.Find the inverse of A if it has one, or state that the inverse does not exist.
Answers
Step-by-step explanation:
Find the discriminant
\( |a| = ad - bc\)
for a matrix
(a b)
(c d)
Here it is
\(( - 6)( - 2) - 0(5)\)
\(12 - 0 = 12\)
The formula for the inverse of a 2 by 2 matrix,
\( \frac{1}{ad - bc} \binom{d}{ - c} \binom{ - b}{a} \)
Note since the discriminant isn't 0, an inverse must exist.
\( \frac{1}{12} \binom{ - 2}{ - 5} \binom{ 0}{ - 6} \)
Multiply everything by the fraction.
\( \binom{ - \frac{1}{6} }{ \frac{ - 5}{12} } \binom{ 0 }{ - \frac{1}{2} } \)
The last matrix is the inverse of A
Related Questions
Is x 8 a factor of the function f x 2x3 17x2 64?
Answers
A factor of a polynomial function is a polynomial that divides the function evenly when using polynomial division.
To check if x - 8 is a factor of the function f(x) =\(2x^3 - 17x^2 + 64\) , we can use polynomial division.
When dividing f(x) by x - 8, the remainder should be zero.
x - 8 is a factor of the function when the remainder is zero.
In this case, the quotient is \(2x^2 + x + 8\) , and the remainder is zero, so x - 8 is a factor of the function f(x) = \(2x^3 - 17x^2 + 64.\).
So we can say \(f(x) = (x-8)(2x^2 + x + 8)\)
Alternatively, we can also use synthetic division to check if x - 8 is a factor of the function.
Synthetic division is a shorthand method of dividing a polynomial by a linear factor such as x - 8.
If the remainder is zero, then x - 8 is a factor of the function.
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Can somebody help me?
Answers
What is the price at which equilibrium is achieved?
$8
$9
$10
$30
Answers
Dasher was super hungry after his big night so he ordered 4 candy cane pies. He ate 3/5 of the pies on the way home. How much pie did he have left when he got home?
Answers
Answer:
He had 3 2/5 pies left when he got home.
Step-by-step explanation:
Hope this helped have an amazing day!
The fraction of candy cane pies that Dasher left is 8/5.
What is a fraction?In such a fraction, the value that appears above the horizontal line is referred to as the numerator.
In another word, if we divide two numbers by each other then the upper word is called the numerator, and the lower word is called the denominator.
As per the given,
Number of candy cane pies that Dasher order = 4
Fraction he ate = 3/5
Left = 4 - ate
Left = 4 - (3/5)4
Left = 4(1 - 3/5)
Left = 4(2/5)
Left = 8/5
Hence "The fraction of candy cane pies that Dasher left is 8/5".
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A manufacturer has a steady annual demand for 15,000 cases of sugar. It costs $10 to store 1 case for 1 year, $30 in set up cost to produce each batch, and $16 to produce each case. Find the number of cases per batch that should be produced to minimize cost.
Answers
The number of cases per batch that should be produced to minimize cost is: 300 units
How to find the economic order quantity?The number of cases per batch that should be produced to minimize cost can be found by using the Economic Order Quantity.
The Economic Order Quantity (EOQ) is a calculation performed by a business that represents the ideal order size that allows the business to meet demand without overspending. The inventory manager calculates her EOQ to minimize storage costs and excess inventory.
Thus:
Number of cases per batch = √((2 * Setup costs * annual demand)/ holding costs for the year)
Solving gives:
√((2 * 30 * 15000)/10)
= √90000
= 300 units
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Which inequality describes all the solutions to (x-2) > 3(x+4)
A. X<-7
B x > -5
C. X>7
D. X<5
Answers
Answer:
x
\( x < - 7\)
Step-by-step explanation:
\(x - 2 > 3 |x + 4| \\ x - 2 > 3x + 12 \\ collect \: like \: term \\ x - 3x > 12 + 2 \\ - 2x > 14 \\ since \: it \: is \: negative \: we \: change \: the \: inequality \: sign \\ - 2x < 14 \\ divide \: both \: sides \: by \: - 2 \\ x < - 7\)
Find the Inverse Laplace transform of the following function: Show Work Please
3)
\[
F(s)=\frac{7}{s^{2}+2}
\]
Answers
The inverse Laplace transform of F(s) is,
⇒ f(t) = 7 / {√2} [ sin(√2t) ].
Now, For the inverse Laplace transform of F(s), we can use the formula:
f(t) = L⁻¹ {F(s)} = 1/2πi ∫_{γ-i∞}^{γ+i∞} \(e^{st}\) F(s) ds
where γ is a real number greater than the real part of all singularities of F(s).
In this case, the denominator of F(s) is s² + 2, which has singularities at s = ±i√2.
Since these are purely imaginary and have negative imaginary parts, we can choose γ = 0.
So, using the formula, we have:
f(t) = 1/2πi ∫ (from {-i∞} to {i∞}) \(e^{st}\) 7 / (s² + 2} ds
To solve this integral, we need to use a complex variable technique called the residue theorem. We can find the residues of the integrand at its poles, which are s = ±i√2.
The residues at each pole are given by:
Res[s = ±i√2] = lim_{s→±i√2} (s - ±i√2) 7 / (s² + 2} = 7 / {2i√2}
Using the residue theorem, we can evaluate the integral as:
f(t) = 2πi [ Res[s = i√2] + Res[s = -i√2] ]
= 2πi [ 7 / (2i√2)\(e^{i\sqrt{2} t}\) + 7 / {-2i√2} \(e^{-i\sqrt{2} t}\) ] = 7/√2} [ sin(√2t) ]
Therefore, the inverse Laplace transform of F(s) is,
⇒ f(t) = 7 / {√2} [ sin(√2t) ].
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Which equation could best be used to determine the value
of f(3) for the function f(x) = 2x + 4?
1) f(3) = 23+ 4
2) f(3) = 2(3)+4
3) f(3) = 3(2x) + 4
4) f(3) = 3(2x + 4)
Answers
Answer:
2) f(3) = 2(3)+4
Step-by-step explanation:
If using the method of completing the square to solve the quadratic equation
x2 - 6x + 6 = 0, which number would have to be added to "complete the square"?
Answers
This is what the steps look like when we complete the square
x^2 - 6x + 6 = 0
x^2 - 6x = -6
x^2 - 6x + 9 = -6+9
(x-3)^2 = 3
We add 9 to both sides so that we can factor x^2-6x+9 into (x-3)(x-3) = (x-3)^2
To get the 9, we cut the -6 (from -6x) in half to get -6/2 = -3. Then we square that to get (-3)^2 = 9
It might help to look at (a+b)^2 = a^2+2ab+b^2 and note the relationship between 2ab and b^2.
A pipe extends from ground level to a location under the ground. Two-fifths of the way down the pipe, there is a leak. The leak's elevation is −6.2 meters.
What is the elevation of the end of the pipe that is under the ground?
Enter your answer as a decimal or as a mixed number in simplest form in the box.
Answers
Answer: -15.5
Step-by-step explanation:
I took the test
\(\to -6.2 \ \text{meters is}\ \frac{2}{5} = 0.4\)
All way down the pipeline Down the line, \(100\% = 1\). The rule of two is\(-6.2\ meters \to 0.4\), and \(x \ meters\to 1\) Applying Cross multiplication:\(\to 0.4 x = -6.2 \\\\\to x = \frac{-6.2}{0.4}\\\\\to x = -15.5\)
As a result, the elevation of the underground pipe end is -15.5 meters.
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for what values of x does 5x^2+4x-4=0
Answers
Answer: See explanation
Step-by-step explanation:
x=-(2-2*the square root of 6)/5, about 0.58
or
x=-(2+2*the square root of 6)/5, about -1.38
The values of the x from equation \(5x^2+4x-4=0\) are x = 0.5798 and -1.38.
Given that:
Equation: \(5x^2+4x-4=0\)
To find the values of x that satisfy the equation \(5x^2+4x-4=0\), use the quadratic formula:
\(x = \dfrac{ -b \± \sqrt{b^2 - 4ac}}{ 2a}\)
Compare the equation with \(ax^2 + bx + c = 0\).
Here, a = 5, b = 4, and c = -4.
Plugging in the values to get,
\(x = \dfrac{-4 \± \sqrt{4^2 - 4 \times 5 \times (-4)}}{2 \times 5} \\x = \dfrac{-4 \± \sqrt{16 +80}}{10} \\x = \dfrac{-4 \± \sqrt{96}}{10}\\x = \dfrac{-4 \± {4\sqrt6}}{10}\)
So the solutions for x are calculates as:
Taking positive sign,
\(x = \dfrac{-4 + {4\sqrt6}}{10}\\x = \dfrac{-4 + {9.798}}{10}\\\)
x = 5.798/10
x = 0.5798
Taking negative sign,
\(x = \dfrac{-4 - {4\sqrt6}}{10}\\x = \dfrac{-4 - {9.798}}{10}\\\)
x = -13.798/10
x = -1.38
Hence, the exact solutions for the equation \(5x^2 + 4x - 4 = 0\) are x = 0.5798 and -1.38.
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A sector with arc measure 72º has an
area of 80 square feet. Determine
the radius of the circle.
A 18 ft
C 12 ft
B 20 ft
D 16ft
Answers
\(\qquad\qquad\huge\underline{{\sf Answer}}♨\)
Let's find the Radius ~
\(\qquad \sf \dashrightarrow \: area = \frac{ \theta}{360} \times \pi {r}^{2} \)
\(\qquad \sf \dashrightarrow \: 80= \frac{ 72}{360} \times \pi {r}^{2} \)
\(\qquad \sf \dashrightarrow \: {r}^{2} = \frac{80 \times 360}{72 \times \pi} \)
\(\qquad \sf \dashrightarrow \: {r}^{2} = \frac{80 \times 5}{ \pi} \)
\(\qquad \sf \dashrightarrow \: {r}^{} = \sqrt\frac{400}{ \pi} \)
\(\qquad \sf \dashrightarrow \:r = \frac{20}{ \sqrt{\pi} } \)
or
\(\qquad \sf \dashrightarrow \:r \approx 11.28 \: ft\)
I hope you understood the whole procedure, let me know if you have doubts ~
➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖
what monomial expression best estimates the behavior of x − 4 as x → ± [infinity] ?
Answers
The monomial expression that best estimates the behavior of x − 4 as x → ± [infinity] is simply x.
An algebraic expression known as a monomial typically has one term, but it can also have several variables and a higher degree.
When 9 is the coefficient, x, y, and z are the variables, and 3 is the degree of the monomial, for instance, 9x3yz is a single term.
This is because as x approaches infinity or negative infinity, the constant term (-4) becomes negligible in comparison to the magnitude of x.
Therefore, the behavior of x − 4 can be approximated by the monomial expression x in the long run.
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Is (3,8) a solution to this system of equations? How do you know?
y=7x + 8
y=x + 1
Answers
Answer:
yes it is
Step-by-step explanation:
The point (3,8) is not a solution of system of equation. Explanation given below.
What is the system of equation?Simultaneous equations are any equations with the same number of unknowns that may be solved at the same time. The system of equations is a simultaneous equation.
Given system of equation,
y=7x + 8
y=x + 1
To find that the point (3,8) is a solution of system of equation,
we substitute the point to both of these equation and verify if that point is satisfying the equation or not.
In the first equation, x=3 and y =8
we get, 8 = 29
which is contradiction.
In the second equation ;
we get, 8 = 4.
Which is also contradiction.
Therefore, the equations do not have the solution (3 , 8).
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what is the answer to 20x + 35y
Answers
The greatest common factor is 5 .
If you factor it out, the expression becomes 5 (4x + 7y) .
Answer: 55x^1 y^1
Step-by-step explanation:
Add 20+35= 55
x & y can't stand by it's self, so x^1 and y^1
So you take those answers and you got 55x^1 y^1
You must keep it in letters in alphabetic order or it's considered wrong.
is there a different way we can do it?
Answers
A new car is available with standard or automatic transmission, two or four doors, and it is available in 10 exterior colors. What is the number of possible outcomes? a.14 b.40 c.44 d.104
Answers
Answer:
40
Step-by-step explanation:
Given that:
Number of options available for transmission = 2 (Standard or Automatic)
Number of options for doors = 2 (2 doors or 4 doors)
Number of exterior colors available = 10
To find:
Total number of outcomes = ?
Solution:
First of all, let us calculate the number of outcomes for the transmission mode and number of doors options.
1. Standard - 2 doors
2. Standard - 4 doors
3. Automatic - 2 doors
4. Automatic - 4 doors
Number of outcomes possible = 4 (which is equal to number of transmission mode available multiplied by number of doors options i.e. 2\(\times 2\))
Now, these 4 will be mapped with 10 different exterior colors.
Therefore total number of outcomes possible :
Number of transmission modes \(\times\) Number of doors options \(\times\) Number of exterior colors
2 \(\times\) 2 \(\times\) 10 = 40
The number of possible outcomes of a new car will be 40. Then the correct option is B.
What is a sample?A sample is a collection of well-defined elements. A sample is represented by a capital letter symbol and the number of elements in the finite sample is shown as a curly bracket {..}.
A new car is available with standard or automatic transmission, two or four doors, and it is available in 10 exterior colors.
Then the number of possible outcomes will be
\(\rm Possible \ outcomes = 2*2*10\\\\Possible \ outcomes = 40\)
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PLEASE HELPP
What is the solution to the following system of equations?
x − 3y = 6
2x + 2y = 4
(-1, 3)
(3, -1)
(1, -3)
(-3, 1)
Answers
Answer:
The solution to the system of equations be:
\(x=3,\:y=-1\)
Hece, option B is correct.
Step-by-step explanation:
Given the system of equations
\(\begin{bmatrix}x-3y=6\\ 2x+2y=4\end{bmatrix}\)
Multiply x − 3y = 6 by 2: 2x-6y=12
\(\begin{bmatrix}2x-6y=12\\ 2x+2y=4\end{bmatrix}\)
so
\(2x+2y=4\)
\(-\)
\(\underline{2x-6y=12}\)
\(8y=-8\)
so the system of equations becomes
\(\begin{bmatrix}2x-6y=12\\ 8y=-8\end{bmatrix}\)
Solve 8y = -8 for y
\(8y=-8\)
Divide both sides by 8
\(\frac{8y}{8}=\frac{-8}{8}\)
\(y=-1\)
\(\mathrm{For\:}2x-6y=12\mathrm{\:plug\:in\:}y=-1\)
\(2x-6\left(-1\right)=12\)
\(2x+6=12\)
subtract 6 from both sides
\(2x+6-6=12-6\)
\(2x=6\)
Divide both sides by 2
\(\frac{2x}{2}=\frac{6}{2}\)
\(x=3\)
Therefore, the solution to the system of equations be:
\(x=3,\:y=-1\)
Hence, option B is correct.
⚽ From Equation (i) we get :
x = 6 + 3y......[Equation (iii)]⚽ Now, Substitute the equation (iii) in equation (ii) we get :
→ 2(6 + 3y) + 2y = 4
→ 12 + 6y + 2y = 4
→ 8y = 4 - 12
→ 8y = -8
→ y = -8 ÷ 8
→ y = -1
⚽ Now Substituting value of y = -1 in equation (iii) we get :
→ x = 6 + 3y
→ x = 6 + 3(-1)
→ x = 6 + (-3)
→ x = 6 - 3
→ x = 3
Hence,the value of x = 3 and value of y = -1.Which statement about geometric figures is true?
O A ray has exactly two endpoints.
O On a coordinate plane, parallel lines have the same slope.
O Two lines on a coordinate plane that do not intersect are perpendicular.
O An angle is formed by a set of points that are all equidistant from a common point.
Answers
Answer:
B
Step-by-step explanation:
On a coordinate plane, parallel lines have the same slope as geometric figures are true. Thus option B is correct.
What are geometric figures?Marks, splines, spheres, and curved are used to build enclosed geometric shapes. We can see these shapes all around us. Illustrations of geometric shapes include circles, rectangles, triangles, and others. Any configuration of points, columns, or angles is a geometry figure.
A point is the most fundamental algebraic concept because it has no boundaries. Simply put, a point upon that plane is a location. A dot is used to symbolize it. A plane can be identified by three non-straight-line points.
As two or much more vectors seldom cross each other and are located on the same dimension, these are said to be parallel. The slope of these lines is constant. Coplanar lines that seem to be parallel are the ones that don't cross. Therefore, option B is the correct option.
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Hey can someone pls help me will give brainliest I will be very grateful thx!!!!
Answers
Answer:
17) 4 inches
18) 14 inches
19) 162 inches
20) 30 inches
Step-by-step explanation:
For the first two, you use the formula π× (radius)². Since you have the area, divide it by two then find the square root. For the second two, you just divide the by pi.
Answer:
17) D=4in
18)D=14yd
19)D=162yd
20)D=30yd
Step-by-step explanation:
17)
\(d = 2 \times \sqrt{4} = 4\)
What is. 3/4x - 2/3x - 4 = 1/6x - 7
I give Brainliest!
Answers
Answer:
x = 36
Step-by-step explanation:
(3/4)*x-((2/3)*x)-4 = (1/6)*x-7 // - (1/6)*x-7
(3/4)*x-((2/3)*x)-((1/6)*x)-4+7 = 0
(3/4)*x+(-2/3)*x+(-1/6)*x-4+7 = 0
3-1/12*x = 0 // - 3
-1/12*x = -3 // : -1/12
x = -3/(-1/12)
x = 36
You recently opened your checking account with reallybigbank on August 1 using $100.00
Answers
What's the question:
whats the question
What's the question:
whats the question
What proportion of these candy bars have fewer than 200 calories? (round your answer to two decimal places.)
Answers
The proportion of candy bars with fewer than 200 calories cannot be determined without additional information.
To calculate the proportion of candy bars with fewer than 200 calories, we need to know the total number of candy bars and the number of candy bars that have fewer than 200 calories.
Without this information, we cannot determine the proportion. The number of candy bars with fewer than 200 calories could vary greatly depending on the specific dataset or context.
The proportion of candy bars with fewer than 200 calories cannot be determined without additional information regarding the total number of candy bars and the number of candy bars that fall into that category.
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Suppose an economy has the following equations:
C =100 + 0.8Yd;
TA = 25 + 0.25Y;
TR = 50;
I = 400 – 10i;
G = 200;
L = Y – 100i;
M/P = 500
Calculate the equilibrium level of income, interest rate, consumption, investments and budget surplus.
Suppose G increases by 100. Find the new values for the investments and budget surplus. Find the crowding out effect that results from the increase in G
Assume that the increase of G by 100 is accompanied by an increase of M/P by 100. What is the equilibrium level of Y and r? What is the crowding out effect in this case? Why?
Expert Answer
Answers
The equilibrium level of income (Y), interest rate (i), consumption (C), investments (I), and budget surplus can be calculated using the given equations and information. When G increases by 100, the new values for investments and budget surplus can be determined. The crowding out effect resulting from the increase in G can also be evaluated. Additionally, if the increase in G is accompanied by an increase in M/P by 100, the equilibrium level of Y and r, as well as the crowding out effect, can be determined and explained.
How can we calculate the equilibrium level of income, interest rate, consumption, investments, and budget surplus in an economy, and analyze the crowding out effect?To calculate the equilibrium level of income (Y), we set the total income (Y) equal to total expenditures (C + I + G), solve the equation, and find the value of Y that satisfies it. Similarly, the equilibrium interest rate (i) can be determined by equating the demand for money (L) with the money supply (M/P). Consumption (C), investments (I), and budget surplus can be calculated using the respective equations provided.
When G increases by 100, we can recalculate the new values for investments and budget surplus by substituting the updated value of G into the equation. The crowding out effect can be assessed by comparing the initial and new values of investments.
If the increase in G is accompanied by an increase in M/P by 100, the equilibrium level of Y and r can be calculated by simultaneously solving the equations for total income (Y) and the interest rate (i). The crowding out effect in this case refers to the reduction in investments resulting from the increase in government spending (G) and its impact on the interest rate (r), which influences private sector investment decisions.
Overall, by analyzing the given equations and their relationships, we can determine the equilibrium levels of various economic variables, evaluate the effects of changes in government spending, and understand the concept of crowding out.
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the high school dropout rate in the united states is greater than 25 percent. c. wright mills would classify this situation as
Answers
The high school dropout rate in the United States is greater than 25 percent. C. Wright Mills would classify this situation as a societal issue.
This is because societal issues are matters of public concern that affect large numbers of people and require collective action to resolve. The high school dropout rate in the United States is an example of a societal issue because it has widespread implications for both individuals and society as a whole.
According to Mills, a societal issue is one that cannot be attributed solely to the actions of individuals. Instead, it is the result of social structures and processes that operate at a broader level.
For example, the high school dropout rate in the United States is influenced by a variety of factors such as poverty, race, and family background. These factors are not solely determined by the actions of individual students but are instead shaped by broader social forces such as economic inequality and institutional racism.
To address the high school dropout rate, therefore, requires collective action at the societal level. This may involve policy changes, community-based initiatives, or other forms of collective action aimed at addressing the underlying social factors that contribute to the problem.
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How could you find the length of the legs of an isosceles right triangle if you are only given the length of the hypotenuse?.
Answers
Answer:So for an isosceles right triangle with side length a , the hypotenuse has a length of a√2 . Similarly, if the hypotenuse of an isosceles right triangle has length of a , the legs have a length of a√2ora√22 each
Step-by-step explanation:
An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as triangles and their side lengths follow a specific pattern that states that one can calculate the length of the legs of an isoceles triangle by dividing the length of the hypotenuse by the square root of 2.
FOR EXAMPLE What is the length of the legs of an isosceles right triangle if the hypotenuse is 8 units?
2 Answers By Expert Tutors. For an isoceles right trangle, the legs each have length x and the hypotenuse has length x √2. Therefore, since the hypotenuse has length 8√2, the legs must each be of length 8.
Use the following system of linear equations to do the three tasks below.
x = y +3
Y=-4x-3
• Part A: Solve the linear system of equations by graphing.
• Part B: Use substitution to solve the linear system of equations.
• Part C: Use addition to solve the linear system of equations.
In your final answer, include all of your work, the graph of the equations and the solution.
Answers
Step-by-step explanation:
where the is y,you put the value of y 4x-3 and where the is x,put the value of x (y+3)
Identify which value represents the sample mean and which value represents the claimed population mean. American households spent an average of about $52 in 2007 on Halloween merchandise such as costumes, decorations and candy. To see if this number had changed, researchers conducted a new survey in 2008 before industry numbers were reported. The survey included 1, 500 households and found that average Halloween spending was $58 per household. The sample mean is dollars, while the claimed population mean is dollars. The average GPA of students in 2001 at a private university was 3.37. A survey on a sample of 203 students from this university yielded an average GPA of 3.59 in Spring semester of 2012. The sample mean is and the claimed population mean is
Answers
It is stated that the population mean is 3.37, which represents the average GPA of all students at the university in that year.
In the given scenarios:
American households' Halloween spending:
Sample Mean: $58 per household
Claimed Population Mean: $52
In this case, the sample mean represents the average spending per household obtained from the survey conducted in 2008 on 1,500 households. This sample mean is $58. It is a representative value calculated from the data collected in the sample of households.
On the other hand, the claimed population mean refers to the average spending per household in the entire population of American households. In this case, it is claimed or known that the population mean is $52, which represents the average Halloween spending in 2007.
The researchers conducted the survey in 2008 to determine if there was any change in Halloween spending compared to the claimed population mean of $52.
Average GPA of students at a private university:
Sample Mean: 3.59
Claimed Population Mean: 3.37
Here, the sample mean is the average GPA calculated from the data collected in the survey on a sample of 203 students from the university in the spring semester of 2012. The sample mean is 3.59, representing the average GPA of the sampled students.
The claimed population mean, on the other hand, refers to the average GPA of all students at the private university in 2001. It is stated that the population mean is 3.37, which represents the average GPA of all students at the university in that year.
The survey conducted in 2012 aimed to determine if there was any change in the average GPA compared to the claimed population mean of 3.37 in 2001.
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PLEASE HELP ASAP WILL GIVE BRAINLIST
Answers
iffx=3x−4findf1x0
Step 1: Add -3x to both sides.
f2ix+−3x=−4df2in+3x+−3x
f2ix−3x=−4df2in
Step 2: Factor out variable x.
x(f2i−3)=−4df2in
Step 3: Divide both sides by f^2i-3.
x(f2i−3)
f2i−3
=
−4df2in
f2i−3
x=
−4df2in
f2i−3
Answer:
x=
−4df2in
f2i−3
the 8000 that he invested in fund a returned a 6% profit.the amount that he invested in fund b returned a 2% profit how much did he invest in fund b if both funds together returned a 3 % profit
Answers
Let
\(\begin{gathered} x=\text{ the amount invested in fund b} \\ m=\text{ the profit from the fund a investment} \\ n=\text{ the profit from the fund b investment} \end{gathered}\)Therefore,
\(\frac{m}{8000}=\frac{6}{100}\)Hence,
\(m=\frac{6}{100}\times8000=6\times80=\text{ \$480}\)\(\begin{gathered} \text{ Also} \\ n=0.02x------(1) \\ \frac{m+n}{8000+x}=0.03 \\ \text{therefore} \\ \frac{n+480}{8000+x}=0.03 \end{gathered}\)Thus
\(n+480=0.03(8000+x)-----------------(2)\)Substituting equation(1) into equation(2), we have
\(\begin{gathered} 0.02x+480=0.03(8000+x) \\ 0.02x+480=240+0.03x \\ \text{ Thus} \\ 0.03x-0.02x=480-240 \\ 0.01x=240 \\ x=\frac{240}{0.01}=2400 \end{gathered}\)Hence, the amount he invested is $2400