1) Examining the function f(x)= x²-3x -10 to get this function reflected over the x axis,
We'll need to multiply the "a" parameter by -1, so that we can get:
f(x) = -x²-3x -10
¨do you know what complex numbers are? Can you divide two complex numbers? Give us an example here!¨
A complex number z is a number of the form z = a + bi where a and b are real numbers, and i is the imaginary number, defined as the solution for i² = - 1.
We can indeed divide complex numbers. Let's take the numbers 1 + i and 1 - 2i for example. Dividing the first number by the second, we have
[tex]\frac{1+i}{1-2i}[/tex]To solve this division, we need to multiply both the numerator and denominator by the complex conjugate of the denominator
[tex]\frac{1+\imaginaryI}{1-2\imaginaryI}=\frac{1+\imaginaryI}{1-2\imaginaryI}\cdot\frac{1+2i}{1+2i}=\frac{(1+i)(1+2i)}{(1-2i)(1+2i)}[/tex]Expanding the products and solving the division, we have
[tex]\frac{(1+\imaginaryI)(1+2\imaginaryI)}{(1-2\imaginaryI)(1+2\imaginaryI)}=\frac{1+3i-2}{1+4}=\frac{-1+3i}{5}=-\frac{1}{5}+\frac{3}{5}i[/tex]And this is the result of our division
[tex]\frac{(1+\imaginaryI)}{(1-2\imaginaryI)}=-\frac{1}{5}+\frac{3}{5}i[/tex]9.03 divided by 0.3
in order to divide
[tex]\frac{9.03}{0.3}[/tex]we must convert the decimal number to intergers. We can do that by multiplying by 100/100.
That is
[tex]\frac{9.03}{0.3}=\frac{9.03}{0.3}\cdot\frac{100}{100}[/tex]hence,
[tex]\frac{9.03}{0.3}=\frac{903}{30}[/tex]and now, we can apply the long division on 903/30:
Therefore, 9.03/0.3=30.1
An aquamum contains dolphins, sharks, andwhales. There are twice as many dolphins as whalesand 8 fewer sharks than dolphins and whales com-bined. If there are w whales, which of the followingrepresents the number of sharks?
Given:
An aquamum contains dolphins, sharks, and whales. There are twice as many dolphins as whales and 8 fewer sharks than dolphins and whales combined.
Required:
If there are w whales, which of the following represents the number of sharks
Explanation:
The question asks for the correct expression of the number of sharks in terms of whales and dolphins . If w represents the number of whales , then the phrase " twice as many dolphins as whales " means that there are 2w dolphins . Therefore , " dolphins and whales combined " is 2w + w , or 3w . Because there are 8 fewer sharks than dolphins and whales combined , you need to subtract 8 from 3w.
you can also answer this question by using the Picking Numbers strategy . Pick a small , positive number , like 5 , for the number of whales . If there are 5 whales and " twice as many dolphins as whales , " then there must be 10 dolphins . Combine the number of whales and dolphins and subtract 8 from that sum to find the number of sharks : ( 5 + 10 ) -8 = 15-8 = 7 . Plug in w = 5 to determine which answer choice gives you a value of 7
Final answer:
B
Write equation of circle in standard form. Quadrant lies in 2 tangent to x=–12 and x=–4
Solution
Explanation:
The diameter of the circle is defined by the distance between (-12, 0) and (-4, 0).
The distance from the mid point of the line joining points (-12, 0) and (-4, 0) to point is the radius of the circle = 4
Find PR.Write your answer as an integer or as a decimal rounded to the nearest tenth. PR = ___
In triangle PQR, RQ is 4 units and angle P is 29 degrees.
Use the trigonometric ratio of tan to find PR as follows:
[tex]\begin{gathered} \tan 29=\frac{RQ}{PR} \\ PR=\frac{RQ}{\tan 29} \\ PR=\frac{4}{0.5543090} \\ PR=7.21619 \\ PR\approx7.2 \end{gathered}[/tex]Hence the value of PR is 7.2 rounded to one decimal place.
A board game of chance costs $2 is play You have a 20% chance dans is the expected value of playing the game you lose your bet 15% of the m
Given
Cost to play game = $2
Find
Expected value of playing
Explanation
10% chance to win 1 = 1 x 10% = $0.1
25% chance to win 2 = 2 x 25% = $0.5
50% chance to win 5 = 5 x 50% = $2.5
15% chance to lose 2(being cost) = 2 x 15% = $0.3
= 1.5 -0.1 - 0.3 = 1.1
Final Answer
The expected value of playing is $1.10
Hence option (d) is correct
f (x)=2^ x -10 and the domain of f(x) is the set of integers from 1 to 3which values are elements of the range of f(x) Select all that apply.a. -12b. -10c. -9d. -6e. -2
We have to find the range of the function f(x).
The definition of f(x) is:
[tex]f(x)=2^x-10[/tex]The domain of this function is defined as: D: {-1, 0, 1, 2, 3}, which represents all the integers from -1 to 3.
Then, we have to find the range by applying the function to each of the elements of the domain:
[tex]f(-1)=2^{-1}-10=\frac{1}{2}-10=-9.5[/tex][tex]f(0)=2^0-10=1-10=-9[/tex][tex]f(1)=2^1-10=2-10=-8[/tex][tex]f(2)=2^2-10=4-10=-6[/tex][tex]f(3)=2^3-10=8-10=-2[/tex]Then, the range of f(x) is R: {-9.5, -9, -8, -6, -2}.
Answer:
The options that apply from the list are -9, -6 and -2. [Options c, d and e]
The table shows the highest maximum temperature for the month of October in Philadelphia Pennsylvania over the yearsPart A identify the independent and dependent quantity in their units of measure?Part B identify the equation of line of best fit using the data table.what is the slope and y-intercept of the line and what do they represent?
Answers:
A. Independent = Year
Dependent = Temperature
B. Temp = 0.6733(Year) - 1293.61
Explanation:
The independent variable is the variable that is not affected by the other, in this case, no matter the temperature, the year is given, so the independent variable is the year and the dependent variable is the highest temperature because it changes depending on the year.
Then, to identify the equation of the line of best fit, we will use the following:
First, we need to calculate the mean of both variables, so:
[tex]\begin{gathered} \text{Mean Year = }\frac{2008+2009+2010+2011+2012+\cdots+2017}{10} \\ \text{Mean Year = }2012.5 \\ \text{Mean Temperature = }\frac{64.9+53.1+61+54+\cdots+66.9}{10} \\ \text{Mean Temperature=}61.47 \end{gathered}[/tex]Then, we need to fill the following table:
Now, the slope of the line can be calculated as the sum of the values in the row (Year - Mean Year) x (Temp - Mean Temp) divided by the sum of the row (Year - Mean Year)^2. So, the slope of the line is:
[tex]m=\frac{55.55}{82.5}=0.6733[/tex]Finally, the y-intercept can be calculated as:
[tex]\begin{gathered} b=\text{Temp Mean - Slope x Year Mean} \\ b=61.47-0.6733(2012.5) \\ b=-1293.61 \end{gathered}[/tex]So, the equation of the line that best fits the data table is:
[tex]\text{Temp}=0.6733(\text{Year)}-1293.61[/tex]After the end of an advertising campaign, the daily sales of a product fell rapidly, with daily sales given by S=3800e−0.05x dollars, where x is the number of days from the end of the campaign.a. What were daily sales when the campaign ended?b. How many days passed after the campaign ended before daily sales were below half of what they were at the end of the campaign?
Since the given equation is
[tex]S=3800e^{-0.05x}[/tex]S is the amount of the daily sales from ending to x days
Since the form of the exponential function is
[tex]y=ae^x[/tex]Where a is the initial amount (value y at x = 0)
Then 3800 represents the daily sales when x = 0
Since x = 0 at the ending of the campaign, then
a. The daily sales when the campaign ended is $3800
Since the daily sales will be below half $3800 after x days
Then find half 3800, then equate S by it, then find x
[tex]\begin{gathered} S=\frac{1}{2}(3800) \\ S=1900 \end{gathered}[/tex][tex]1900=3800e^{-0.05x}[/tex]Divide both sides by 3800
[tex]\begin{gathered} \frac{1900}{3800}=\frac{3800}{3800}e^{-0.05x} \\ \frac{1}{2}=e^{-0.05x} \end{gathered}[/tex]Insert ln for both sides
[tex]\ln (\frac{1}{2})=\ln (e^{-0.05x})[/tex]Use the rule
[tex]\ln (e^n)=n[/tex][tex]\ln (\frac{1}{2})=-0.05x[/tex]Divide both sides by -0.05 to find x
[tex]\begin{gathered} \frac{\ln (\frac{1}{2})}{-0.05}=\frac{-0.05x}{-0.05} \\ 13.86294=x \end{gathered}[/tex]Since we need it below half 3800, then we round the number up to the nearest whole number
Then x = 14 days
b. 14 days will pass after the campaign ended
Which statement about this figure is true ? ○ it has rotational symmetry with an angle of 45°.○ it has no reflectional symmetry.○ it has reflectional symmetry with one line of symmetry. ○ it has point symmetry
The figure has different measure in all their parts, then, it has no reflectional symmetry.
For a point symmetry evey part has matching part, the same distance form a central point but our figure has the same 2 ellipses but with different measure. The figure dont have rotational symmetry because there is a little ellipse in the middle of the others.
. A plant grows 4 centimeters in two month. How many centimeters does it grow in one week?
it is given that
in a month the plant grows = 4 cm
and there are four complete weeks in a month
so, in four weeks the plant grows = 4 cm
in 1 weel the growth of the plant is 4/4 = 1 cm
so in a week, the plant grows 1 cm
You ordered from an online company. The original price of the item is $65. Theitem is on sale for 10%, and you have a coupon for an additional 15%. Applying onediscount at a time, what is the final price?$46.96$49.73$49.47$45.45
Given:
The original price, CP=$65.
The initial discount on sale, D1=10%.
The additional discount, D2=15%.
If the cost price(CP) of an item is given, then the selling price after the first discount is applied is,
[tex]SP=CP\times(\frac{1-First\text{ }Discount\text{ Percentage}}{100})[/tex]The additional discount is applied to the price after the first discount is applied. So, the final price after applying the second discount is,
[tex]SP^{\prime}=SP\times(\frac{1-Second\text{ }Discount\text{ Percentage}}{100})[/tex]Applying the first discount on the original price, the selling price is,
[tex]\begin{gathered} SP=CP\times\frac{(100-D1)}{100} \\ =65\times\frac{(100-10)}{100} \\ =58.5 \end{gathered}[/tex]Applying the second discount on the selling price, the final selling price is,
[tex]\begin{gathered} SP^{\prime}=SP_{}\times\frac{(100-D2_{})}{100} \\ =58.5_{}\times\frac{(100-15_{})}{100} \\ \cong49.73 \end{gathered}[/tex]Therefore, the final price is $49.73.
Note:
The direct formula for the final price if two successive discounts D1 and D2 are applied to a cost price CP is,
[tex]SP=CP\times(\frac{100-D1}{100})(\frac{100-D2}{100})[/tex]The diameter of a grain of sand measures at about 0.0046 inches, while the diameter of a dust particle measured at about 0.00005 inches. About how many times larger is the diameter of a grain of sand than a dust particle? Estimate the following problem using powers of 10.
We have the following:
To know how many times one grain is bigger than the other, we must calculate the quotient of them, as follows:
[tex]\frac{0.0046}{0.00005}=\frac{4.6\cdot10^{-3}}{5\cdot10^{-5}}=0.92\cdot10^{-3-(-5)}=0.92\cdot10^2=92[/tex]Therefore it is 92 times larger
Lotsa Boats requires 75$ plus payment of 10$ an hour for each hour for which the boated is rented.Which equation could be used to find the number of hours h the johnsons rented the boat for if they paid 125$ need answer helpp.
The required equation will be 75 +10 [tex]x[/tex] =125
and the value of x = 5 hours
Linear equation in one variable:
Equation having one variable and degree of the equation is one, called linear equation in one variable.
Example: 3x+2 =5
Given,
Base price of boat is 75$
charge per hour is 10$
johnsons rented the boat and he paid 125$
let,
he has taken the boat for rent for x hours
then,
according to question,
75 +10 [tex]x[/tex] =125
now solving the equation to get the value of x
10x = 125 - 75
10x = 50
x = 50/10
x = 5 hours
Hence,
The required equation will be 75 +10 [tex]x[/tex] =125
and the value of x = 5 hours
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10Determine which of the following are the solutions to the equation below.I2 = 5OA.5OB.V10O C.10ODEV5
LCF 4 10 2
2 5 2
1 5 5
1
LCF = 2 least common fa
Business is projected to be booming afterthe latest release of The Fast and theFurious 3.14159265359... Carver's AutoCustom must determine how many cansof paint and rims to stock at theirShanghai location.The Carver Family did choose WarehouseSpace A. The warehouse includes 8000sq. ft. of showroom and workshop space.One half of this warehouse space will beused to stock paint cans and rims. Thewarehouse has a height of 20 ft.Calculate the maximum numberof cylindrical paint cans thatCarver's Auto Custom can stock,if the paint comes in a 2-packhazmat box that measures 15<=>
Total Area = 8000 ft^2
Stock Area = 4000 ft^2
h = 20 ft
Stock's Volume = 4000*20 = 80000 ft^3
Boxes's Volume = 15*7*6 = 630 ft^3
Now he need to find how many boxes can stock in the warehouse space
80000/630 = 127 boxes. And if each box has two cans:
127 * 2 = 254 cans can be stored.
What number must be added to the expression below to complete the
square?
x²-x
O A. -1/1
4
O B. /
O C. - 12/2
O D.
-12
Answer:
1/4
Step-by-step explanation:
Completing the square
x^2 -x
Take the coefficient of the x term
-1
Divide by 2
-1/2
Square it
(-1/2) ^2 = 1/4
Add this to each side
Subtract 5y^2-6y-11 from 6y^2+2y+5?
Subtract 5y^2 - 6y - 11 from 6y^2 + 2y + 5
They are both quadratic expression
6y^2 + 2y + 5 - (5y^2 - 6y - 11)
Firtstly, use the negative sign to open the parentheses
6y^2 + 2y + 5 - 5y^2 + 6y + 11
Secondly, collect the like terms
6y^2 - 5y^2 + 2y + 6y + 5 + 11
since y^2 is common, factorize it out
y^2(6 - 5) + y(2 + 6) + 26
y^2(1) + y(8) + 26
y^2 + 8y + 26
The answer is y^2 + 8y + 26
When factored completely, which is a factor of 3x3 − 9x2 − 12x A. L(x − 3) B. (x − 4) C. (3x − 1) D. <(3x − 4)
Solution:
[tex]3x^3-9x^2-12x[/tex]Step 1:
Factor out the common term
The common term is 3x
By doing this, we will have
[tex]\begin{gathered} 3x^3-9x^2-12x=3x(\frac{3x^3}{3x}-\frac{9x^2}{3x}-\frac{12x}{3x}) \\ =3x(x^2-3x-4) \end{gathered}[/tex]Step 2:
Factorise the quadratic expression in the bracket
[tex]3x(x^2-3x-4)[/tex]By doing this, we will have to look for two factors to multiply to give i4 and if we add them together, we will have -3
The two factors are -4 and +1
therefore,
Replace -3x with -4x + x
[tex]\begin{gathered} 3x(x^2-3x-4) \\ =3x(x^2-4x_{}+x-4) \\ =3x(x(x-4)+1(x-4) \\ =3x(x+1)(x-4) \end{gathered}[/tex]Hence,
The final answer is = (x-4)
Please help 50 points!
1. A cylindrical jar has a radius of 6 inches and a height of 10inches. The jar is filled with marbles that have a volume of 20 in3. Use 3.14 for pi. Show work. Complete sentences.
What is the volume of the jar?
The volume of the jar is 1130.4 in³. The number of marbles that is filled the jar is 56.
What is the cylindrical shape?
The three-dimensional shape of a cylinder is made up of two parallel circular bases connected by a curved surface. The right cylinder is created when the centers of the circular bases cross each other. The axis, which represents the height of the cylinder, is the line segment that connects the two centers.
Given that the radius of cylindrical jar is 6 inches and the height is 10 inches.
The volume of a cylindrical shape is [tex]\pi r^2h[/tex].
Where r is the radius of the cylinder and h is the height of the cylinder.
Given that, the radius of the cylindrical jar is 6 inches and height is 10 inches.
The volume of a cylindrical shape is 3.14 × 6² × 10 = 1130.4 in³.
The number of marbles by which the jar can be filled is 1130.4/20 = 56.52 = 56 (approx.)
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7. Solve the following set of equations: 3x - 7y=-4 and 2x - 5y = -3a. (1, 2)b. (2, 1)c. (-2,-1)d. (1, 1)e. (-1,-1)
We will have the following:
First, we solve both expressions for "y", that is:
[tex]\begin{gathered} 3x-7y=-4\Rightarrow-7y=-3x-4 \\ \Rightarrow y=\frac{3}{7}x+\frac{4}{7} \\ \\ and \\ \\ 2x-5y=-3\Rightarrow-5y=-2x-3 \\ \Rightarrow y=\frac{2}{5}x+\frac{3}{5} \end{gathered}[/tex]Now, we equal both expressions:
[tex]\begin{gathered} \frac{3}{7}x+\frac{4}{7}=\frac{2}{5}x+\frac{3}{5}\Rightarrow\frac{1}{35}x=\frac{1}{35} \\ \\ \Rightarrow x=1 \end{gathered}[/tex]Now, we determine the value of y:
[tex]y=\frac{2}{5}(1)+\frac{3}{5}\Rightarrow y=1[/tex]So, the solution is:
[tex](1,1)[/tex]solve for x. then find the missing piece(s) of the parallelogram for #6.
Solution
Recall
[tex]\begin{gathered} 50x+130x=180\text{ \lparen supplementary\rparen} \\ 180x=180 \\ divide\text{ both sides by 180} \\ \frac{180x}{180}=\frac{180}{180} \\ \\ x=1 \\ \end{gathered}[/tex]The final answer
[tex]x=1[/tex]the volume of a right cone is 27 π units^3. if its height is 9 units find its circumference in terms of π.
Given:
the volume of a right cone is 27 π units³
And the height of the cone = h = 9 units
First, we will find the radius of the base (r) using the formula of the volume.
[tex]V=\frac{1}{3}\pi r^2h[/tex]Substitute V = 27π and h = 9
[tex]27π=\frac{1}{3}πr^2(9)[/tex]Solve the equation to find (r)
[tex]\begin{gathered} r^2=\frac{3*27}{9}=9 \\ r=\sqrt{9}=3 \end{gathered}[/tex]Now, we will find the circumference using the following formula:
[tex]circumference=2πr[/tex]substitute r = 3
[tex]circumference=2π(3)=6π[/tex]So, the answer will be: Circumference = 6π units
does the number line below present the solution to the equivalent X < -1?
On a number line, if we choose a number a, all the numbers to the right of a are greater than a, while all the numbers to the left are lower than a.
In this case, the inequality is:
[tex]x<-1[/tex]So, the solution in the number line would be all the numbers to the left of -1.
Therefore, this number line does not represent the solution to x<-1 because everything is to the right instead of to the left.
Simplified What is the ratio of Π to Δ ? = What is the ratio of A to ( +Δ) ? ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ ΔΔΔΔΔΔΔΔΔ
• The ratio of ⬛ to △.
First count the number of ⬛ and the number of △ given.
We have:
Number of ⬛ = 45
Number of △ = 81
The ratio of ⬛ to △ = Number of ⬛ : Number of △
= 45 : 81
To Simplify the ratio, divide by the GCF which is 9.
= 45 ÷ 9 : 81÷9 = 5 : 9
Therefore the simplified ratio of ⬛ to △ = 5 : 9
• The ratio of △ to (⬛ + △)
81 : (45 + 81)
= 81 : 126
Simplify the ratio by dividing it with its GCF which is 9:
81 ÷ 9 : 126 ÷ 9 = 9 : 14
Therefore, the simplified ratio of △ to (⬛ + △) = 9 : 14
ANSWER:
⬛ to △ = 45 : 81 Simplified = 5 : 9
△ to (⬛ + △) = 81 : 126 Simplified = 9 : 14
Lila's retirement party will cost $8 if she invites 4 guests. If there are 9 guests, how much will Lila's retirement party cost? Solve using unit rates.
We are assuming that the party cost is directly proportional to the number of guests.
Then, if it will cost $8 for 4 guests, the unit rate is:
[tex]c=\frac{8\text{ dollars}}{4\text{ guests}}=2\text{ dollars per guest}[/tex]Then, we can use this unit rate to calculate the cost for 9 guests:
[tex]C(9)=2\frac{\text{ dollars}}{\text{ guest}}\cdot9\text{ guests}=18\text{ dollars}[/tex]The cost for 9 guests is $18.
Conner left his house and rode his bike into town at 6mph. Along the way he got a flat tire so he had to turn around and walk his bike to his house traveling 3 mph. If the trip down and back took 15 hours, how far did he get before his tire went flat?Conner went ___ miles before his tire went flat.
The main point in this question, that the distance of the first part = the distance of the second part
[tex]d_1=d_2[/tex]Since the speed of the first part is 6 mph
Let the time of it be t1
Since distance = speed x time, then
[tex]\begin{gathered} d_1=v_{1_{}}\times t_1_{} \\ d_1=6\times t_1 \\ d_1=6t_1 \end{gathered}[/tex]Since the speed of the second part is 3 mph
Let the time of it be t2, then
[tex]\begin{gathered} d_2=3\times t_2 \\ d_2=3t_2 \end{gathered}[/tex]Equate d1 and d2 to find t2 in terms of t1
[tex]3t_2=6t_1[/tex]Divide both sides by 3
[tex]\begin{gathered} \frac{3t_2}{3}=\frac{6t_1}{3} \\ t_2=2t_1\rightarrow(1) \end{gathered}[/tex]Since the total time of the two parts is 15 hours, then
[tex]t_1+t_2=15\rightarrow(2)[/tex]Substitute (1) in (2)
[tex]\begin{gathered} t_1+2t_1=15 \\ 3t_1=15 \end{gathered}[/tex]Divide both sides by 3
[tex]\begin{gathered} \frac{3t_1}{3}=\frac{15}{3} \\ t_1=5 \end{gathered}[/tex]Now, let us find d1
[tex]\begin{gathered} d_1=6\times5 \\ d_1=30 \end{gathered}[/tex]Conner went 30 miles before his tire went flat
Show your steps when solving the problem below. Container A has 800 mL of water and is leaking 6 mL per minute. Container B has 1,000 mL of water and is leaking minute. How many minutes will it take for the two containers to have the same amount of water?
Step 1 : Let's review the information given to us to answer the problem correctly:
• Container A = 800 ml - 6 ml per minute
,• Container B = 1,000 ml - 10ml per minute
Step 2: Let's write the equation to solve the problem, as follows:
Let x to represent the number of minutes both containers have the same amount of water
Container A = Container B
800 - 6x = 1,000 - 10x
Like terms:
-6x + 10x = 1,000 - 800
4x = 200
Dividing by 4 at both sides:
4x/4 = 200/4
x = ?
I think you can calculate the value of x without problems.
Write the system below in the form AX=B. Then solve the system by entering A and B into a graphing utility and computing
We are given the system
[tex]\begin{gathered} x\text{ -3y+z=8} \\ 3x+4y+2z=\text{ -17} \\ 4x\text{ -4y +2z= -2} \end{gathered}[/tex]to write this system of the form
[tex]Ax=b[/tex]where A is a matrix, x is a vector and b is another vector, we simply take each equation and write it in matrix form. The first equation is
[tex]x\text{ -3y+z=8}[/tex]so, we will take a look at the left hand side of the equality sign. We have
[tex]x\text{ -3y+z}[/tex]we will take a look at the coefficients of each variable and write that as the first row of the matrix. That would be the row 1 -3 1 as the coefficient of x and z is 1 and the coefficient of y is -3. For b, the first row would be simply the number 8. So, if we do the same with the other two equations, we have
[tex]A=\begin{bmatrix}{1} & {\text{ -3}} & {1} \\ {3} & {4} & {2} \\ {4} & {\placeholder{⬚}\text{ -4}} & {2}\end{bmatrix}[/tex]and
[tex]b=\begin{bmatrix}{8} & {\placeholder{⬚}} & {\placeholder{⬚}} \\ {\placeholder{⬚}\text{ -17}} & {\placeholder{⬚}} & {\placeholder{⬚}} \\ {\text{ -2}} & {\placeholder{⬚}} & {\placeholder{⬚}}\end{bmatrix}[/tex]By using any of the two methods of the question (the use of software is beyond the scope of the session) we get that the solution is
[tex]\begin{gathered} x=\frac{\placeholder{⬚}\text{ -19}}{3} \\ y=\frac{\text{ -}8}{3} \\ z=\frac{19}{3} \end{gathered}[/tex]how long must $1000 be invested at an annual interest rate of 3% to earn $300 in sinple interest?