
x = 10/3
y = 0
z = 0
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Related Questions
Pls help me I will make you brain plssss
Answers
Answer:
whatsup. ......
Step-by-step explanation:
its A :)
Rachel has two identical basil plants and an aloe plant. She also has two identical white lamps and two identical red lamps she can put each plant under (she can put more than one plant under a lamp, but each plant is under exactly one lamp). How many ways are there for Rachel to put her plants under her lamps
Answers
The answer is 14.
We can divide this into cases.
Consider the situation in which all three plants are illuminated by the same color lamp. Either all three plants are illuminated by the same light, both basil plants are illuminated by one lamp and the aloe plant is illuminated by the other, or the aloe plant and one basil plant are illuminated by one lamp and the other basil plant is illuminated by the other. This scenario provides us three options for each lamp color, for a total of six options.
Consider the situation in which the aloe plant is exposed to a different hue of lamp than the two basil plants. Because the two lights of the same hue that the aloe plant can be placed under are similar, it makes no difference which one the aloe plant is placed under. The basil plants can be placed under the same lamp or under separate lamps. This scenario provides us two options when the aloe is placed under a white lamp and two options when the aloe is placed under a red lamp, for a total of four options.
Finally, examine the scenario in which the basil plants are each lit by a different colored lamp. For a total of four options, the aloe plant can be placed under the same white lamp as a basil plant, the same red lamp as a basil plant, a different white lamp from the basil plant, or a different red lamp from the basil plant.
In all, there are 6+4+4=14 possibilities.
What is probability?
Probability is simply the likelihood of something occurring. When we are uncertain about the result of an event, we might discuss the probabilities of several outcomes—how likely they are. Statistics is the study of occurrences guided by probability.To learn more about Probability visit:
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x:y = 9:7 Form an equation that links x and y
Answers
Equations are expressions with equal values
The equation that links x and y is \(7x = 9y\)
How to determine the equationThe ratio is given as:
\(x : y = 9 : 7\)
Express the ratio, as a fraction
\(\frac xy = \frac 97\)
Cross multiply
\(7 *x = 9 * y\)
Evaluate the product
\(7x = 9y\)
Hence, the equation that links x and y is \(7x = 9y\)
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GIVING BRAINLEIST
b – 12 = –8
WHAT IS THIS THE CHOICES ARE
A -20
B 4
C 20
D -4
Answers
Answer:
B 4 is the correct answer
Step-by-step explanation:
b - 12 = - 8
b = 12 - 8
b = 4
Jamal is 12 years older than twice Stephanie’s age. The sum of their ages is 45. A. If Stephanie’s age is represented by x, write an expression for Jamal’s age in terms of Stephanie’s age. _________________________ B. Write an equation that you can use to find the ages of both Stephanie and Jamal. Solution: ________________ D. Stephanie’s age: ___________ Jamal’s age: PLEASE PLEASE HELPPPP IMPORTANNNTTTT
Answers
Answer:
Jamal's age = 2×x+12= 2x+12
sum og ages = 45
2x+12+x= 45
=3x+12=45
=45-12= 3x
= 32=3x
x= 32/3= 10.67
2x+13 =21.34+12= 33.34
hope.ot helps
plz mark as brainliest
What is the answer of 10-squareroot of 4 divided by 2
Answers
Answer:
10
Step-by-step explanation:
sqrt(4)=2
2/2=1
10×1=10
A group of nine friends po to a baseball game. Each friend buys either a hamburger or hot dag
at the game
The group buy 6 hot dogs and 3 hamburgers for a total of $27. 75.
• A hamburger costs $1. 75 more than a hot dog.
What is the cost of a hamburger?
Answers
$4.25 is the cost of a hamburger
Let the cost of the hotdog be x
Since a hamburger costs $1.75 more than a hot dog. Then, the cost of the hamburger will be:
= x + 1.75
We are further told that the group buys 6 hot dogs and 3 hamburgers for a total of $27.75. This can then be represented in an equation:
6 hot Dogs + 3 hamburger = $27.75
(6 × x) + 3 × (x + 1.75) = 27.75
6x + 3x + 5.25 = 27.75
9x = 27.75 - 5.25
9x = 22.50
x = 22.50/9
x = 2.50
The hotdog cost $2.50
Cost of hamburger = x + 1.75
= $2.50 + $1.75
= $4.25
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Katalin drove 180 miles on her vacation. She drove an average of 1.5 times faster on the second 90 miles of her trip than she did on the first 90 miles of her trip. Which expression represents the time she spent driving? Let x = her speed on the first half of the trip.
Answers
===================================================
Work Shown:
x = speed on 1st half of the trip
distance = rate*time
90 = x*t
t = 90/x
She spent 90/x hours driving on the 1st half of the trip
1.5x = speed on the 2nd half of the trip
d = r*t
t = d/r
t = 90/(1.5x)
t = 60/x
She spent 60/x hours driving on the 2nd half of the trip
Overall, she spent 90/x+60/x hours driving the entire 180 mile trip.
Edit:
That expression simplifies to 150/x since 90+60 = 150
A bird on a long migration flies 63 kilometers per hour for 2900 kilometers how long does he fly Not simplified
Answers
Answer:
Simplified - 46
Not Simplified - 46.03174603
Step-by-step explanation:
2900 divided by 63 (2900/63)
The Simplified form would be; 46 and Not Simplified - 46.03174603
What is simplification of an expression?Usually, simplification involves proceeding with the pending operations in the expression. Like, 5 + 2 is an expression whose simplified form can be obtained by doing the pending addition, which results in 7 as its simplified form. Simplification usually involves making the expression simple and easy to use later.
We are given that bird on a long migration flies 63 kilometers per hour for 2900 kilometers.
Then we need to solve as;
2900 divided by 63
(2900/63) = 46.03174603
This is not simplified.
The simplified form will be 46.
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Angie and Kelly are playing a card game. Angie has 37 points. She has 5 points less than Kelly. How many points does Kelly have?
A) 42
B)45
C)32
Answers
Answer:
42 a
Step-by-step explanation:
brailest plz
In the picture below, polygon ABCD ~ polygon WXYZ. Solve for m.
A
13
D 10 C
12
B
W
24
Z 15 Y
m
X
m =
Answers
Since polygon ABCD is similar to polygon WXYZ, the corresponding sides are proportional.
That means:
AB/WX = BC/XY = CD/YZ = AD/WZ
We can use this fact to set up the following equations:
AB/WX = 13/24
CD/YZ = 12/15 = 4/5
AD/WZ = 10/m
We are given that AB = 13 and WX = 24, so we can substitute those values in the first equation:
13/24 = BC/XY
We are also given that CD = 12 and YZ = 15, so we can substitute those values in the second equation:
4/5 = BC/XY
Since both equations equal BC/XY, we can set them equal to each other:
13/24 = 4/5
To solve for m, we can use the third equation:
10/m = AD/WZ
We know that AD = AB + BC = 13 + BC, and WZ = WX + XY = 24 + XY. Since BC/XY is the same in both polygons, we can use the results from our previous equations to find that BC/XY = 4/5.
So we have:
AD/WZ = (13 + BC)/(24 + XY) = (13 + (4/5)XY)/(24 + XY) = 10/m
Now we can solve for XY:
13 + (4/5)XY = (10/m)(24 + XY)
Multiplying both sides by m(24 + XY), we get:
13m(24 + XY)/5 + mXY(24 + XY) = 10(13m + 10XY)
Expanding and simplifying, we get:
312m/5 + 13mXY/5 + mXY^2 = 130m + 100XY
Rearranging and simplifying further, we get:
mXY^2 - 87mXY + 650m - 1560 = 0
We can use the quadratic formula to solve for XY:
XY = [87m ± sqrt((87m)^2 - 4(650m - 1560)m)] / 2m
Simplifying under the square root:
XY = [87m ± sqrt(7569m^2 - 2600m)] / 2m
XY = [87m ± sqrt(529m^2)] / 2m
XY = (87 ± 23m) / 2
Since XY must be positive, we can use the positive solution:
XY = (87 + 23m) / 2
Now we can substitute this value for XY in the equation we derived earlier:
13 + (4/5)XY = (10/m)(24 + XY)
13 + (4/5)((87 + 23m) / 2)= (10/m)(24 + (87 + 23m) / 2)
Multiplying both sides by 10m, we get:
130m + 52(87 + 23m) / 10 = (240 + 87m) / 2
Simplifying and solving for m, we get:
1300m + 52(87 + 23m) = 240 + 87m
1300m + 4524 + 1196m = 240 + 87m
2403m = -4284
m = -4284 / 2403
m ≈ -1.78
Therefore, the value of m is approximately -1.78.
Uitute Start Video Share Content Participant original $14 markdown= 15% ole TOTAL original Mark down
Answers
Jenelle bought a home for $460,000, paying 16as a down payment, and financing the rest at 5.4% Interest for 30 years. Round your answers to the nearest cent How much money did Jenelle pay as a down payment? $ What was the original amount financed? $ What is her monthly payment? $ If Jenelle makes these payments every month for thirty years, determine the total amount of money she will spend on this home. Include the down payment in your answer. $
Answers
To find the down payment, we need to multiply the price of the house by the percentage paid as a down payment:
Down payment = 16% × $460,000 = $73,600
The original amount financed is the price of the house minus the down payment:
Original amount financed = $460,000 - $73,600 = $386,400
To find the monthly payment, we can use the formula for a fixed-payment loan:
Monthly payment = (P × r) / (1 - (1 + r)^(-n))
where P is the principal (the original amount financed), r is the monthly interest rate (5.4% divided by 12), and n is the number of monthly payments (30 years times 12 months per year, or 360).
Monthly payment = ($386,400 × 0.0045) / (1 - (1 + 0.0045)^(-360)) = $2,187.28
To find the total amount of money Jenelle will spend on the home, we can multiply the monthly payment by the number of payments:
Total amount spent = (Monthly payment) × (Number of payments)
Number of payments = 30 years × 12 months per year = 360
Total amount spent = $2,187.28 × 360 = $788,020.80
Adding the down payment, the total cost of the home for Jenelle will be:
Total cost = $460,000 + $788,020.80 = $1,248,020.80
Given two points A (0, 4) and B (3, 7), what is the angle of inclination that the line segment A makes with the positive x-axis? A. 90° B. 60° C. 45° D. 30°
Answers
The angle of inclination that the line segment A makes with the positive x-axis is 45° (option C).
To determine the angle of inclination that the line segment A makes with the positive x-axis, we can use the slope of the line. The slope is given by the formula:
slope = (change in y)/(change in x)
In this case, the change in y is 7 - 4 = 3, and the change in x is 3 - 0 = 3. Thus, the slope of the line is:
slope = 3/3 = 1
The angle of inclination θ can be found using the inverse tangent function:
θ = tan^(-1)(slope)
Substituting the slope value of 1 into the equation, we have:
θ = tan^(-1)(1) ≈ 45°
Therefore, the angle of inclination that the line segment A makes with the positive x-axis is 45°.
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12
Calculate the area of the given segment. Round your answer to the nearest tenth, if necessary.
60
8 in.
Answers
Check the picture below.
\(\textit{area of a segment of a circle}\\\\ A=\cfrac{r^2}{2}\left( ~~ \cfrac{\pi \theta }{180}-\sin(\theta ) ~~ \right) \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=8\\ \theta =60 \end{cases} \\\\\\ A=\cfrac{8^2}{2}\left( ~~ \cfrac{\pi (60) }{180}-\sin(60^o ) ~~ \right)\implies A=32\left( ~~ \cfrac{\pi }{3}-\sin(60^o ) ~~ \right) \\\\\\ A=32\left( ~~ \cfrac{\pi }{3}-\cfrac{\sqrt{3}}{2} ~~ \right)\implies A=\cfrac{32\pi }{3}-16\sqrt{3}\implies A\approx 5.8~in^2\)
PLS HELP !!!! 100 POINTS ON THE LINE !!!! 6TH GRADE MATH
JUST ANSWER FIRST 2 QUESTIONS THE OPTIONS FOR THE BLANKS for 1 are: PQ, PR, QR.
FOR THE SECOND QUESTION THE BLANKS COULD BE: ST, SU, TU.
Answers
Answer:
Slope ratio of first triangle: 3:2
6/4x is the slope but its ratio is the same as the other triangle 3:2
Step-by-step explanation:
You are trying to find the time it will take you to drive from Kansas City to Des Moines. You know the distance you travelled is 193 miles 25 miles. Your average speed will be 65 mph 8 mph. How long will it take you to make the drive
Answers
To drive from Kansas City to Des Moines, with a distance of 193.25 miles and an average speed of 65.8 mph, it will take approximately 2 hours and 56 minutes.
To calculate the time it will take to make the drive, we can use the formula: time = distance / speed. Given that the distance is 193.25 miles and the average speed is 65.8 mph, we can substitute these values into the formula.
Dividing the distance by the speed, we get: 193.25 miles / 65.8 mph = 2.94 hours.
Since time is typically expressed in hours and minutes, we need to convert the decimal portion (0.94) of the hours into minutes. Multiplying 0.94 by 60, we find that the decimal portion is equivalent to approximately 56 minutes.
Therefore, the total time it will take to drive from Kansas City to Des Moines is approximately 2 hours and 56 minutes.
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find an equation of the line tangent to the curve at the point corresponding to the given value of t. x=t^2-23, y=t^3 + t; t=5
Answers
The equation of the line tangent to the curve at the point corresponding to the given value of t.
x = t²-23 and y = t³ + t, at t = 5 is
38x - 5y + 574 = 0
Given, a curve with the points represented by
x = t²-23 and y = t³ + t, at t = 5
we have to find an equation of the line tangent to the curve at the given point on the curve.
so, the given point is (x , y) = (5² - 23 , 5³ + 5)
(x , y) = (2 , 130)
Now, the slope of the curve at that point be,
dy/dx = (3t² + 1)/(2t)
dy/dx = 76/10
Now, on using the slope-intercept form, we get
(y - 130)/(x - 2) = 38/5
5(y - 130) = 38(x - 2)
5y - 650 = 38x - 76
38x - 5y + 574 = 0
Hence, the equation of the line tangent to the curve at the point corresponding to the given value of t.
x = t²-23 and y = t³ + t, at t = 5 is
38x - 5y + 574 = 0
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Mrs. Johnson spent $611 buying lunch for 78 students. If all the lunches cost the same, about how much did she spend on each lunch?
Answers
Answer:
$7.83
Step-by-step explanation:
Answer:
7.83
Step-by-step explanation:
611 ÷ 78 = 7.83
Evalute the expression X=6 y=-4 -3y+x
Answers
Answer:
y= 1/2
Step-by-step explanation:
y=-4 -3y+x
<=> y + 3y = -4 + x
<=> 4y = -4 + 6
<=> 4y = 2
<=> y = 1/2
Could two samples have the same range but different means? Explain. surements: nedian: n (ne The standard deviation indicates how measurements vary about the mean. The standard deviation is easy t0 cal- culate. Begin by calculating the mean, measuring the devia- tion of each sample from the mean, squaring each deviation and tnen summing the deviations. This summation results in the sum of squared deviations. For example, consider group of shrimp that are 22, 19, 18, and 21 cm long: The mean length of these shrimp is 20 cm: leaf eaf was Mean Deviation (Deviation} Sample Value
Answers
Yes, two samples can have the same range but different means. The range of a dataset is a measure of the spread or dispersion and is determined by the difference between the maximum and minimum values. The mean, on the other hand, represents the average value of the dataset.
To illustrate this concept, let's consider two different samples:
Sample 1: 1, 5, 6, 9, 10
Sample 2: 2, 4, 7, 8, 10
Both samples have a range of 9 (10 - 1).
However, the means of the samples are different. The mean of Sample 1 is
(1 + 5 + 6 + 9 + 10) / 5 = 6.2,
while the mean of Sample 2 is
(2 + 4 + 7 + 8 + 10) / 5 = 6.2.
Despite having the same range, the means differ between the two samples.
The range only considers the extreme values of the dataset, whereas the mean takes into account all values and provides an average measure. It is possible for the individual values within the samples to be distributed differently, resulting in different means, even if the range remains the same.
Therefore, the range and the mean are distinct measures that capture different aspects of the dataset. While it is possible for two samples to have the same range, they can still have different means based on the specific values and their distribution within each sample.
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If Kailyn ran 3 miles in 20 minutes, how many miles would she run in 1 hour?
1 hour = how many minutes?
Answers
Answer:
1 hr =60 mins kailyn ran 9 miles in 1 hour
Step-by-step explanation:
Hope this helps:)
A radio transmission tower is 160 feet tall. How long should a guy wire be if it is to be attached 13 feet from the top and is to make an angle of 29\deg with the ground? Give your answer to the nearest tenth of a foot.
Answers
x = 147 / 0.5446 ≈ 270.2 ft
To find the length of the guy wire for a radio transmission tower, trigonometry concepts are applied. Given a tower height of 160 feet, with the wire attached 13 feet from the top and making an angle of 29° with the ground, we can solve for the length of the guy wire, represented by x.
Using the Pythagorean theorem and considering the right triangle formed by the tower height, the wire attachment point, and the ground, we can set up the equation:
x = √((160 - 13)² + x²)
Next, we apply the tangent function to the given angle:
tan(29°) = (160 - 13) / x
Simplifying, we have:
0.5446 = 147 / x
To solve for x, we rearrange the equation:
x = 147 / 0.5446 ≈ 270.2 ft
Rounding to the nearest tenth of a foot, the length of the guy wire required is approximately 270.2 feet. This wire is attached 13 feet from the top of the tower and makes a 29° angle with the ground.
Trigonometry plays a crucial role in solving real-world problems involving angles and distances. It provides a mathematical framework for calculating unknown values based on known information, enabling accurate measurements and constructions.
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I’ll more u brainlies plz
The property for -y/2 -6+6=15+6 and -2•-y/2=-2•21
Answers
Answer:
Step-by-step explanation:
step 2, addition
step 4 multiplication
Please help me!!!!!!
Answers
Answer:
I am sorry but this is too difficult
Step-by-step explanation:
I recommend asking a teacher for help not to be rude
Beth is making fruit salad. She adds 4 grapes for every 2 strawberries. If she uses 12 strawberries, how many grapes will she use?
Answers
Answer:
24
Step-by-step explanation:
12/2=6
6*4=24
Answer:
Beth will use 24 grapes.
Step-by-step explanation:
Not going to lie I forgot how i got my answer but ik its good.
Use the rules of deduction in the Predicate Calculus to find a formal proof for the following sequent (without invoking sequent or theorem introduction): (Wr)((G() VH (ar)) = K(z)), (2x) ~ K(x) + (Ex)
Answers
To find a formal proof for the given sequent, we will use the rules of deduction in the Predicate Calculus.
1. (Wr)((G() VH (ar)) = K(z)) premise
2. (2x) ~ K(x) + (Ex) premise
3. ~K(a) assumption
4. (Ea) assumption
5. ~K(a) + (Ea) conjunction introduction (3, 4)
6. (2x) ~ K(x) premise
7. ~K(b) assumption
8. ~K(b) + (Eb) conjunction introduction (7, 4)
9. ~(G(b) V H(b)) = K(z) universal elimination (1)
10. ~(G(b) V H(b)) equivalence elimination (9)
11. ~(G(b) V H(b)) equivalence elimination (9)
12. ~G(b) /\ ~H(b) de Morgan's law (10)
13. ~G(b) conjunction elimination (12)
14. H(b) -> K(z) equivalence elimination (10)
15. H(b) -> K(z) equivalence elimination (10)
16. ~H(b) V K(z) implication elimination (14)
17. ~H(b) V K(z) implication elimination (14)
18. K(z) disjunction elimination (16, 17)
19. ~K(a) + K(z) universal elimination (1)
20. K(z) disjunction elimination (18, 19)
21. ~K(a) -> K(z) implication introduction (3-20)
22. K(a) assumption
23. ~K(a) negation introduction (3-21)
24. (Ex) existential introduction (22)
25. ~K(b) -> (Ex) implication introduction (7-24)
26. (Ex) disjunction elimination (25, 5)
27. (2x) ~ K(x) -> (Ex) implication introduction (6-26)
28. (Ex) universal elimination (27)
29. (Ex) existential elimination (2, 28)
Therefore, we have proved the given sequent using the rules of deduction in the Predicate Calculus.
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Liam deposits $3,500 in a saving account that pays 1.5% interest, compounded quarterly.
a.Find the balance after a year.
B. Compare the interest compounded annually.
Answers
Therefore the balance after a year comes out to be $3552.80 and the balance comes out to be after compounded annually is $3552.50
What is compound interest ?Compound interest is the interest on savings that is computed using both the original principal and the interest accrued over time.
It is believed that Italy in the 17th century was where the idea of "interest on interest" or compound interest first emerged. It will accelerate the growth of a sum more quickly than simple interest, which is only applied to the principal sum.
Money multiplies more quickly through compounding, and the higher the
Here,
The Liam deposits $3,500 in a saving account that pays 1.5% interest,
Thus rate=1.5% and principle amount =3500
Therefore it is compounded quartely ,
The balance after year=
C=P\(e^{rt}\)
C=3500*\(e^{1.5*4}\)
C=3552.80
Therefore the balance after a year comes out to be $3552.80
Therefore the balance comes out to be after compounded annually is
$3552.50
So there is only $0.30 difference between the amounts
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ILL MARK AS BRAINLESS PLS HELP
Answers
hippo- kilometers
fingernail clipping- millimeter
whiteboard- meters
the border between the U.S and Canada- miles
hope this helped; have a great day :)
The length of a rectangle is 12cm.its with is 6cm calculate the perimeter of the rectangle.
Answers
The perimeter of the rectangle is 36 cm.
To calculate the perimeter of a rectangle, you need to add the lengths of all its sides. In this case, the length is given as 12 cm and the width as 6 cm.
A rectangle has two pairs of equal sides. The length and width are opposite sides and each pair is equal in length. Therefore, to find the perimeter, we can use the formula:
Perimeter = 2 * (length + width)
Substituting the given values:
Perimeter = 2 * (12 cm + 6 cm)
Perimeter = 2 * 18 cm
Perimeter = 36 cm
Therefore, the perimeter of the rectangle is 36 cm.
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