To put
[tex]\frac{5}{7-i}[/tex]In the standard form, we must multiply the numerator and denominator by the complex conjugate of (7-i), it means
[tex]\frac{5}{7-i}\cdot\frac{7+i}{7+i}[/tex]And now we solve it, therefore
[tex]\frac{5}{7-i}\cdot\frac{7+i}{7+i}=\frac{5(7+i)}{7^2-i^2}[/tex]Remember that
[tex]i^2=-1[/tex]Then
[tex]\begin{gathered} \frac{5(7+i)}{7^2-i^2}=\frac{5(7+i)}{49+1} \\ \\ \frac{5(7+i)}{49+1}=\frac{5(7+i)}{50} \end{gathered}[/tex]Now we can simplify it
[tex]\frac{5(7+i)}{50}=\frac{7+i}{10}[/tex]And we have it in the standard form
[tex]\frac{7}{10}+\frac{1}{10}i[/tex]Hi, can you help me to evaluate (if possible) thesix trigonometric functions of the real number.Please.
Okay, here we have this:
Considering the provided angle, we are going to evaluate the trigonometric functions, so we obtain the following:
Sine:
[tex]\begin{gathered} \sin (-\frac{2\pi}{3}) \\ =-\sin (\frac{2\pi}{3}) \\ =-\cos \mleft(\frac{\pi}{2}-\frac{2\pi}{3}\mright) \\ =-\cos \mleft(-\frac{\pi}{6}\mright) \\ =-\cos \mleft(\frac{\pi}{6}\mright) \\ =-\frac{\sqrt{3}}{2} \end{gathered}[/tex]Cos:
[tex]\begin{gathered} cos\mleft(-\frac{2\pi}{3}\mright) \\ =\cos \mleft(\frac{2\pi}{3}\mright) \\ =\sin \mleft(\frac{\pi}{2}-\frac{2\pi}{3}\mright) \\ =\sin \mleft(-\frac{\pi}{6}\mright) \\ =-\sin \mleft(\frac{\pi}{6}\mright) \\ =-\frac{1}{2} \end{gathered}[/tex]Tan:
[tex]\begin{gathered} tan\mleft(-\frac{2\pi\:}{3}\mright) \\ =\frac{\sin (-\frac{2\pi\: }{3})}{\cos (-\frac{2\pi\: }{3})} \\ =\frac{-\frac{\sqrt[]{3}}{2}}{-\frac{1}{2}} \\ =\sqrt[]{3} \end{gathered}[/tex]Csc:
[tex]\begin{gathered} \csc \mleft(-\frac{2\pi}{3}\mright) \\ =\frac{1}{\sin\left(-\frac{2\pi}{3}\right)} \\ =-\frac{1}{\frac{\sqrt{3}}{2}} \\ =-\frac{2\sqrt{3}}{3} \end{gathered}[/tex]Sec:
[tex]\begin{gathered} \sec \mleft(-\frac{2\pi}{3}\mright) \\ =\frac{1}{\cos\left(-\frac{2\pi}{3}\right)} \\ =\frac{1}{-\frac{1}{2}} \\ =-2 \end{gathered}[/tex]Cot:
[tex]\begin{gathered} \cot \mleft(-\frac{2\pi}{3}\mright) \\ =\frac{1}{\tan (-\frac{2\pi}{3})} \\ =\frac{1}{\sqrt[]{3}} \\ =\frac{\sqrt{3}}{3} \end{gathered}[/tex]The length of sides of a triangle are xem, (x + 1)cm and (x + 2)cm. Determine x so that this triangle is a right- angled triangle.
The value of x = 3
3cm, 4cm, and 5cm are the sides of a right-angled triangle.
What is Pythagoras theorem?
In a right-angled triangle, the hypotenuse is the largest of the three sides, so hypotenuse is (x +2).
As a result of Pythagoras' theorem,
hyp² = base² + alt²
here ,
x² + (x+1)² = (x+2)²
by simplifying the equation,
x² + x² + 2x +1 = x² + 4 + 4x
=> 2x² + 2x + 1 = x² + 4x + 4
=> x² - 2x -3 = 0
=> x² - 3x + x - 3 = 0
=> x(x-3) + 1(x-3) = 0
=> (x+1) (x-3) = 0
so, x+1 = 0 or x-3 = 0
x = -1 or x = 3
since length cannot be negative,
x = -1 is not considered .
so x= 3.
value of x is 3.
So the triangle's three sides are 3cm, 4cm, and 5cm.
to double-check the answer
Pythagoras' theorem substitute values
hyp² = b²+a²
hyp = [tex]\sqrt{3^{2}+4^{2} }[/tex]
= [tex]\sqrt{9+16}[/tex]
= [tex]\sqrt{25}[/tex]
= 5
As a result, the three sides of a right-angled triangle are 3cm, 4cm, and 5cm.
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Solve the following system of equations using the substitution method.
–6x + 2y = 8
y = 3x + 4
Answer:
Infinite solutions or some courses say all real numbers
Step-by-step explanation:
-6x + 2(3x + 4) = 8 substitute 3x + 4 for y
-6x + 6x + 8 = 8 Distribute the 2
8=8 The x's cancel out leaving a true statement. This means that there are infinite solutions.
A. Let h be the number of hours Chris worked. How many hours did Mark work?
a. We start by noting the number of hours Mark worked, given that h is the number of hours Chris worked.
To get this, we look for the relationship between the number of hours Mark worked and the number of hours Chris worked
From the question, we are told that they worked the same umber of hours, so the number of hours Mark worked is also h hours since they worked equal number of hours.
b. We shll now use the relationship with Kari for both Chris annd Mark to write two equations.
For Chris, we rae told that Kari worked twicw as many hours as he worked.
So if Chris worked h hoirs, then Kari worked 2 * h = 2h hours
For Mark, we are told that Kari worked 10 less than 3 times the number of hours Mark worked
The number of hours Mark worked was h hours, 3 times this is 3 * h = 3h
10 less than this would be (3h - 10) hours
c. An equation describing the expression above;
Since each expressin represents the number o hours Kari worked, then the two expressions must be equal.
Thus;
2h = 3h - 10
7 times blank equals 1
Answer: 0.1429.
Step-by-step explanation: To double-check our work, multiply 0.1429 by 7 to see that it equals 1.
Why was the Battle of Fallen Timbers important for the new democracy?
Please write 3 paragraphs, stating 3 different reasons why the battle was important. (Paragraphs should be 4-6 sentences each
The Whiskey Rebellion put an end as a consequence of the Battle of Fallen Timbers; that had a significant outcome.
Why was Battle of Fallen Timbers important to the US?After the Battle of Fallen Timbers, the Indians agreed to the Treaty of Greenville, which gave the United States authority over most of the river crossings in the Old Northwest Territory as well as vital sites like Detroit.
The Battle of Fallen Timbers is considered to be the last battle of the American Revolution and which also helped the new country start expanding the territory to the west.
This will effectively ensured American dominance over the Indian tribes. The Whiskey Rebellion; which was the first significant confrontation between the nascent American government and it's inhabitants, which was put a stop to by it, and peace was restored.
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PLEASEEE HELP Find the rate of change of the line that contains the two points (1, 4) and (5, -4). Be sure to show all calculations and reduce your slope to a fraction in simplest form.
The rate of change of the line or the slope of the line that contains the two points (1, 4) and (5, -4) is m = 2
As per the question statement, we are supposed to find the rate of change of the line or the slope of the line that contains the two points (1, 4) and (5, -4).
We know slope [tex]m = \frac{y_{2}-y_{1}}{x_{2} -x_{1}}[/tex] for the line passing though the points (x1,y1) and (x2, y2)
Using the same formula and finding out the rate of change of the line or the slope of the line that contains the two points (1, 4) and (5, -4).
slope m = (-4-4)/(5-1)
m=-8/4
m = -2
Hence the rate of change of the line or the slope of the line that contains the two points (1, 4) and (5, -4) is m = 2
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f(x)=3x+2 and g(x)=2x2−5
Given also that fg(x)=ax2+bx+c work out the value of a+b−c
Answer:
a= -4
b -15
c -10
Step-by-step explanation:
by avoiding x^3 you can find your problem
Cynthia Besch wants to buy a rug for a room that is 25 ft wide and 31 ft long. She wants to leave a uniform strip
of floor around the rug. She can afford to buy 567 square feet of carpeting. What dimensions should the rug
have?
Answer:
21 ft by 27 ft
Step-by-step explanation:
You want the dimensions of a rug with an area of 567 square feet such that it fits in a 25 ft by 31 ft room with a uniform space all around.
SetupWe note the room is 31-25=6 ft longer than it is wide. Since the rug has a uniform border around it, the rug dimensions will be 6 ft longer than wide. We want the rug area to be 567 square feet, so for width w we have ...
w(w+6) = 567
Solutionw² +6w +9 = 576 . . . . . . add 9 to complete the square
(w +3)² = 24² . . . . . . . . express as squares
w +3 = 24 . . . . . . . . . positive square root
w = 21 . . . . . . . . . . subtract 3 to find the width
w+6 = 27 . . . . . . add 6 to find the length
The rug should have dimensions 21 ft wide by 27 ft long.
__
Additional comment
The uniform strip of floor around the rug will be 2 feet wide.
Find the vertex of the absolute value function y = 7|x – 3| - 2.(-3, -2)(3, 2)(3, -2)(3, -1/2)
By the first term of the given function 7lx - 3l you can notice that there is a translation 3 units to the right of the pattern function lxl. This allows one to conclude that the x-coordinate of the vertex is 3.
Now, for such a value of x the function value is 7l3 - 3l - 2 = -2. Then, the y-coordinate of the vertex is -2.
Hence, the vertex of the function is (3,-2)
Isaac creates a scatter plot showing the association between the height and weight of his maleclassmates. He models the association with the equation w = 6.5h - 275, where h is theclassmate's height in inches and w is the classmate's weight in pounds. What is the meaning of theslope in this equation?
solution
for this case we have the following equation:
w = 6.5h -275
For this case the slope is the number next to the h so then m = 6.5
And the best answer would be:
D) For every 6.5 inch in height , 275 pounds is subtracted to get the weight
Line CD passes through (0,1) and is parallel to x+y=3. Write the standard form of the equation of line CD
The standard form of the equation of line CD is x + y = 1
What is standard form of equation of a line?The standard form ;of equation of s line is written in the form
Ax + By = C
Line that are parallel has equal slope hence solving for the slope of line
x + y = 3 we have
y = -x + 3
The slope is -1
Equation of line CD passing through points ( 0, 1 )
( y - 1 ) = -1 * ( x - 0 )
y - 1 = -1 * x
y = -x + 1
writing to standard form
x + y = 1
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Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use.noyes; ASA Congruence Theoremyes; AAS Congruence Theorem
The diagramof the triangles is shown below
Looking at the two traingles, they meet at a vertex. Vertically opposite angles are equal.
Evaluate the expression for j = 8.
-7 + 2j =
Answer:
The answer tp this question is: 9.
Charlie has 5/6 pounds of dog food. She wants to put equal portions into 3 bowls how much food will go into each bowl?
Use point-slope form to write the equation of a line that passes through the point (-5,7)(−5,7) with slope -5−5
The equation of the line that passes through the point (-5,7) with slope -5 in the point-slope form is 5x + y = -18 .
The Point - Slope form of the line passing through (x₁,y₁) with slope m is given by the equation
(y-y₁)= m(x-x₁)
In the question ,
it is given that the required line passes through the point(-5,7) and have the slope = -5 .
the point is (-5,7)
so x₁= -5 and y₁=7 and m = -5
Substituting the value in the equation of point slope form , we get
(y-y₁)= m(x-x₁)
(y-7)= (-5)(x-(-5))
simplifying further , we get
(y-7)= (-5)(x+5)
y-7 = -5x -25
5x + y = -25 +7
5x + y = -18
Therefore , the equation of the line that passes through the point (-5,7) with slope -5 in the point-slope form is 5x + y = -18 .
The given question is incomplete , the complete question is
Use point-slope form to write the equation of a line that passes through the point (-5,7) with slope -5 .
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Your lunch in the hospital cafeteria cost $6.50. Based on working 50 weeks per year, how much will you spend on lunch per year?
The total amount spent on lunch per year would be 325 dollars.
What is the fundamental principle of multiplication?Multiplication is the mathematical operation that is used to determine the product of two or more numbers. If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
We have been given that Your lunch in the hospital cafeteria cost $6.50. Based on working 50 weeks per year, then;
The lunch in the hospital cafeteria cost = $6.50.
The total amount spend on lunch per year;
6.50 x 50
325
Hence, the total amount spent on lunch per year would be 325 dollars.
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(50 x 2) plus (15 x 2)
Answer:
130
Step-by-step explanation:
100+30
Answer:
130
Step-by-step explanation:
100+30=130
Tools Question 2 Nicolas makes toys at a toy shop. The graph represents the relationship between the number of toys (1) that Nicolas makes and the number of hours Toy Making 50 45 40 35 30 Number of Toys 25 20 15 10 2 6 10 Number of Hours Which of the following equations represents a toy-making rate, in toys per hour, that is HALF that of Nicolas's toy-making rate?
The equation of a proportional relationship between two variables x and y with a constant of proportionality k, is:
[tex]y=kx[/tex]If y represents the number of toys and x represents the number of hours, substitute the corresponding values of x and y to find the constant of proportionality k. Use, for instance, the fact that Nicholas made 40 toys in 10 hours:
[tex]40=k\cdot10[/tex]Divide both sides of the equation by 10:
[tex]k=4[/tex]Since Nicholas's toy-making rate is 4 toys per hour, half that rate would be 2 toys per hour. Then, out equation would become:
[tex]y=2x[/tex]Using the letter "t" for toys instead of y and "h" for hours instead of x, then:
[tex]t=2h[/tex]I'll give brainliest!
Answer: D
Step-by-step explanation:
Answer:
D
Step-by-step explanation:
...................
Write the equations for the lines parallel and
perpendicular to the given line j that passes
through Q. SEE EXAMPLE 4
26. y = -4x + 1; Q(6, -1)
27. y = ¾x + 4; Q(−1, 1)
LESSON 2-4 Slopes of Parallel and Perpendicular Lines 97
y = -4x + 1 and y = ¾x + 4, Clearly by values we can see that they are neither parallel nor perpendicular.
What is a slope?The ratio of the "vertical change" to the "horizontal change" between (any) two unique points on a line is used to compute slope. The ratio can also be written as a quotient ("rise over run"), which produces the same number for every two distinct points on the same line. A declining line has a negative "rise." The line might be useful, as determined by a road surveyor, or it might appear in a diagram that represents a road or a roof as a description or a design.
The slope's absolute value serves as a gauge for a line's steepness, incline, or grade. A steeper line is indicated by a slope with a higher absolute value. A line can be drawn with one of four directions: upward, downward, horizontal, or vertical.
If a line rises from left to right, it is said to be growing. The slope is upward, or m>0.If a line slopes downward from left to right, it is diminishing. The slope, m0, is negative.The slope of a line is 0 if it is horizontal. This function is constant.A line's slope is ambiguous if it is vertical.Clearly by values we can see that they are neither parallel nor perpendicular.
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how many weeks would be spent per year if you spend 2 hours tutoring per week, for the last 3 years?
Answer: .619 weeks per year
Step-by-step explanation:
52 weeks * 2 hours * 3 years = 312 hours tutored over 3 years
24 hours in a day * 7 days in a week = 168 hours in a week
312 / 168 = 1.857 weeks tutored over 3 years
1.857 / 3 = .619 weeks tutored per year
[tex] \frac{x}{3} + 1 = \frac{2}{3} [/tex]find the value of x
Given:
[tex]\frac{x}{3}+1=\frac{2}{3}[/tex]Find: x
Explanation:
[tex]\begin{gathered} \frac{x}{3}+1=\frac{2}{3} \\ \frac{x}{3}=\frac{2}{3}-1 \\ \frac{x}{3}=\frac{-1}{3} \\ x=-1 \end{gathered}[/tex]Final answer: the required value of x is -1
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9.823 x 10^-9 = 9.823/ 10^ 9
9.823/ 1000000000 = 0.000000009823
the long form has the same number of zeros that 10^-9, then it have 9 zeros.
if the company decides to produce 2,000 containers of regular crunchy peanut butter, how many containers of new extra crunchy peanut butter would it produce?
The number of extra crunchy peanut butter that the company would produce (current production units) is 1,700 containers.
How is the number determined?Using the mathematical operation of subtraction, the number of units to be produced can be determined.
First, we have that the ending inventory for the last period was 300 containers.
This ending inventory is subtracted from the required production level, to determine the number of containers to produce for this period.
Thus, to meet the production requirements, the company would produce additional 1,700 containers of crunchy peanut butter.
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Question Completion:Ending inventory of regular crunchy peanut butter from last period = 300 containers.
Janie has $3dollar sign, 3. She earns \$1.20$1.20dollar sign, 1, point, 20 for each chore she does and can do fractions of chores. She wants to earn enough money to buy a CD for $13.50
The inequality to represent the situation for Janie will be 3 + 1.2c > 13.50.
How to calculate the inequality?From the information, it was illustrated that Janie has $3 and that she earns $1.20 for every chore.
Let the total number of chores that she can do be represented as c.
It should be noted that she also wants about $13.50 to buy her CD. Therefore, the inequality can be represented as:
3 + (1.2 × c) > 13.50
3 + 1.2c > 13.50
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Solve each system using substitution show your answer as an ordered pair no solution or infinite solution
1) Given y = x+2 and y = -4x-8.
Since the left hand sides of both equations are same, equate the right hand side of both the equations.
[tex]x+2=-4x-8[/tex]Add 4x on both sides.
[tex]\begin{gathered} x+2+4x=-4x-8+4x \\ 5x+2=-8 \end{gathered}[/tex]Add -2 on both sides.
[tex]\begin{gathered} 5x+2-2=-8-2 \\ 5x=-10 \end{gathered}[/tex]Divide by 5 on both sides.
[tex]\begin{gathered} x=-\frac{10}{5} \\ =-2 \end{gathered}[/tex]Substitute the value of x into y = x+2.
[tex]\begin{gathered} y=-2+2 \\ =0 \end{gathered}[/tex]Solution is (-2,0).
2) Given y = 3x+1 and y = -2x+6.
Since the left hand sides of both equations are same, equate the right hand side of both the equations.
[tex]3x+1=-2x+6[/tex]Add 2x on both sides.
[tex]\begin{gathered} 3x+1+2x=-2x+6+2x \\ 5x+1=6 \end{gathered}[/tex]Add -1 on both sides.
[tex]\begin{gathered} 5x+1-1=6-1 \\ 5x=5 \end{gathered}[/tex]Divide by 5 on both sides.
[tex]\begin{gathered} x=\frac{5}{5} \\ =1 \end{gathered}[/tex]Substitute the value of x into y = 3x+1.
[tex]\begin{gathered} y=3\cdot1+1 \\ =4 \end{gathered}[/tex]Solution is (1, 4).
3) Given y = -3x-6 and 6x+2y = -2.
Substitute -3x-6 for y into 6x+2y = -2.
[tex]\begin{gathered} 6x+2(-3x-6)=-2 \\ 6x-6x-12=-2 \\ -12=-2 \end{gathered}[/tex]which is not possible. Hence the given system of equations has no solution.
4) Given y = -5 and -8x+4=-20.
From the second equation, -8x+4 = -20, solve for x.
Add -4 on both sides.
[tex]\begin{gathered} -8x+4-4=-20-4 \\ -8x=-24 \end{gathered}[/tex]Divide by -8 on both sides.
[tex]\begin{gathered} x=\frac{-24}{-8} \\ =3 \end{gathered}[/tex]Solution is (-5, 3).
Javier exercises for 2 1/2 hours every Saturday. His exercisesincludes two parts. If he spends the same amount of timeon each part how many hours does he spend weightlifting?
According to the given data we have the following:
Javier exercises for 2 1/2 hours every Saturday, hence, he exercises for 2.5 hours.
In order to calculate the amount of time he spend weightlifting we would make the following calculation:
amount of time he spend weightlifting=total training per day/2
amount of time he spend weightlifting=2.5 hours/2
amount of time he spend weightlifting=1.25 hours
Therefore, he spends weightlifting 1 hour and 25 minutes every saturday
Solve the quadratic equation x2=2536.What are the solutions of the equation x2=2536?
Given
[tex]x^2=2536[/tex]Answer
[tex]\begin{gathered} x^2=2536 \\ x=5036 \end{gathered}[/tex]Kuta Software - Infinite Algebra 2 Graphing Absolute Value Equations Graph each equation. 1-1-11 Name Date > 1512020 2) y-lx-a 13
we have the equation
[tex]y=-3\lvert-2x+4\rvert+3[/tex]using a grahing tool
see the attached figure