Solve the system of equations using a matrix.
x + 3y = 22 2x - y = 2
Answers
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The system of equations using a matrix can be solved, we can represent the coefficients of the variables and the constant terms in a matrix form.
Let's call this matrix A.
The given system of equations is:
x + 3y = 22 (Equation 1)
2x - y = 2 (Equation 2)
To write this in matrix form, we can arrange the coefficients of the variables in a matrix, as well as the constant terms in another matrix. Let's call the matrix of coefficients B and the matrix of constant terms C.
B = | 1 3 |
| 2 -1 |
C = | 22 |
| 2 |
Now, we can write the system of equations in matrix form as:
AX = C
where X is a matrix of variables, and A is the matrix of coefficients.
To solve for X, we can multiply both sides of the equation by the inverse of matrix A. Let's call the inverse of matrix A as A⁻¹.
A⁻¹(AX) = A⁻¹C
Since A⁻¹A is the identity matrix, we have:
X = A⁻¹C
To find the inverse of matrix A, we can use the formula:
A⁻¹ = (1/|A|)adj(A)
where |A| is the determinant of matrix A, and adj(A) is the adjugate of matrix A.
By calculating the determinant of matrix A, we find:
|A| = (1)(-1) - (3)(2) = -1 - 6 = -7
Next, we can find the adjugate of matrix A by swapping the elements of the main diagonal (1 and -1) and changing the sign of the other two elements:
adj(A) = | -1 -3 |
| -2 1 |
Now, we can calculate A⁻¹ using the formula mentioned earlier:
A⁻¹ = (1/(-7)) * | -1 -3 |
| -2 1 |
Simplifying this expression, we get:
A⁻¹ = | 1/7 3/7 |
| 2/7 -1/7 |
Finally, we can calculate X by multiplying A⁻¹ and C:
X = A⁻¹C
| x | | 1/7 3/7 | | 22 |
| | = | | * | |
| y | | 2/7 -1/7 | | 2 |
Simplifying this expression, we get:
| x | | 1/7 * 22 + 3/7 * 2 |
| | = | |
| y | | 2/7 * 22 - 1/7 * 2 |
Calculating further, we find:
| x | | 22/7 + 6/7 |
| | = | |
| y | | 44/7 - 2/7 |
Simplifying this expression, we get the final solution:
| x | | 28/7 |
| | = | |
| y | | 42/7 |
Therefore, the solution to the system of equations is:
x = 4
y = 6
Note: The solution can also be represented as a fraction, where x = 4 and y = 6.
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The solution to the given system of equations is x = -20/7 and y = 6/7.
To solve the system of equations using a matrix, we can represent the equations in the form of a matrix equation.
Let's write the given system of equations in matrix form:
```
| 1 3 | | x | | 22 |
| 2 -1 | | y | = | 2 |
```
We can write this equation as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Now, to solve for X, we can use the inverse of matrix A. The inverse of A is denoted as \(A^{(-1)}\).
Multiplying both sides of the equation AX = B by \(A^{(-1)}\), we have:
\(A^{(-1)}\) * AX = \(A^{(-1)}\) * B
Since A^(-1) * A gives the identity matrix I, we are left with:
IX = \(A^{(-1)}\) * B
Simplifying, we have:
X =\(A^{(-1)}\) * B
To find \(A^{(-1)}\), we need to calculate the determinant of A. The determinant is found by subtracting the product of the diagonals going from top left to bottom right from the product of the diagonals going from top right to bottom left. In this case, the determinant of A is (1 * (-1)) - (3 * 2) = -7.
To find \(A^{(-1)}\), we need to divide each element of the matrix by the determinant:
```
| 1/(-7) -3/(-7) | | -1/(-7) 3/(-7) |
| 2/(-7) -1/(-7) | = | 2/(-7) -1/(-7) |
```
Simplifying further, we have:
```
| -1/7 3/7 |
| -2/7 1/7 |
```
Now, we can calculate \(A^{(-1)}\) * B to find X:
```
| -1/7 3/7 | | 22 | | x |
| -2/7 1/7 | * | 2 | = | y |
``
Multiplying, we get:
```
-1/7 * 22 + 3/7 * 2 = x
-2/7 * 22 + 1/7 * 2 = y
```
Simplifying further, we have:
```
x = -20/7
y = 6/7
```
Remember to check your solution by substituting these values back into the original equations to ensure they satisfy both equations.
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Related Questions
plzzzzz help me 50points plzzz help me
Answers
Answer:
the answer will be
area is= 10
maximum area is 100
Step-by-step explanation:
hello,
to find the area of the figure:
A = b x h
b = w ; h = 20-w
A = w(20-w)
A = 20w - w^2 OR A = -w^2+20w
to find the maximum w value:
so this equation when graphed would make a parabola. the maximum area is found using the vertex of the parabola.
you can graph it to find the vertex or the vertex can be found with the formula -b/2a
our b is 20 and our a is -1
-b/2a = -20/2(-1) = 10
10 is the w value
to find the area we substitute all the w's in the equation for 10
A = 20w-w^2
A = 20(10) - (10)^2
A = 200 - 100
A = 100
so to sum up:
the equation for the area is A = 20w - w^2 OR A = -w^2+20w (whichever one should work)
the maxiumum area is 100 square units
hope this helped :)
Kyle purchased a plant that was 2 inches tall. The function h(t)=0.75t models the height of the plant t months after kyle purchased the plant. What is the meaning of h(11)
Answers
Scarlett usually rides her bike about 1 and 1/5 hours every day. The distance between the library and school is 7/8 mile. Yesterday the bike had a problem and Scarlett only rode her bike 2/3 of the way from school to the library and walked the rest of the way. How far did she ride her bike
Answers
Scarlett typically rides her bike for around 1 and a half hours per day, thus she covered a distance of 7/12.
what is distance ?The overall uninspired velocity of an object will be its distance. Distance is the amount of space an object has traveled, regardless of where it began or ended. Distance would be the measurement of said size or extent of the separation between two places. It is important to keep in mind that the distance between two spots and the traveled between them aren't the same thing. Distance traveled is the sum of a line's length joining various places. Distance refers to the actual travel path of a body. Path length is another term for it. For instance, if an object is traveling from board to point P, the path's length will be similar to OP = 360 m.
given
We are aware of: t = 1 1/5 d = 7/8
Write down and replace 7/8 = v * (1* 1/5)
Remove the connecting element: 7/4 * 1/3
As a single fraction, write:
7 / 4*3
Determine the quotient or product: 7/12
Response: 7/12
Scarlett typically rides her bike for around 1 and a half hours per day, thus she covered a distance of 7/12.
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Complete a function table from an equation
Complete the table.
Answers
a professor gives a special tutoring session to students who scored less than 60 percent on the first test. on the second test, their mean score rises to 70 percent. what conclusion can we draw, if any, and why?
Answers
We can conclude that the special tutoring session seems to have had a positive impact on the students' performance, as their mean score increased from below 60% to 70%.
Determine the conclusionWe can conclude that the special tutoring session has been effective since the students' mean score increased from 60 percent to 70 percent on the second test.
The students who scored less than 60 percent on the first test were given a special tutoring session, which resulted in their mean score increasing to 70 percent on the second test.
Therefore, we can conclude that the special tutoring session has been effective in improving the students' performance on the second test.
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I NEED AN ANSWER ASAP PLEASE
Answers
Step-by-step explanation:
the correct answer is option a 6a-7
please help me with these questions
Problem 1: Find the measure of each marked angle. 2. (7x+19) (2x-1)º "V Vest (-3x+5)° (-8x+30) 5. 6. (32-2x)" (10x-10) (2x+18) (8x+14) (12x+40) (20x + 10) mand n are parallel. Problem 2: Identify th
Answers
In Problem 1, the measure of each marked angle is as follows:292º, -112º, -282º, -46º, 380º, 96º, 326º, 508º, and 790º.In Problem 2, the angles indicated by the letters in the given figure are as follows:c = 65º, d = 95º, e = 65º, f = 95º, g = 85º, and h = 85º.
Problem 1:The measures of the marked angles are as follows:(7x + 19)º and (-3x + 5)º are supplementary angles since they are the interior angles on the same side of the transversal "V Vest".
Therefore, we can say: (7x + 19)º + (-3x + 5)º = 180º Simplifying, 7x + 19 - 3x + 5 = 180
Combine like terms and solve for x: 4x + 24 = 180 4x = 180 - 24 4x = 156 x = 39 Now substitute x = 39 in the given expressions and find the value of each angle.
(7x + 19)º = (7 × 39 + 19)º = 292º(-3x + 5)º
= (-3 × 39 + 5)º = -112º(-8x + 30)º = (-8 × 39 + 30)º
= -282º(32 - 2x)º = (32 - 2 × 39)º = -46º(10x - 10)º
= (10 × 39 - 10)º = 380º(2x + 18)º = (2 × 39 + 18)º = 96º(8x + 14)º
= (8 × 39 + 14)º = 326º(12x + 40)º = (12 × 39 + 40)º
= 508º(20x + 10)º = (20 × 39 + 10)º = 790º
Therefore, the measures of the marked angles are:292º, -112º, -282º, -46º, 380º, 96º, 326º, 508º, and 790º.Problem 2:The angles indicated by the letters in the given figure are as follows: Angle c: Corresponding angles with respect to the parallel lines n and m are equal. Therefore, we can say: c = 65º.Angle d: Vertically opposite angles are equal. Therefore, we can say: d = 95º.
Angle e: Alternate interior angles with respect to the parallel lines n and m are equal. Therefore, we can say: e = 65º.Angle f: Corresponding angles with respect to the parallel lines n and m are equal. Therefore, we can say: f = 95º.Angle g: Interior angles on the same side of the transversal are supplementary. Therefore, we can say: g = 180º - 95º = 85º.Angle h: Vertically opposite angles are equal. Therefore, we can say: h = 85º.
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What is the decimal equivalent of 6/13 ? Round to tenths place. If answer doesn't
have a tenths place then use zero on the end to make the answer have a tenths place
to be counted correct.
Answers
Answer:
0.5
Step-by-step explanation:
6/13 is the same as saying 6 ÷ 13
6 ÷ 13 = 0.4615384615
The first position after the decimal is the tenths column. Round up, because 6>5.
Discuss the population scenario of Dhaka City. How do you want to restructure the population of Dhaka City to mitigate the present traffic jam situation? \( (3+7) \)
Answers
To mitigate the present traffic jam situation in Dhaka City, it is important to consider restructuring the population distribution and implementing effective urban planning strategies.
Here are some possible approaches:
Decentralization: Encourage the development of satellite towns and economic centers outside the central areas of Dhaka City. This can help disperse the population and economic activities, reducing the strain on the city's infrastructure and transportation systems.Improved public transportation: Enhance the public transportation network by expanding the coverage, increasing the frequency of services, and improving the quality of transportation modes such as buses, metro rail, and waterways. This can encourage more people to rely on public transport, reducing the number of private vehicles on the roads.Mixed-use development: Promote mixed-use development in the city by integrating residential, commercial, and recreational areas. This can reduce the need for long commutes and decrease traffic congestion during peak hours.Traffic management and infrastructure improvement: Implement effective traffic management strategies, including the development of intelligent transportation systems, traffic signal synchronization, and efficient road network planning. Additionally, invest in improving road infrastructure, constructing new roads, flyovers, and pedestrian-friendly infrastructure to accommodate the growing population and enhance traffic flow.Encourage alternative modes of transport: Promote and incentivize the use of alternative modes of transport such as cycling, walking, and carpooling. Establish dedicated cycling lanes, pedestrian-friendly sidewalks, and carpooling initiatives to reduce the reliance on private vehicles.Urban planning and zoning regulations: Enforce strict urban planning and zoning regulations to control haphazard urban growth and prevent the concentration of population in specific areas. Encourage the development of mixed-income neighborhoods and provide affordable housing options in various parts of the city.Telecommuting and flexible working arrangements: Encourage businesses and organizations to adopt telecommuting and flexible working arrangements to reduce peak-hour traffic congestion. This can be achieved by promoting remote work options and implementing policies that support flexible working hours.In conclusion, mitigating the traffic jam situation in Dhaka City requires a comprehensive approach that includes restructuring the population distribution, improving public transportation, implementing effective traffic management strategies, and promoting alternative modes of transport. These measures, combined with urban planning initiatives and flexible working arrangements, can help alleviate congestion and create a more sustainable and livable city.
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Can you please help me answer this picture
Answers
hope this helps !!
a line ray or segment that divides a segment into two congruent parts at a 90° angle
Answers
A line, line segment, or ray that divides an angle into two congruent angles is called an angle bisector.
What are supplementary angles example?
Supplementary angles are those angles that sum up to 180 degrees. For example, angle 130° and angle 50° are supplementary angles because sum of 130° and 50° is equal to 180°. Similarly, complementary angles add up to 90 degrees.
What do you mean by angle bisector?
If the angle is divided by the line and the angle is divided such that the angles are equal to each other then the line is called the angle bisector.
The word congruent means equal of the things. In this, the angles are congruent to each other which means the angles are equal to each other.
Thus, the black space is to be filled by the angle bisector.
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how do i distribute
x+y=6
x-y=2
Answers
x=4
y=2
solve the equations together to get value of x and y
Answer:
The value are
x=4
y=2
Step-by-step explanation:
x+y=6 1
x-y=2 2
Using Substitution Method
From equation 2
x=y+2 3
input equation 3 into equation 1
(y+2)+y=6
y+y+2=6
2y+2=6
2y=6-2
2y=4
divide both sides by 2
2y/2=4/2
y=2
input into equation 3
x=y+2
x=2+2
x=4
Which function is a quadratic function?
Answers
Answer:
\(12-t+6t^2\)
Step-by-step explanation:
Find the one with the highest exponent 2.
\(12-t+6t^2\) is the only one
Answer:
h(t) = 12 -t+6t^2
Step-by-step explanation:
A quadratic function has the highest power of the variable as 2 and must have a power of 2
h(t) has a power of 3
g(t) has a power of 3
k(t) has a power of 4
Amy is helping plan her school's new basketball court. The west edge of the basketball court is located on the line y = 5x + 2. The east edge cannot intersect with the west edge. On which line could the east edge be located? (1 point)
−y − 5x = 100
y + 5x = 100
−5x − y = 50
5x − y = 50
Answers
Based on the analysis, the east edge of the basketball court could be located on the line given by either −y − 5x = 100, y + 5x = 100, or −5x − y = 50, as these lines do not intersect with the west edge.
To determine on which line the east edge of the basketball court could be located, we need to find a line that does not intersect with the west edge represented by the equation y = 5x + 2.
The slope-intercept form of a line is given by y = mx + b, where m is the slope of the line and b is the y-intercept.
Comparing the equation y = 5x + 2 with the given options, we can observe that the slope of the west edge is 5.
Now let's analyze the options:
Option 1: −y − 5x = 100
By rearranging the equation to slope-intercept form, we get y = -5x - 100. The slope of this line is -5, which is not equal to the slope of the west edge (5).
Therefore, this line could be the east edge of the basketball court since it does not intersect with the west edge.
Option 2: y + 5x = 100
Rearranging the equation to slope-intercept form, we get y = -5x + 100. The slope of this line is -5, which is not equal to the slope of the west edge (5).
Thus, this line could be the east edge of the basketball court since it does not intersect with the west edge.
Option 3: −5x − y = 50
Rearranging the equation to slope-intercept form, we get y = -5x - 50. The slope of this line is -5, which is not equal to the slope of the west edge (5).
Hence, this line could be the east edge of the basketball court since it does not intersect with the west edge.
Option 4: 5x − y = 50
By rearranging the equation to slope-intercept form, we get y = 5x - 50. The slope of this line is 5, which is equal to the slope of the west edge (5).
Therefore, this line cannot be the east edge of the basketball court as it intersects with the west edge.
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(-12) Divided by 6 +2
Answers
Answer:
0
I hope its zero.........
Answer:
-1.5
Step-by-step explanation:
-12/6+2
-12/8
-1.5
Solve the equation 6(2x+4)^2=(2x+4)+2 a x=-5/3 or -9/4
Answers
Answer:
x=5/3
Step-by-step explanation:
Answer:
-5/3 or -9/4 is the answer
Step-by-step explanation:
Just completed it
Miles family is going on a road trip. They are planning to cover the distance of 1700 miles. If each day Miles are planning to cover 300 miles, write the expression showing how many miles they will have left to cover after d number of days.
Answers
From the given scenario, the expression must show that from a total of 1,700 miles distance to be covered, it will be deducted by 300 miles multiplied by the number of days Miles travels since she is planning to cover 300 miles each day.
The expression will be,
\(\text{ 1,700 - 3}00d\)Where d = number of days.
Option 2: Increase by 10% each year
linear and exponential growth graph
Type 2:
d =
e =
Answers
Answer:
Type 2: exponential growth
d = 121
e = 133
Answer:
Exponential growth
d: 121
e: 133.1
Step-by-step explanation:
edge 23
What is 7.5 divided by .05 PLEASE SHOW WORK ASAP!!
Answers
Answer:
the answer to that is 150
I dont even know if I did the first one right. May you please help me with my little sisters Homework, Idk what her teacher gives her
Answers
Answer:
we cant see it
Step-by-step explanation:
Answer:
Addition Property
Step-by-step explanation:
a house painter charges each customer a $50 estimation fee and then $30 per hour. write and solve an equation to find the number of hours the house painter worked if she earned $1250 at the last job
Answers
Answer:
Step-by-step explanation:
The graphs of f(x) and g(x) are shown below:
Graph of function f of x open upward and has its vertex at negative 7, 0. Graph of function g of x opens upward and has its vertex at negative 7, negative 2.
If f(x) = (x + 7)2, which of the following is g(x), based on the translation?
g(x) = (x + 9)2
g(x) = (x + 5)2
g(x) = (x + 7)2 + 2
g(x) = (x + 7)2 − 2
Answers
Answer: g(x)=(x+7)^2-2
Step-by-step explanation:
g(-7)=f(-7)-2
g(-7)=(-7+7)^2-2
g(x)=(x+7)^2-2
Write an equation in slope-intercept form (y=mx+b) of the line that passes through (2,3) and (0,7)
Answers
How fast is the baseball moving horizontally, rounded to the nearest hundredth? m/s.
Answers
The baseball's speed is 0.82 m/s, rounded to the closest tenth.
Explain the term Gravitational Acceleration?The acceleration that an object experiences due to gravity force that acts on it is known as gravitational acceleration. Its symbol is "g," and its unit is "m/s²."We'll employ a motion equation:
h = ut + 1/2 gt²
The starting height is h.Initial vertical velocity is given by u = 0.g = Gravitational Accelerationt = TimeA baseball is thrown horizontally from the a height of 1.8 metres, as stated.
Therefore, replace h = 1.8 m.
1.8 = (0)t + 1/2*9.8*t²
3.6 = (9.8)t²
t = 0.61
This baseball travels 0.5 metres before it lands, according to the information provided.
Distance is currently equal to 0.5 metres.
Time: 0.61 second
Finding speed
Speed = distance/Time
Speed = 0.5/0.61
Speed = 0.82
Thus, Speed = 0.82 m/s.
Hence, the baseball travels with a speed of 0.82 m/s, rounded to the closest tenth.
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The complete question is-
A baseball is launched horizontally from a height of 1.8 m. The baseball travels 0.5 m before hitting the ground.
How fast is the baseball moving horizontally, rounded to the nearest hundredth? m/s.
The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
Answers
y2 – 5y = 750
750 – y(y – 5) = 0
(y + 25)(y – 30) = 0
I need help pls and no files ok pls no files
Answers
Answer:
$2,725
Step-by-step explanation:
The maximum number of days is 7.
The cost per days is $375.
The cost for 7 days is 7 * $375 = $2,625.
The owner applies a $100 fee for cleaning which must be added to the cost of the days.
$2,625 + $100 = $2,725
The greatest value for the range is the greatest cost there can be which is $2,725.
Which ordered pairs are solutions to the inequality y - 41 > -5?
Answers
Answer:
y - 41 > -5
y > +41 - 5
y > 36
the solutions for inequality is y > 36
Which expression is the simplest form of 4(3x + y) + 2(x - 5y) + x??O A. x2 + 14x - 6yO B. x2 + 13x - 9yc. x2 + 14x - 9yO D. x + 14x-y
Answers
In order to simplify the expression 4(3x + y) + 2(x - 5y) + x² we just have to get rid of the parenthesis by distributing the products and then combine like terms, like this:
1. Get rid of the parenthesis
4(3x + y) + 2(x - 5y) + x² = 4×3x + 4×y + 2×x - 2×5y + x²
4×3x + 4×y + 2×x - 2×5y + x² = 12x + 4y + 2x - 10y + x²
2. Combine like terms
12x + 4y + 2x - 10y + x² = 14x - 6y + x² = x² + 14x - 6y
By simplifying the expression 4(3x + y) + 2(x - 5y) + x², we get x² + 14x - 6y, then the first option is the correct answer.
find an equation of the tangent line to the curve at the given point. y = 2ex cos(x), (0, 2)
Answers
The equation of the tangent line to the curve `y = 2ex cos(x)` at the point (0,2) is given by `y = 2ex + 2`.
To find an equation of the tangent line to the curve at the given point (0,2) whose equation is given by `y = 2ex cos(x)`, we need to determine the derivative `y'` of `y = 2ex cos(x)` first. Using the product rule, we have;
`y = 2ex cos(x)`...let `u = 2ex` and `v = cos(x)`, then `u' = 2ex` and `v' = -sin(x)`.`y' = u'v + uv'` `= 2ex cos(x) - 2ex sin(x)` `= 2ex(cos(x) - sin(x))`
Therefore, the derivative of `y = 2ex cos(x)` is `y' = 2ex(cos(x) - sin(x))`.
The equation of the tangent line to the curve at the point (0,2) is obtained by using the point-slope formula, which is given by: `y - y1 = m(x - x1)`where `(x1,y1)` is the point of tangency, `m` is the slope of the tangent line.
Substituting the values of `m`, `x1` and `y1`, we obtain: `m = y' |(0,2)` `= 2e(1 - 0)` `= 2e`Using the point-slope formula with `(x1,y1) = (0,2)` and `m = 2e`, we have: `y - 2 = 2e(x - 0)` `y - 2 = 2ex` `y = 2ex + 2`
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Roman Derleth
Write Single Inequality from Context
Sep 21, 7:53:35 PM
?
Jonathan and his children went into a restaurant and will buy hamburgers and tacos.
Each hamburger costs $5 and each taco costs $2.50. Jonathan has a total of $60 to
spend on hamburgers and tacos. Write an inequality that would represent the
possible values for the number of hamburgers purchased, h, and the number of tacos
purchased, t.
Answer:
Submit Answer
Answers
The inequalities that represent the problem is 5h + 2.50t ≤ 60
How to find the inequality that represent the problem?Each hamburger costs $5 and each taco costs $2.50.
He has a total of $60 to spend on hamburgers and tacos.
The inequalities that would be used to represent the possible values for the number of hamburgers purchased, h, and the number of tacos purchased, t is as follows:
Therefore,
h = number of hamburgers purchased
t = the number of tacos purchased
Hence,
5h + 2.50t ≤ 60
Therefore, the inequalities that represent the problem is 5h + 2.50t ≤ 60
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