The general solution is [tex]$$\begin{pmatrix}x \\ y\end{pmatrix} = c_1e^{(7+\sqrt{3})t}\begin{pmatrix}5 \\ 1+\sqrt{3}\end{pmatrix} + c_2e^{(7-\sqrt{3})t}\begin{pmatrix}5 \\ 1-\sqrt{3}\end{pmatrix}$$[/tex]
How to find the general solution of the given system?We can write the system of differential equations in matrix form as:
[tex]\frac{d}{dt}\begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}6 & -5 \\ -2 & 8\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}[/tex]
To find the general solution, we first need to find the eigenvalues and eigenvectors of the coefficient matrix:
[tex]$$\begin{pmatrix}6-\lambda & -5 \\ -2 & 8-\lambda\end{pmatrix} = 0$$[/tex]
Solving the determinant, we get:
[tex]$$(6-\lambda)(8-\lambda) - (-2)(-5) = 0$$[/tex]
Simplifying, we get [tex]$\lambda^2 - 14\lambda + 46 = 0$[/tex]. Using the quadratic formula, we get:
[tex]$$\lambda = \frac{14 \pm \sqrt{(-14)^2 - 4(1)(46)}}{2} = 7 \pm \sqrt{3}$$[/tex]
Thus, the eigenvalues are [tex]\lambda_1 = 7 + \sqrt{3}$ and $\lambda_2 = 7 - \sqrt{3}[/tex]
To find the eigenvectors, we solve the system of equations[tex]$(A - \lambda I)\mathbf{v} = \mathbf{0}$[/tex] for each eigenvalue. For[tex]$\lambda_1 = 7 + \sqrt{3}$[/tex], we have:
[tex]$$\begin{pmatrix}-1-\sqrt{3} & -5 \\ -2 & 1-\sqrt{3}\end{pmatrix}\begin{pmatrix}v_1 \\ v_2\end{pmatrix} = \begin{pmatrix}0 \\ 0\end{pmatrix}$$[/tex]
Solving this system, we get the eigenvector [tex]$\mathbf{v}_1 = \begin{pmatrix}5 \\ 1+\sqrt{3}\end{pmatrix}$[/tex].
Similarly, for [tex]$\lambda_2 = 7 - \sqrt{3}$[/tex], we have:
[tex]$$\begin{pmatrix}-1+\sqrt{3} & -5 \\ -2 & 1+\sqrt{3}\end{pmatrix}\begin{pmatrix}v_1 \\ v_2\end{pmatrix} = \begin{pmatrix}0 \\ 0\end{pmatrix}$$[/tex]
Solving this system, we get the eigenvector[tex]$\mathbf{v}_2 = \begin{pmatrix}5 \\ 1-\sqrt{3}\end{pmatrix}$.[/tex]
Therefore, the general solution is:
[tex]$$\begin{pmatrix}x \\ y\end{pmatrix} = c_1e^{(7+\sqrt{3})t}\begin{pmatrix}5 \\ 1+\sqrt{3}\end{pmatrix} + c_2e^{(7-\sqrt{3})t}\begin{pmatrix}5 \\ 1-\sqrt{3}\end{pmatrix}$$[/tex]
where [tex]$c_1$[/tex] and [tex]$c_2$[/tex] are constants determined by the initial conditions.
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True or False: an equilateral triangle is always an acute triangle, but an acute triangle is never an equilateral triangle.
True
False
Answer: False.
Step-by-step explanation: An equilateral triangle is always an acute triangle because all of its angles are congruent (equal) to 60 degrees. An acute triangle is a triangle with all its angles are less than 90 degrees, so an equilateral triangle with all angles of 60 degrees is an acute triangle. An acute triangle doesn't necessarily have to be an equilateral triangle.
answer : the answer is false
pls helppppppppppppppppppp
Answer: look it up
Step-by-step explanation:
Find the distance between the points (2,10) and (10,10).
Answer: The distance is 8.
Help me please (geometry)
Answer:
x = 3
y = 4
Step-by-step explanation:
The opposite sides of a parallelogram are always equal.
From the above statement, we get two equations-
[tex]4x+6 = 7x-3[/tex] and [tex]4y-3=3y+1[/tex]
Solving for x:
[tex]4x+6=7x-3\\6=3x-3\\9=3x\\3=x[/tex]
Solving for y:
[tex]4y-3=3y+1\\y-3=1\\y=4[/tex]
The total measure of the angles is (n−2)·180°, where n=8. The term regular means all interior angle measures are the same. How can you determine each angle measure from the total angle measure?
Each measure of the angle when the total measure of the angles is (n−2)·180° and n is 8 will be 235°.
How to calculate the angle?From the information, the total measure of the angles is (n−2)·180°, where n=8. The term regular means all interior angle measures are the same.
It should be noted that in Mathematics, the value of the sum of angles is Illustrated in this case. We can determine each angle measure from the total angle measure thus:
(n - 2) × 180
n = 8
It should be noted that n in this case simply means the number of sides
= (n - 2) × 180
= (8 - 2) × 180
= 1080°
Each measure of the angle will be:
= 1080° / 8
= 135°
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Multiplying Polynomials
3a³(8a + b)
On solving the provided question, we can say that multiplication of polynomial 3(8 + ) is 24+3
What is Polynomial?A polynomial is an algebraic expression in mathematics that solely uses the operations of addition, subtraction, multiplication, and powers of positive-integer variables. It consists of variables and coefficients.
For example, in the expression , x is the variable and coefficients of and are 1 and 2 respectively.
When we are asked to do multiplication of polynomials, we multiply term by term in the brackets until every term is multiplies with another in different bracket.
Like in 3(8 + ), we first multiply 3 with the first term inside the bracket, which is 8 and then we multiply 3 with the second term, which is b and maintaining the plus sign as it is.
So the answer is
(3)(8)+(3)()
=24+3
Hence, the answer is 24+3.
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Luca i making 12 jumbo muffin to ell at a bake ale. He ha 2 bowl full of batter what fraction of a bowl of batter hould Luca put in each muffin cup
Since 1 bowl can fill 6 muffins, it's fraction is 1/6 would Luca put in each muffin cup
Luca i making 12 jumbo muffin to ell at a bake ale.
He ha 2 bowl full of batter
Number of sections batter was divided = 12
We need to find the fraction of a cup of muffin content in each muffin.
Now according to given data Batter was divided in 12 sections of the pan.
So we can say the number of muffins baked = 12
Now we know that;
bowl content in each muffin can be calculated by dividing bowl content in the batter with the number of sections batter was divided.
Find out how much muffins is needed for 1 bowl.
2 bowls = 12 muffins
1 bowl = 12 ÷ 2 = 6 muffins
Since 1 bowl can fill 6 muffins, it's 1/6 r would Luca put in each muffin cup
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9th and 10th Graders 11th and 12th Graders
Turkey 112 150
Ham 64 58
Vegetarian 24 32
A segmented bar chart was created showing the lunch preference by grades. What percent would the bar for 11th and 12th graders show for preferring a vegetarian sandwich?
Answer:
12.73%
Step-by-step explanation:
24 + 32 = 56(veggie sandwich)
112 + 150 = 262
262 + 64 = 326
326 + 58 = 384
384 + 56(veg sand) = 440
440 = total number of students
Divide 100 by 440 and you get 0.23
Multiply that by 56 and you get 12.73%
:)
What is 1/3 with an exponent of 2?
Which best describes the triangle or triangles if anything that can be formed with sides measuring 60 cm 100 cm times 40 cm? 
The statement which best describes the triangle or triangles, if any, that can be formed with the sides measuring 60 centimeters, 100 centimeters, and 40 centimeters include the following: A. No triangle.
What is the Triangle Inequality Theorem?In Mathematics, the Triangle Inequality Theorem is represented by this mathematical expression:
b - c < a < b + c
Where:
a, b, and c represent the side lengths of this triangle.
By applying the Triangle Inequality Theorem to the side lengths of this triangle, the number of triangles that can be formed would be determined as follows:
60 + 40 > 100 (False and no triangle).
In conclusion, we can logically deduce that the two (2) short side lengths does not exceed the long side lengths and as such no triangle can be formed with the sides measuring 60 centimeters, 100 centimeters, and 40 centimeter.
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Complete Question:
Which best describes the triangle or triangles, if any, that can be formed with the sides measuring 60 centimeters, 100 centimeters and 40 centimeter?
A. No triangle
B. One unique scalene triangle
C. infinitely small similar scalene triangles
D. Two unique scalene triangles.
What is the distance between (7, 4) and (3, 4)
The distance between points (7, 4) and (3, 4) is 4 units.
What is the distance formula between two points?The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by,
[tex]D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2[/tex].
Given, Are two points (7, 4) and (3, 4).
Therefore, By the distance formula the distance between the two points
(7, 4) and (3, 4) would be,
[tex]D = \sqrt{(3 - 7)^2 + (4 - 4)^2[/tex].
[tex]D = \sqrt{(-4)^2 + (0)^2[/tex].
[tex]D = \sqrt{(16[/tex].
D = 4. (negative 4 is inadmissible).
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Solve for x and find DFG
Since FE bisects <DFG, then the value of x and measure of <DFG are:
a. x = 24
b. m<DFG = 106^o
What is an angle bisector?An angle bisector is a straight line constructed in such a way that it divides a given angle into two equal measures.
Thus given that FE bisects <DFG, this implies that;
i. m<DFE ≅ m<EFG
So that,
3x - 19 = 2x + 5
collect like terms to have;
3x - 2x = 19 + 5
x = 24
To solve for x, the value of x is 24^o.
ii. To determine the measure of DFG;
m<DFG = (3x - 19) + (2x + 5)
= (3(24) - 19) + (2(24) + 5)
= 53 + 53
= 106
m<DFG = 106^o
The measure of <DFG is 106^o.
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what is 9 + 5 equals to?
Answer: 34
Step-by-step explanation:
Answer: 14
Step-by-step explanation:
Write in a verbal expression for each algebraic expression
c+2d
Therefore , the solution of the given problem of expression comes out to be the first number plus twice the second number, or the summation of the numbers c and d.
How does one choose an expression?An algebraic expression must be checked by replacing each variable with a number and performing arithmetic operations. In the example above, the answer variable is identical to 6, since 7 + 8 = 15. If we know the values of the variables, we can replace them with those values and evaluate the expression.
Here,
Assuming that:
The formula is as follows:
= c + 2d
Here, the real numbers are c and d.
c should be the first number.
The following number is d.
A digit is a mental construct that can be employed in the counting, measuring, and naming of objects. The numbers include 1, 2, 56, etc. as an example.
the two-fold sum between the first and second numbers.
In term of c and d, or
the product of c and d divided by two
The first number plus twice the second number, or the summation of the numbers c and d, is what the mathematical equation c + 2d means in verbal form.
Therefore , the solution of the given problem of expression comes out to be the first number plus twice the second number, or the summation of the numbers c and d.
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What is 20 in exponential form?
20 cannot be represented in exponential form as it's a whole number and it's not in the form of a base number raised to a power.
Exponential notation is typically used to express a number as a product of the base number raised to a power. For example, the number 8 can be written as 2³ (2 raised to the power of 3) or 8 can be written as [tex]10^{0.8[/tex].
In mathematics, exponential notation is used to represent large or small numbers in a more manageable form. The number 20 is not considered large or small as it's a whole number, it is not in the exponential form
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edward deposited 9000 into a savings account 5 years ago the simple interest rate is 2 how much did edward earn interest
Answer:
900
Step-by-step explanation:
[tex]I=Prt=(9000)(0.02)(5)=900[/tex]
PLEASE HELP ME it’s sooo many questions
Answer:
D
The parallelogram proves that D is correct
7. Mrs. Tenon buys 25 party
favors for $5. Find the constant of
proportionality. Write an equation.
At his rate how would she pay for
40 party favors?
The amount of money you must pay at the gas station could be represented using the variables y = price in dollars, x = gas in gallons, and a proportionality constant, k: y = kx.
Mrs. Tenon buys 25 party favors for $5. Find the constant ofproportionality?Mrs. Tenon has spent $5 on 25 party favours.
Equation form of proportional relationships is written as y=kx.
If you substitute $5 for y and $25 for x in the formula for the proportional relationship above, you can find the value of the proportionality constant, or k.
$5=k(25)\sk=$5/25\s=0.2
0.2 is the proportionality constant. Thus, each party favour costs $0.20.
Let's say there are r party favours and the cost is c. c=0.2r
In the equation above, 40 should be substituted for r to get the price of 40 party favours: c=0.2 (60) = 12.00.
As a result, the cost for 40 guests is $ 8.00, and the cost for 60 party favours is $ 12.00.
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Solve for x when lines are parallel
Answer:
4x-10=3x+12
4x+3x= 12+10
7x= 32
x= 32÷7
x=4.571
What does n² mean in maths?
[tex]\displaystyle n\hat A[/tex] is the number of squares of arrays present.
An arrangement of objects, images, or numbers in rows and columns is called an array. Arrays are a convenient representation of the concept of multiplication. This array has 4 rows and 3 columns. This can also be described as a 4 x 3 array. Arrays are a way of representing multiplication and division using rows and columns. Rows represent the number of groups. The columns represent the number of each group or the size of each group. The number 24 can be represented as a 6-by-4 array (6-by-4) because 6-by-4 = 24. It can also be represented as a 2 by 12 array, since 2 by 12 = 24.
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PLS ANSWER ASAP ILL GIVE BRAINLIEST AND A LOT OF POINTS PLEASEEE THANK YOU
Answer:
25√x^21
Step-by-step explanation:
x 21/25
What is the range of y = log subscript 8 baseline x?all real numbers less than 0all real numbers greater than 0all real numbers not equal to 0all real numbers.
The range of y = log subscript 8 baseline x is all real numbers.
How to find the range of a function?The range is a set of all the defined values of y-correspond to the domain. In this case, to find the range of the function y = log subscript 8 baseline x, we need to draw the graph of the function.
The graph is attached.
Now, because of the base of the logarithm is 8 > 1; so, the graph is raising and passes through the point (1,0).
So, the domain of this function is:
x > 0
And, the range of this function is:
y ∈ (–∞,∞)
or all real numbers.
Hence, the range of y = log subscript 8 baseline x is all real numbers.
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Answer:
All real answers
Step-by-step explanation:
edge
a right triangle that has a base of 9 and a height of 16
Que knows that the required hypotenuse of the given right triangle is 18.35 units from the Pythagorean theorem.
What is the Pythagorean theorem?To determine the undiscovered side of a right-angled triangle, utilize the Pythagoras theorem.
The hypotenuse (third side) of a right-angled triangle, for instance, can be determined using the formula c2 = a2 + b2, where 'c' stands for the hypotenuse and 'a' and 'b' are the two legs.
So, we have:
Height of 6 and bas of 9 units of the right triangle.
Pythagorean formula:
c=√a²+b²
Now, insert values and calculate as follows:
c = √a²+b²
c = √9²+16²
c = 18.35 units
Therefore, we know that the required hypotenuse of the given right triangle is 18.35 units from the Pythagorean theorem.
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Complete question:
A right triangle has a base of 9 and a height of 16. Calculate hypotenuse.
Graph the line with slope -3 passing through the point (4, - 1)
A graph of the line with slope -3 passing through the point (4, -1) has been plotted in the image attached below.
What is the point-slope form?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y are the points.From the information provided above, we have the following slope and data points on the line:
Points = (4, -1).Slope, m = -3At point (4, -1), a linear equation of this line can be calculated in point-slope form as follows:
y - y₁ = m(x - x₁)
y - (-1) = -3(x - 4)
y + 1 = -3x + 12
y = -3x + 12 - 1
y = -3x + 11
Next, we would use an online graphing calculator to plot the above linear equation in slope-intercept form as shown in the graph attached below.
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Quadrilateral ABCD is transformed into quadrilateral A'B'C'D'.
Name two transformations which were involved in the sequence. Explain how you know those transformations were used.
Your explanation does not have to include specific measurements (rotate 90 degrees), it can be more general but you must support the
reason you chose each transformation.
A
D
2
6
2
2
-4
B
B A
C
C
2 4
D
6
B was moved down by -2 and A was moved down by-2 and right by 6
So on solving the provided question we can say that here all angles of of a quadrilateral = 30, so, each will be 90
what is quadrilateral?
A quadrilateral is a four-sided polygon in geometry that has four edges and four corners. The word is a derivative of the Latin words quadri and latus (meaning "side"). Having four sides, four vertices, and four corners, a rectangle is a two-dimensional form. Concave and convex come in primarily two varieties. Additionally, there are several subclasses of convex quadrilaterals, including trapezoids, parallelograms, rectangles, rhombuses, and squares. Four straight sides make up a rectangle, which is a two-dimensional form. There are several different types of quadrilaterals, including parallelograms, trapezoids, rectangles, kites, squares, and rhombuses.
here all angles of of a quadrilateral = 30
so, each will be 90
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#6: Emma takes care of dogs while their owners are on vacation. She charges $35 for each of the first 3 days she takes care of a dog and $25 for each day thereafter. Which equation can be used to find the total cost (y), in dollars, to have Emma take care of a dog for x days, where x is a whole number greater than 3? A.y = 75 + 35(x - 3) B.y = 105 + 25(x - 3) C.y = 25 + 35x D.y = 35 + 25x +
The requried equation used to find the total cost of care of dogs is given as, 1. y = 35x for 0 ≤ x ≤3 2. y = 35 + 25x for 3 < x. None of the op[tions are correct.
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
Here,
Emma takes care of dogs while their owners are on vacation. She charges $35 for each of the first 3 days she takes care of a dog and $25 for each day thereafter.
Two equations will be formed,
1. y = 35x for 0 ≤ x ≤3
2. y = 35 + 25x for 3 < x
Thus, the requried equation used to find the total cost of care of dogs is given as, 1. y = 35x for 0 ≤ x ≤3 2. y = 35 + 25x for 3 < x. None of the op[tions are correct.
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The length of a rectangular garden is 3 m more than its breadth. Find the area of garden, if the perimeter is 54 m.
180
since it's a rectangle it wil have 4 sides hence to find breadth we need to make it a square so all 4 sides will be equal hence 54 minus 3 times 2 and the length of the rectangle is 3 cm longer than the breadth of the rectangle making the perimeter into 48 after the equation 48 divide by 4 equal 12 . so now we know that the breadth of the rectangle is 12 we can add 3 to 12 to find the length of the rectangle and use 12 times 15 to find the area of the rectangle
What is not an ordered pair?
The pairs which cannot be written or expressed in the notation of (a,b) and do not follow a specific order are not ordered pairs .
What are Ordered Pairs ?
The Ordered Pair is defined as a pair of elements that have specific importance for order of their placements. Ordered pairs are used in coordinate geometry to represent a point on a coordinate plane.
So , the numbers that do not follow the specific order of placements cannot be called as Ordered Pairs .
Two Ordered Pairs [tex](a_{1} ,b_{1} )[/tex] and [tex](a_{2} ,b_{2} )[/tex] can be considered as equal if and only if [tex]a_{1} =a_{2}[/tex] and [tex]b_{1} =b_{2}[/tex] .
Every point on Coordinate Plane is represented by ordered pair (x, y) where "first element x" is called x coordinate and "second element y" is called y-coordinate.
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If it takes 1 4 cup of flour to make 1 6 of a cake, how many cups of flour are needed to make a whole cake?
Answer:
1 1/2 cups of flour.
Step-by-step explanation:
You can work this as a proportion:
1/4 cup → 1/6 of a cake
x cups → 1 cake
Cross-multiply.
(1/4 cup)(1 cake) = (x cups)(1/6 cake)
Solve for x.
x = 6(1/4) = 1 1/2 cups
1+3(4f-7)=28 disrtibutive property
Answer:
F=4
Step-by-step explanation:
1+3(4f-7)=28
3×4f= 12f
3×-7= -21
ADD THE LIKE TERMS
1+-21= -20
now you have 12f-20=28
at 20 to both sides
12f=48
divide 12 from both sides
48/12=4
F=4