See below for the proof of the equation [tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex]
How to prove the equation?The equation is given as:
[tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex]
Take the LCM
[tex]\frac{(ax + by)(x - y) -(ax - by)(x + y)}{(x + y)(x - y)}= 2(a - b)[/tex]
Expand
[tex]\frac{ax^2 - axy + bxy - by^2 -ax^2 - axy + bxy + by^2}{(x + y)(x - y)}= 2(a - b)[/tex]
Evaluate the like terms
[tex]\frac{-2axy + 2bxy }{(x + y)(x - y)}= 2(a - b)[/tex]
Rewrite as:
[tex]\frac{-2axy + 2bxy }{x^2 - y^2}= 2(a - b)[/tex]
Factorize the numerator
[tex]\frac{2(a - b)(x^2 - y^2)}{x^2 - y^2}= 2(a - b)[/tex]
Divide
2(a - b)= 2(a - b)
Both sides are equal
Hence, the equation [tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex] has been proved
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50 POINTS HELP NOW
Find the logarithm base 10 of each number:
10
The logarithm base 10 of the number 10 is 1
How to solve logarithm ?We have to find logarithm base 10 of number 10.
Therefore, using logarithm rule
logₐ a = 1
Hence,
log₁₀ 10 = 1
Therefore, logarithm base 10 of the number 10 is 1
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Answer:
1
Step-by-step explanation:
The supplement of a 162 degree angle is an angle that measures
Answer:
18 degrees
Step-by-step explanation:
Supplementary angles are angles that when added together equal 180 degrees.
A + 162 = 180
- 162 -162
A = 18
Answer:
18
Step-by-step explanation:
When your are asked the supplement you minus 162 from 180. When you join SU from supplement and add a 1 behind SU you'll get 180. [1SU TO 180]
180-162
18
For how many values of x does the graph f(x)= x^2 -3 intersects the x-axis?
Answer:
cocomelon . 3/5 - 4
Step-by-step explanation:
Answer:
x=[tex]\sqrt3[/tex] and -[tex]\sqrt3\\[/tex]
Step-by-step explanation:
the roots(zeros) are the value of x where the graph intersects the x-axis.
to find the roots:
x^2-3=0
x^2=3
x=+-[tex]\sqrt3[/tex]
Which of the following lines will have a negative slope? Select all that apply
Answer:
1, 5
Step-by-step explanation:
-16/4=-4
10/5=2
1/2=1/2
-1/-3=1/3
-1/4/1/2=-1/2
The negative ones are your answer.
4.751 4.373 4.177 4.676 (a) Construct a confidence interval for the mean rate. Round the answer to at least four decimal places. A confidence interval for the mean rate is
98% confidence interval for the mean rate = [4.1437 , 4.4983]
We are given the interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon, Georgia for one day ;
4.751, 4.373, 4.177, 4.676, 4.425, 4.228, 4.125, 4.251, 3.951, 4.192, 4.291, 4.414
Now, Firstly we will find the Mean of the above data, Xbar ;
Mean, Xbar = ∑ x ÷ n =
4.751 + 4.373 + 4.177 + 4.676 + 4.425 + 4.228 + 4.125 + 4.251 + 3.951 + 4.192 + 4.291 + 4.414 ÷ 12 = 4.321
Standard deviation, s = √ ∑(x - x bar)²÷ n-1 = 0.226.
Now, the pivotal quantity for the 98% confidence interval for the mean rate is;
P.Q = x bar - ц ÷ n - 1 ≈ tn - 1
where, Xbar = sample mean
s = sample standard deviation
n = sample size = 12
So, 98% confidence interval for the mean rate, μ is ;
P(-2.718 < t₁₁ <2.718) = 0.98
P(-2.718 <Xbar - μ σ√ⁿ < 2.718) = 0.98
P(Xbar - 2.718 * ₈÷ √ⁿ < μ Xbar + 2.718 * ₈÷ √ⁿ ) = 0.98
98% confidence interval for μ = (Xbar - 2.718 * ₈÷ √ⁿ < μ Xbar + 2.718 * ₈÷ √ⁿ ) = 0.98
[4.321 - 2.718 * 0.226 ÷ √₁₂ , 4.321 + 2.718 * 0.226 ÷ √₁₂
= [4.1437 , 4.4983]
Therefore, 98% confidence interval for the mean rate = [4.1437 , 4.4983]
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Find the value of x.
10
135-
Step-by-step explanation:
[tex] \frac{x}{12} = \frac{x + 4}{5 + 12} \\ [/tex]
[tex] \frac{x}{12} = \frac{x + 4}{17} \\ [/tex]
Now cross multiply...
[tex]12(x + 4) = 17x[/tex]
[tex]12x + 48 = 17x[/tex]
[tex]12x - 17x = - 48[/tex]
[tex] - 5x = - 48[/tex]
Now divide both term by same number
[tex] \frac{ - 5x}{ - 5} = \frac{ - 48}{ - 5} \\ [/tex]
[tex]x = 9.6[/tex]
Solve the equation and provide all steps ¾ x+2¼ =9
Answer:
x=9
Step-by-step explanation:
3/4x+2 1/4=9
3/4x=9-2 1/4
3/4x=6 3/4
4(3/4x)=4(6 3/4)
3x=27
x=9
Answer:
x = 9
Step-by-step explanation:
3/4 x + 2 1/4 = 9 subtract 2 1/4 from both sides of the equation
3/4 x = 6 3/4 now divide both sides by 3/4
x = 9
Emil is running a lemonade stand. He sells half of his lemonade in the morning, a quarter of what was left at lunchtime, and a third of what was left in the afternoon. When he closes for the day, he has 4 liters of lemonade left. How many liters of lemonade did he start the day with?
Answer:
16 Liters of the correct answer.
Emil started his lemonade stand with 12 litres of Lemonade.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication and division.
Given that:-Emil is running a lemonade stand. He sells half of his lemonade in the morning, a quarter of what was left at lunchtime, and a third of what was left in the afternoon.
When he closes for the day, he has 4 litres of lemonade left. Let the total lemonade he started with is x litres. Now we will form an expression from the given data to calculate the value of x.
x - (x/2) -(1/4)(x/2) - ( 1/4)(x/2)(1/3) = 4
x - ( x/2) - (x/8) - ( x/24) = 4
[ 24x -12x -3x - x][24] = 4
8x = 96
x = 12 liters
Therefore Emil started his lemonade stand with 12 liters of Lemonade.
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You are standing 22 feet from a tall building that has a glass elevator on the exterior wall facing you.
You watch the elevator as it ascends to the top of the building. The height h (in feet) of the elevator
above the ground can be modeled by / = 22 tan e, where © is the angle of elevation. Graph the
function. Describe what happens to © as h increases.
a. Find the graph in the attachment
b. As h increases, e tends to 90°
a. How to graph the function?
Since you are standing 22 feet from a tall building that has a glass elevator on the exterior wall facing you. You watch the elevator as it ascends to the top of the building.
The height h (in feet) of the elevator above the ground can be modeled by h = 22tane.
To plot the graph, we see that the function is the tangent function.
The tangent function has the value -∞ ≤ tanx ≤ +∞ for -90° ≤ x ≤ 90°
Since e is our angle of elevation and e ≥ 0, that is 0 ≤ e ≤ 90°. So, we choose the range of values for which tane ≥ 0. So, 0 ≤ tane ≤ +∞
So, we plot h = 22tane for 0 ≤ e ≤ 90°
Find the graph in the attachment
b. What happens to e as h increases?From the graph, we seet that at high h or as h increases, e tends to 90°
So, as h increases, e tends to 90°
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What type of triangle is shown below?
31°
28.5
19.2
109°
40%
991
Answer:
could u send a photo please
Find the equation of the line that passes through (0, -3) and is parallel to
the line joining the points (9, 2) and (3, -5).
Answer:
y = [tex]\frac{7}{6}[/tex] x - 3
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (9, 2 ) and (x₂, y₂ ) = (3, - 5 )
m = [tex]\frac{-5-2}{3-9}[/tex] = [tex]\frac{-7}{-6}[/tex] = [tex]\frac{7}{6}[/tex]
• Parallel lines have equal slopes , so
m = [tex]\frac{7}{6}[/tex] is the slope of the parallel line
the line crosses the y- axis at (0, - 3 ) ⇒ c = - 3
y = [tex]\frac{7}{6}[/tex] x - 3 ← equation of parallel line
Hey there!
[tex] \\ [/tex]
Answer:[tex] \green{\boxed{\red{\bold{\sf{y = \dfrac{7}{6}x - 3}}}}}[/tex]
[tex] \\ [/tex]
Explanation:To find the equation of a line, we first have to determine its slope knowing that parallel lines have the same slope.
Let the line that we are trying to determine its equation be [tex] \: \sf{d_1} \: [/tex] and the line that is parallel to [tex] \: \sf{d_1} \: [/tex] be [tex] \: \sf{d_2} \: [/tex] .
[tex] \sf{d_2} \:[/tex] passes through the points (9 , 2) and (3 , -5) which means that we can find its slope using the slope formula:
[tex] \sf{m = \dfrac{\Delta y}{\Delta x} = \dfrac{\green{y_2} - \orange{y_1}}{\red{x_2} - \blue{x_1 }}} [/tex]
[tex] \\ [/tex]
⇒Subtitute the values :
[tex] \sf{(\overbrace{\blue{9}}^{\blue{x_1}}\: , \: \overbrace{\orange{2}}^{\orange{y_1}}) \: \: and \: \: (\overbrace{\red{3}}^{\red{x_2}} \: , \: \overbrace{\green{-5}}^{\green{y_2}} )} [/tex]
[tex] \implies \sf{m = \dfrac{\Delta y}{\Delta x} = \dfrac{\green{-5} - \orange{2}}{\red{ \: \: 3} - \blue{9 }} = \dfrac{ - 7}{ - 6} = \boxed{ \bold{\dfrac{7}{6} }}}[/tex]
[tex] \sf{\bold{The \: slope \: of \: both \: lines \: is \: \dfrac{7}{6}}} [/tex].
Assuming that we want to get the equation in Slope-Intercept Form, let's substitute m = 7/6:
Slope-Intercept Form:
[tex] \sf{y = mx + b} \\ \sf{Where \: m \: is \: the \: slope \: of \: the \: line \: and \: b \: is \: the \: y-intercept.} [/tex]
[tex] \implies \sf{y = \bold{\dfrac{7}{6}}x + b} [/tex] [tex] \\ [/tex]
We know that the coordinates of the point (0 , -3) verify the equation since it is on the line [tex] \: \sf{d_1} \: [/tex]. Now, replace y with -3 and x with 0:
[tex] \implies \sf{\overbrace{-3}^{y} = \dfrac{7}{8} \times \overbrace{0}^{x} + b} \\ \\ \implies \sf{-3 = 0 + b} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \sf{\boxed{\bold{b = -3}} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
Therefore, the equation of the line [tex] \: \bold{d_1} \: [/tex] is [tex] \green{\boxed{\red{\bold{\sf{y = \dfrac{7}{6}x - 3}}}}} [/tex]
[tex] \\ [/tex]
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Wonder waves aquarium just opened a new tropics wing, with a large main tank and a small observation tank. there are 75 fish in the large tank, 25 of which are angelfish. there are 25 fish in the small tank, 5 of which are angelfish. do the tanks have the same ratio of angelfish to total fish?
Answer:
No, the ratio is not the same.
Step-by-step explanation:
75 total (large tank) / 25 total (small tank) = 3
25 angelfish (large tank) / 5 angelfish (small tank) = 5
The ratio of angelfish to total fish is not the same. When we scale up from the small tank to large tank in terms of total fish, we have a factor of 3. However, when we scale up from the small tank to large tank in terms of angelfish, we have a factor of 5. For the ratio to be the same, these factors would need to be identical.
In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle
The largest possible error in the area of the circle is of 44%.
What is the area of a circle?The area of a circle of radius r is given by:
[tex]A = \pi r^2[/tex]
With a diameter of 20 cm = radius of 10 cm, the area in cm² is given by:
[tex]A = \pi \times 10^2 = 100\pi[/tex]
With the error, with a diameter of 20 x 1.2 = 24 cm = radius of 12 cm, the area in cm² is given by:
[tex]A = \pi \times 12^2 = 144\pi[/tex]
The error is given by:
[tex]E = 144\pi - 100\pi = 44\pi[/tex]
The percent error is the error divided by the initial amount, multiplied by 100%, hence:
[tex]P = \frac{44\pi}{100\pi} \times 100\% = 44\%[/tex]
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PLEASE HURRY!!!!!!!!
Mr. Jackson has decided to open a new health club in his town. To decide whether he will have enough members to be successful, he hires a researcher to conduct a survey. What is the population from which the researcher should choose a sample?
A. People who exercise regularly.
B. Members of other health clubs in the town.
C. Trainers at other health clubs.
D. His friends and family.
Answer:
I would say b since this would tell him how many people are wiling to join a health club
Hope This Helps!!!
The main floor of Kinsey's home is 1,425 square feet. The upper level is 875 square feet with one unfinished bedroom that measures 150 square feet. The finished basement is 1,425 square feet with 7.5-foot ceilings. Kinsey's MLS listing includes below-grade square footage in livable area. What's the reportable square footage for Kinsey's home
The reportable square footage for Kinsey's home = 2150square feet.
The main floor of Kinsey's home is 1,425 square feet.
The upper level is 875 square feet with one unfinished bedroom that measures 150 square feet.
The finished basement is 1,425 square feet with 7.5-foot ceilings.
That means the square footage of the upper floor plus the main floor (minus the unfinished laundry area).
Kinsey should report square feet (1425+875- 150)
= 2150square feet.
Hence, The reportable square footage for Kinsey's home = 2150square feet.
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. A housing down payment is money that a
prospective buyer provides up front when
purchasing a home and is usually a per-
cent of the purchase price of the home. A
lender typically requires private mortgage
insurance (PMI) when the buyer's down
payment is less than 20% of the purchase
price. To secure a mortgage, buyers also
need to have additional cash on hand
for closing costs and prepaid property
tax. Suppose a buyer wants to purchase
a $375,000 house and must have $7,200
on hand for closing costs and property
tax. Which of the following inequalities
represents the total funds (f) the buyer
must have on hand to secure the mortgage
without having to pay PMI?
A) f≤0.2(375,000) + 7,200
B) f≥0.2(375,000) +7,200
C) f≤0.2(375,000 +7,200)
D) f≥0.2(375,000 +7,200)
Answer:
B) f≥0.2(375,000) +7,200
Step-by-step explanation:
≥ means 'at least'
Identifying Transformations On a coordinate plane, triangle A has points (3, 3), (6, 3), (6, 9). Triangle A prime has points (negative 2, negative 1), (negative 2, negative 2), (negative 3, negative 2). Determine all the transformations needed to transform the pre-image, A, to the image, A'. Select all that apply. dilation by a scale factor of 1/2 dilation by a scale factor of 1/3 reflection over the x-axis translation to the left 4 units translation down 3 units
Triangle A was dilated by a scale factor of 1/3 and reflected over the x axis to form triangle A'
What is a transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformation are translation, rotation, reflection and dilation.
Dilation is the increase or decrease in the size of a figure.
Triangle A was dilated by a scale factor of 1/3 and reflected over the x axis to form triangle A'
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Answer:
b d e
Step-by-step explanation:
CAN SOMEONE HELP PLEASE
a = 3, b = 5, and c = 7. What is m&B° to the nearest tenth?
Answer: 38.2 degrees
Step-by-step explanation:
[tex]b^2 = a^2 + c^2 - 2ac \cos B\\\\5^2 = 3^2 + 7^2 - 2(3)(7) \cos B\\\\25 = 58-42 \cos B\\\\-42 \cos B=-33\\\\\cos B=\frac{33}{42}\\\\B \approx 38.2^{\circ}[/tex]
What is the value of (-5)^4
Answer:
-20
Step-by-step explanation:
-5-5-5-5=-20
help me answer this table problem please
By using the concepts of relations and functions and based on the table given, the function presents the following relationship:
x → f(x)
7 → 95
How to evaluate a function according to a table
Functions are relations between an input set named domain (x) and an output set named range (f(x)), that is:
x → f(x)
After a quick observation, for x = 7, then f(7) = 95. In more graphically manner, the function present the respective relation:
7 → 95
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5. A bus covers a certain distance in 45 minutes if it runs at a Speed of 60km/hr. What must be the speed of the bus in order to reduce the time by 20 minutes.
Answer:
108 km/hr
Step-by-step explanation:
45 minutes = 3/4 hours
1) Distance = speed × time
= 60 × 3/4
= 45 km
2) 20 minutes = 1/3 hours. So, 3/4 - 1/3 = 5/12 hours.
speed = distance / time
= 45 / 5/12
= 108 km/hr
an elevator has a maximum load restriction of 2.5 tons. is it safe for two tile layers weighing 175 ib and 240 ib plus 80 boxes of tile weighing 50 ib each
Using proportions, it is found that the elevator will be safe, as the total weight of 2.2075 tons is less than the maximum load restriction of 2.5 tons.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
Each lb has 0,0005 tons. Hence the total weight in tons that we want to put in the elevator is:
T = (175 + 240 + 80 x 50) x 0.0005 = 2.2075 tons.
Hence, the elevator will be safe, as the total weight of 2.2075 tons is less than the maximum load restriction of 2.5 tons.
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Collect Data from Our Solar System
Fill in the table. To obtain each planet's distance from the sun
Answer:
jupiter, Saturn, Uranus, neptune are all distance from sun
Step-by-step explanation:
Answer:
AU:
Merc-0.39
Venus-0.72
Earth- 1
Mars- 1.52
Jupiter - 5.20
Uranus - 19.1
Neptune - 30
Years:
Merc-0.24
Venus-0.61
Earth- 1
Mars- 1.88
Uranus - 84?(maybe)
Neptune - 165
Step-by-step explanation:
Edge
PLEASE HELP ME I WILL MARK CORRECT ANSWER BRAINLIEST
Answer:
36 units
Step-by-step explanation:
to do this you must first find the lengths of all the sides with pythagorean theorem
for a to b you use
3^2 + 4^2 = c^2
9 + 16 = c^2
25 = c^2
c = 5
5 is one of the lengths
b to c uses the same equation
so
5 is another one of the lengths
for d to c your equation is
5^2 + 12^2 = c^2
25 + 144 = c^2
169 = c^2
13 is another of the lengths
d to a has the same length as c to d so there is another 13 unit length
so
5 + 5 + 13 + 13 is your equation to find the perimeter
5 + 5 + 13 + 13 = 36
that is your answer
Can you help me please
Answer: Option D
[tex]\displaystyle d=\frac{C}{\pi}[/tex]
Step-by-step explanation:
To solve the formula for d, we will isolate the d variable.
Given:
C = πd
Divide both sides of the equation by π:
[tex]\displaystyle\frac{C}{\pi}=\frac{\pi d}{\pi}[/tex], [tex]\displaystyle\frac{C}{\pi}=d[/tex]
Reflexive property
[tex]\displaystyle d=\frac{C}{\pi}[/tex]
Evaluate the expression using x=-1 and y=3 (6^4y^3)
The value of the expression when evaluated is: 162.
How to Evaluate an Expression?To evaluate the expression, 6x^4y^3, substitute x = -1 and y = 3 into the expression and solve.
6(-1^4) × 3^3
6(1) × 27
6 × 27
162
The value of the expression, 6x^4y^3, is: 162.
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The function g(x) is graphed.
On a coordinate plane, a curved line enters the plane at point (negative 2.3, 5), crosses the x- and y-axis at (0, 0), and leaves the plane at point (2.3, 5).
Which statements about the function are true? Choose three options.
The statements about the function that are true include:
g(0) = 0
g(1) = -1
g(-1) = 1
How to depict the function?From the information, the points include:
(x1, y1) = (-2, 2, 5).
(x2, y2) = (0, 0)
(x3, y3) = (2, 3, 5).
It should be noted that (x2, y2) implies that g(0) = 0. In this case, the statements about the function that are true include g(0) = 0, g(1) = -1, and g(-1) = 1. This is illustrated in the graph.
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4 Quick algebra 1 question for 50 points!
Only answer if you know the answer, tysm!
Define the domain and range of each function: (in the pictures)
[tex]f(x )= - 2x[/tex]
Domain: ( - ∞ , ∞ ) or ] -∞ , ∞ [Range: ( - ∞ , ∞ ) or ] -∞ , ∞ [B)[tex]g(x) = \frac{1}{2} x {}^{2} + 5[/tex]
Domain: ( - ∞ , ∞ ) or ] -∞ , ∞ [Range: [ 5 , ∞ [ or [ 5 , ∞ )C)Domain: ( - ∞ , ∞ ) or ] -∞ , ∞ [Range: ( - ∞ , ∞ ) or ] -∞ , ∞ [D)Domain: ( - ∞ , ∞ ) or ] -∞ , ∞ [Range: ( - ∞ , - 3 ] or ] - ∞ , - 3 ]#1
y=-2xDomain
(-oo,oo)Range
(-oo,oo)#2
y=0.5x²+5Domain
(-oo,oo)Range
[5,oo)#3
Its graphed
Domain=Range=(-oo,oo)
#4
Domain
(-oo,oo)Range
[-3,-oo) or (-oo,-3]
Weekly wages at a certain factory are
normally distributed with a mean of
$400 and a standard deviation of $50.
Find the probability that a worker
selected at random makes between
$300 and $350.
[? ]%
Answer:
0.1359
Step-by-step explanation:
For the following exercises, find the x- and y-intercepts of the graphs of each function.
26. f(x) = 4| x − 3 | + 4
For the function f(x) = 4|x - 3| + 4, we have that:
The function has no x-intercepts.The y-intercept is of y = 16.What is the x-intercept of a function?The x-intercept of a function is the value of x when y = f(x) = 0, hence:
0 = 4|x - 3| + 4
4|x - 4| = -4
|x - 4| = -1
The absolute value cannot be negative, hence the function has no x-intercepts.
What is the y-intercept of a function?The x-intercept of a function is the value of y = f(x) when x = 0, hence:
f(0) = 4|0 - 3| + 4 = 12 + 4 = 16.
The y-intercept is of y = 16.
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