Answers
Answer:
y = -2x -3
Step-by-step explanation:
To write an equation we need two points that the line cuts through.
we then use the formula: Gradient = Rise/Run
the points chosen are (0,-3) and (1,-5)
Gradient = (-5--3)/(1-0)
= -2
to write the equation we need one known point, gradient and one unknown point say (x,y).
we therefore use the formula for gradient/slope
G = Rise/Run
-2 = y+3/x-0
-2x = y + 3
y = -2x -3
Answer:
y = - 2x - 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 ) = (0, - 3) and (x₂, y₂ ) = (1, - 5) ← 2 points on the line
m = [tex]\frac{-5-(-3)}{1-0}[/tex] = [tex]\frac{-5+3}{1}[/tex] = [tex]\frac{-2}{1}[/tex] = - 2
the line crosses the y- axis at (0, - 3 ) ⇒ c = - 3
y = - 2x - 3 ← equation of line
Related Questions
suppose that, in the base period, a college student buys 20 gallons of gasoline at $2 per gallon, 2 mugs for $13 each, and 4 movie tickets for $7 each. in the next month, the price of gasoline is $2.25 per gallon, mugs cost $12.50 each, and the price of a movie ticket is $7.50. the change in the price level from the first to the second month is:
Answers
The change in the price level from the first to the second month is 106.4
What is meant by price index?Price index: Price index, measure of relative price changes, consisting of a series of numbers arranged so that a comparison between the values for any two periods or places will show the average change in prices between periods or the average difference in prices between places.
A college student purchases 4 movie tickets for $7 each, 2 CDs for $13 each, and 20 gallons of gasoline at $2 a gallon during the base period.
Gas will cost $2.25 per gallon in the upcoming month, CDs will cost $12.50 each, and cinema tickets will cost $7.50
The price index
=(price in current year/price in last year) x 100
=100/94 x 100
=106.38 or 106.4
The change in the price level from the first to the second month is 106.4
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What is the solution to -5 < 9 - 6x
Answers
Answer:
x < 7/3
Step-by-step explanation:
[tex] - 5 < 9 - 6x[/tex]
[tex] - 14 < - 6x[/tex]
[tex] \frac{7}{3} > x[/tex]
[tex]x < \frac{7}{3} [/tex]
What is the minimum value of the function g(x) = |x – 4| + 8 on the interval [-2, 1]?
A. 14
B. 5
C. -2
D. 11
Answers
[tex]|x-4|[/tex] is non-negative, and vanishes when [tex]x=4[/tex].
For all [tex]x\in[-2,1][/tex], we have [tex]x<4[/tex], so by definition of absolute value,
[tex]g(x) = |x-4| + 8 = -(x-4)+8 = -x+4[/tex]
which is strictly decreasing on [-2, 1] since [tex]g'(x)=-1<0[/tex]. Then the minimum value of [tex]g[/tex] on [-2, 1] occurs at the endpoint [tex]x=1[/tex], so that
[tex]\min\left\{g(x) \mid -2\le x\le1\right\} = g(1) = |1-4| + 8 = \boxed{11}[/tex]
The standard form for a parabola with vertex (h,k) and an axis of symmetry of x=h is:(x-h)^2=4p(y-k)The equation below is for a parabola. Write it in standard form. When answering the questions type coordinates with parentheses and separated by a comma like this (x,y). If a value is a non-integer then type is a decimal rounded to the nearest hundredth. 5x^2-50x-4y+113=0 The value for p is: AnswerThe vertex is the point: AnswerThe focus is the point: AnswerThe directrix is the line y=Answer
Answers
Given the equation of a parabola:
[tex]5x^2-50x-4y+113=0[/tex]• You can rewrite it in Standard Form by following these steps:
1. Add the y-term to both sides of the equation:
[tex]\begin{gathered} 5x^2-50x-4y+113+(4y)=0+(4y) \\ \\ 5x^2-50x+113=4y \end{gathered}[/tex]2. Subtract the Constant Term from both sides of the equation:
[tex]\begin{gathered} 5x^2-50x+113-(113)=4y-(113) \\ \\ 5x^2-50x=4y-113 \end{gathered}[/tex]3. Divide both sides of the equation by 5 (the leading coefficient)
[tex]x^2-10x=\frac{4}{5}y-\frac{113}{5}[/tex]4. The coefficient of the x-term is:
[tex]b=-10[/tex]Then, you need to add this value to both sides:
[tex](\frac{-10}{2})^2=5^2[/tex]Therefore.
[tex]\begin{gathered} x^2-10x+5^2=\frac{4}{5}y-\frac{113}{5}+5^2 \\ \\ x^2-10x+5^2=\frac{4}{5}y+\frac{12}{5} \end{gathered}[/tex]5. Rewrite the equation as follows:
[tex]\begin{gathered} (x-5)^2=\frac{4}{5}(y+3) \\ \\ (x-5)^2=0.8(y+3) \end{gathered}[/tex]• Having the equation written in Standard Form:
[tex](x-h)^2=4p(y-k)[/tex]You can identify that:
[tex]4p=\frac{4}{5}[/tex]Solving for "p", you get:
[tex]p=\frac{4}{5\cdot4}=\frac{1}{5}=0.2[/tex]• You can identify that:
[tex]\begin{gathered} h=5 \\ k=-3 \end{gathered}[/tex]Therefore, the Vertex is:
[tex](5,-3)[/tex]• By definition, the Focus of a parabola that opens upward is given by:
[tex](h,k+p)[/tex]Then, in this case, this is:
[tex](5,-3+0.2)=(5,-2.8)[/tex]• By definition, the Directrix for a parabola that opens upward is given by:
[tex]y=k-p[/tex]Then, in this case, you get:
[tex]\begin{gathered} y=-3-0.2 \\ \\ y=-3.2 \end{gathered}[/tex]Hence, the answers are:
• Standard Form:
[tex](x-5)^2=0.8(y+3)[/tex]• Value for "p":
[tex]p=0.2[/tex]• Vertex:
[tex](5,-3)[/tex]• Focus:
[tex](5,-2.8)[/tex]• Directrix:
[tex]y=-3.2[/tex]
A copy machine makes 126 copies in 3 minutes and 30 seconds. How many copies does it make per minute?
Answers
Answer: 36 copies per minute
Step-by-step explanation:
126 copies in 3.5 minutes
We want x number of copies per 1 minute
[tex]\frac{126copies}{3.5minutes} = \frac{xcopies}{1minute}[/tex]
Cross multiply and we get
126 = 3.5x
Solve for x.
x = 126/3.5
x = 36
36 copies per minute
If a=6 and b=4, solve for c using the following equation:
c=√a² + b²
c=√/52
c=20
c=√/20
c=52
Answers
Answer:
22
Step-by-step explanation:
1. √a square+b square2.√6 square +4 square3.√36 +164.6+165.22Anya has 115 mangoes lauren has five times the the number of mangoes Anya has they put them together and sold them in the market at $450 to $15 the sold all the mangoes how much money was collected at the end of the sale
Answers
The total amount of money collected if Anya and Lauren sold their mangoes at $15 is $10,350
How to find total revenue made from selling the mangoes?Number of mangoes Anya has = 115 mangoesNumber of mangoes Lauren has = 5 × 115= 575 mangoes
Total number of mangoes = Number of mangoes Anya has + Number of mangoes Lauren has
= 115 + 575
= 690 mangoes
Amount each mangoes was sold = $15
Total amount of money collected = Total number of mangoes × Amount each mangoes was sold
= 690 × 15
= $10,350
So therefore, Anya and Lauren sold their total mangoes for $10,350
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What is an expression equivalent to 5(32) using the distributive property?
Drag and drop the appropriate number into each box.
2 5 10 12 30 500
Answers
Answer:
The answer is 5(30+2)!
The 20% off sale is a better deal than the $200 rebate or
$150 coupon for the $1,500 dining set. The Porters
budgeted $1,250 for a new dining room set. Explain why
this is the best deal for them. Are they under budget?
Answers
Answer:
20% off is better and it is the only offer which is under the budget.
20% off of the $1,500 dining set is $300, and the dining set price will be 1500 - 300 = $1200.
And by applying the $150 coupon, the dining set price will be $1500 - $150 = $1350.
Therefore 20% off is better and it is the only offer which is under the budget.
Step-by-step explanation:
A recent survey was given to determine students favorite school subjects. 6 students chose Math, 3 students chose Science, 4 students chose English and 7 students chose History. Predict the number of students out of 400 that would pick math as their favorite subject
Answers
Out of 400 students, 120 students would pick math as their favorite subject.
Data provided to us
fav subject students
Math 6
Science 3
English 4
History 7
Hence the total becomes 20 students
we have to calculate students whose favorite subject is math from 400 students
students with math as fav subject out of 20 students = 6
students with math as fav subject out of 400 students = (6 / 20) x 400
= 120 students
we can verify it as
%age of students out of 20 , who like math = 6/20 x 100
= 30 %
similarly, %age of students out of 400, who like math= 120/400 x 100
= 30 %
As we can see both the percentages are same. and the answer is verified .
So 120 students would pick math as their favorite subject out of 400 students.
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Shirley assumes that none of the marks had rubbed off between Monday and Tuesday.
(b) If Shirley's assumption is wrong, explain what effect this would have on your answer
to part (a).
Answers
Using algebra, we know that the total number of bees in her hive is 540.
What is Algebra?One of the many branches of mathematics is algebra. Algebra, which is a common thread throughout practically all of mathematics, is broadly defined as the study of mathematical symbols and the rules for using these symbols in formulas.
So,
Total no. of bees caught on Monday = 90Total no. of bees caught on Tuesday = 120The bees she marked = 20Then,
(A) Work out an estimate for the total number of bees in her hive
90 / x = 20 / 12090 / x = 1 / 690 x 6 = x540 = xThe estimated total no. of bees will be 540 bees.
(B) If Shirley’s assumption is wrong, explain what effect this would have on your answer to part (A).If Shirley's assumption is wrong, the answer would be too high and it would be overestimated.If the marks are rubbed off Shirley will have overestimated the number of bees (there will be fewer bees).Therefore, using algebra, we know that the total number of bees in her hive is 540.
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Use the digits 1-9 to create 3 equivalent ratios. Each digit can only be used once. _:_=_ _:_=_ _:_ _
Answers
Equivalent ratios are the ratios that are the same when we compare them. Three equivalent ratios are 1/2, 4/8, 3/6 in digits from 1 to 9.
What is Ratio?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other
Equivalent ratios are the ratios that are the same when we compare them.
The digits from 1 to 9 include 1,2,3,4,5,6,7,8,9.
Ratios are nothing but fractions.
Let 1/2 (one by two) is the first ratio.
Now 3/6 (three by six) is the second ratio
4/8 (four by eight) is the third ratio.
If we calculate the value of 1/2 = 0.5
The value of 3/6=1/2=0.5
The value of 4/8= 1/2=0.5
we get the same answer for all the ratios 1/2, 4/8 and 3/6 that is 0.5
That is 0.5 or 1/2.
Hence the three equivalent ratios are 1/2, 4/8, 3/6.
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Question 1
In each pair, some of the angles of two triangles in degrees are given. Use the
information to decide if the triangles are similar or not. Explain how you know.
Triangle A: 53, 71,; Triangle B: 53, 71,
• Triangle C: 90, 37,______; Triangle D: 90, 53,
• Triangle E: 63, 45,; Triangle F: 14, 71,
Triangle G: 121,____________; Triangle H: 70,
●
●
-
Answers
The equality of the respective angles and the proportionality of the corresponding sides determine if two triangles are comparable.
Triangle A: 53, 71, Triangle B: 53, 71; Similar
Triangle C: 90, 37, Triangle D: 90, 53; Similar
Triangle E: 63, 45 Triangle F:14, 71; Not similar
Triangle G, 121, Triangles H: 70, Not similar
If two of one triangle's angles are congruent with two of the
angles of a different triangle, then the two triangles are comparable.
Similarity hypothesis, angle, and AA
Given that: The interior product of the AA similarity hypothesis
The angle sum property of a triangle states that the angles of a triangle are 180°,
Consequently, the third angle is also congruent when two angles are
congruent
53, 71 in Triangle A and 53, 71 in Triangle B; similar
Based on the AA similarity postulate, the triangles are similar.
Triangles C (90, 37) and D (90, 53) have similar angles.
Supplied that the given angles are identical to each of their complimentary angles, the triangle is similar.
Triangle F: 14, 71 Triangle E: 63, 45; not comparable
There are no parallel triangles.
Triangles H: 70, not identical to Triangle G's 121,
The details are insufficient.
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What is the volume of this sphere? Use : 3.14 and round your answer to the nearest hundredth. 9 yd cubic yards
Answers
3,058.08 cubic yards
Explanation:radius = 9 yd
π = 3.14
Volume of a sphere = 4/3 πr³
Volume of sphere = 4/3 × 3.14 × 9³
Volume of sphere = 3052.08
To the nearest hundredth, the volume 3,058.08 cubic yards
Please help it’s due tomorrow morning
Explain the step that is used to solve the equation
Answers
x/2 = 8
•2 •2
x = 16
Factorise:
x² + 6x - 16
Answers
Answer:
(x-2)(x+8)
Step-by-step explanation:
can anyone help me pls?
1. Which kind of model do you think best represents the profit for each business?
2. Find a function that models the profit you can expect, based on your projections, for each business. You can round your values to the nearest thousandth, as needed.
Answers
Lawn care is kind of model do you think best represents the profit for each business.
What is profit?
The financial gain a business has when revenue exceeds costs and expenses is repeatedly alluded to as profit. One cup of lemonade, for instance, costs a child at a lemonade stand one quarter. She then charges $2 for the drink. She made $1.75 in profit just on cup of lemonade.Dog walker = 5 + 12 + 23 + 50 + 62 + 82 = $234
House sitter = 11 + 19 + 29 + 38 + 50 + 73 = $220
Lawn care = 142 + 150 + 165 + 141 + 120 + 91 = $809
Lawn care is kind of model do you think best represents the profit for each business.
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Which are the foci of the ellipse represented by 25x^2 + 16y² = 400?
O (-3,0) and (3, 0)
O(-5, 0) and (5, 0)
O (0, 3) and (0, -3)
O (0, 5) and (0, -5)
Answers
The foci of the given ellipse are (D) (0, 5) and (0, -5).
What is an ellipse?In mathematics, an ellipse is a planar curve with two focal points, where the total of the two distances from any point on the curve to the focal points is a constant. It generalizes the shape of a circle, a unique variety of ellipse in which the two focus points coincide. For an ellipse with its center at (0,0) and its axes parallel to the coordinate axes, one point can be used to infer three more points. The other three points are (a,b), (a,b), and (a,b) if your point is (a,b).So, the foci of the ellipse are:
Equation is 25x² + 16y² = 400.Now, plot it on the graph as follows:(Refer to the graph attached below)
We can easily see that the coordinates on the y-axis are (0, 5) and (0, -5).Therefore, the foci of the given ellipse are (D) (0, 5) and (0, -5).
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The correct question is shown below:
Which are the foci of the ellipse represented by 25x² + 16y² = 400?
A. (-3,0) and (3, 0)
B. (-5, 0) and (5, 0)
C. (0, 3) and (0, -3)
D. (0, 5) and (0, -5)
Express (23)−2 using exponents
Answers
The expression as an exponent is 21^1
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to evaluate the expression?The expression is given as
(23) - 2
Open the bracket in the above expression
So, we have the following equation
(23) - 2 = 23 - 2
Evaluate the difference in the above expression
So, we have the following equation
(23) - 2 = 21
The above expression can be rewritten as
(23) - 2 = 21^1
Hence, the equivalent expression is 21^1
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✨Hello! I need some help it's very urgent I will be marking brainliest✨
first blank: never, sometimes, always
second blank: less than or equal to negative 3, between -3 and 3, greater than or equal to 3, greater than or equal to -3, less than or equal to 3.
third blank: positive number, less than or equal to zero, negative number, negative number, a number greater than or equal to zero.
Answers
Answer:
Sometimes, Never, and Always.
(Sounds like a good movie title).
Step-by-step explanation:
See attached image.
-13x-2x + 3 = -5x - 17
Answers
Answer:
-15x - 3
I hope this helps :)
If f(x)= x2-5 and g(x)= -3x + 2, find f(g(x))
Answers
Answer:
D. 9x² - 12x - 1
Explanation:
To find f(g(x)), we need to replace x by g(x) as follows:
f(x) = x² - 5
f(g(x)) = (g(x))² - 5
Then, g(x) is equal to -3x + 2, so replacing g(x) by (-3x + 2), we get:
f(g(x)) = (g(x))² - 5
f(g(x)) = (-3x + 2)² - 5
f(g(x)) = (-3x)² + 2(-3x)(2) + 2² - 5
f(g(x)) = 9x² - 12x + 4 - 5
f(g(x)) = 9x² - 12x - 1
Therefore, the answer is D. 9x² - 12x - 1
Find m∠1 and m∠2. Tell which theorem can be used.
Answers
Answer:
m∠1 = 140 by the Alternate Interior Angles Theorem
m∠2 = 40 by the Consecutive Interior Angles Theorem
Step-by-step explanation:
solve for w -5.36=10+w/6
Answers
An expression is a collection of variables and constants that are bound by some mathematical operation the value for w for the given expression w -5.36=10+w/6 is 18.432.
What is an expression?A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
A statement expressing the equality of two mathematical expressions is known as an equation.
As per the given expression,
w - 5.36 = 10 + w/6
Subtract w/6 both sides,
w - w/6 - 5.36 = 10 + w/6 - w/6
5w/6 = 10 + 5.36
w = 18.432
Hence "The value for w for the given expression w -5.36=10+w/6 is 18.432".
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A. 20 B. 30 C. 23 D. 39 I need a little help with this question
Answers
Given: A table that shows the result of a survey from 35 randomly selected students at Courtney's school.
Required: To determine the number of votes Rockzilla is expected to receive from a population of 115 students at Courtney's school.
This is achieved thus:
First, we determine the probability of Rockzilla's votes as follows:
[tex]\begin{gathered} Prob.=\frac{Number\text{ }of\text{ }required\text{ }outcome}{Number\text{ }of\text{ }total\text{ }outcome} \\ Prob.=\frac{6}{35} \end{gathered}[/tex]Next, we determine the number of votes Rockzilla is expected to receive as follows:
[tex]\begin{gathered} Expected\text{ }votes=\frac{6}{35}\times115 \\ Expected\text{ }votes\approx20 \end{gathered}[/tex]Hence, the answer is:
[tex]20[/tex]Can someone please help me with this? THANKS!
Answers
Answer:
I think it's 171.5 in^3 but not sure :)
Step-by-step explanation:
Pls help I need work and a answer I have given an exsample
Answers
The original price of the jacket before sale amount is $45.
Original price of the jacket will be calculated by forming the expression relating mentioned price. Let us assume the original price of the jacket to be x.
Writing the equation representing original price, percentage and price after sale
80%×x = 36
Converting percentage to fraction
80/100×x = 36
Shifting the fraction to Right Hand Side of the equation
x = 36×100/80
Cancelling zero as it is common in numerator and denominator and performing division
x = 45
Therefore, according to the sale price, the original price of jacket is $45.
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Solve the equation for v.
v over 8 minus 4 equals negative 12
v = −128
v = −64
v = 16
v = 92
Answers
[tex] \frac{v}{8} - 4 = - 12 \\ \frac{v}{8} = - 12 + 4 \\ \frac{v}{8} = - 8[/tex]
Now I will use CROSS MULTIPLICATION
[tex]v = 8( - 8) \\ v = -6 4[/tex]
ATTACHED IS THE SOLUTION
Answer: v/8 - 4 = (-12)
-12 + 4 = (-8)
what divided by 8 is 8?
8/8 = 1 so 8 x 8 = 64
this means v = -64
64/8 - 4 = -12
- 64/8 is -8 and that minus 4 is -12
help help me pleaseeee pleaseeeee.
WITH SOLUTION.
Answers
Answer:
[tex]1) \:\:\: \dfrac{14x}{y}[/tex]
[tex]2) \:\:\:\dfrac{16x^2}{9y^2}[/tex]
[tex]3)\:\:\:10[/tex]
[tex]4)\:\:\:\dfrac{x-2}{x+4}[/tex]
Step-by-step explanation:
I will provide detailed explanation for the first problem and quicker steps to solve the others. You should be able to figure them out
Dividing a number(integer or fraction) by a fraction is equivalent to multiplying that number by the reciprocal of the divisor fraction
So if you have an expression
[tex]\dfrac{a}{b} \div \dfrac{c}{d}\\\\\implies \dfrac{a}{b} \times \dfrac{d}{c}[/tex]
Question 1
[tex]\dfrac{14x^5}{5y^2}\:\div \dfrac{2x^4}{10y}\\\\= \dfrac{14x^5}{5y^2} \times \dfrac{10y}{2x^4}\\\\[/tex]
[tex]=\dfrac{14x^5\cdot \:10y}{10y^2x^4}[/tex]
10 in the numerator and 10 in the denominator cancel:
[tex]=\dfrac{14x^5y}{y^2x^4}[/tex]
[tex]\dfrac{x^5}{x^4} = x^{5-4} = x \;\;\;\;\;\;\text{Since }\dfrac{x^a}{x^b}=x^{a-b}}[/tex]
Likewise
[tex]\dfrac{y}{y^2} = y^{1-2} = y ^{-1} = \dfrac{1}{y}[/tex]
Therefore
[tex]=\dfrac{14x^5\cdot \:10y}{10y^2x^4} = 14 \dfrac{x}{y} = \dfrac{14x}{y}[/tex]
Answer: Question 1
Question 2
[tex]\left(\dfrac{x}{y}\right)\left(\dfrac{4y}{3x}\right)\:\div \dfrac{6y^2}{8x^2}[/tex]
[tex]\dfrac{x}{y} \cdot \dfrac{4y}{3x} = \dfrac{4}{3}[/tex]
[tex]\dfrac{6y^2}{8x^2} = \dfrac{4y^2}{3x^2}[/tex]'
[tex]\left(\dfrac{x}{y}\right)\left(\dfrac{4y}{3x}\right)\:\div \dfrac{6y^2}{8x^2} \\\\= \dfrac{4}{3} \times \dfrac{3x^2}{4y^2}\\\\= \dfrac{16x^2}{9y^2}[/tex]
Question 3
[tex]\dfrac{\left(5\left(y+6\right)\right)}{2\left(xy-3\right)}\:\div \dfrac{\left(y\:+\:6\right)}{4\left(xy\:-\:3\right)}\\\\\\\dfrac{5(y + 6)}{2(xy-3)} \times \dfrac{4(xy-3)}{y+ 6}\\\\[/tex]
The y + 6 terms cancel out from numerator and denominator as do the xy - 3terms
[tex]= \dfrac{5}{2} \times \dfrac{4}{1}\\\\= \dfrac{20}{2} \\\\= 10[/tex]
Question 4
[tex]\dfrac{x^2-4}{x^2+5x+4}\:\div \dfrac{x+2}{x+1}\\\\=\dfrac{x^2-4}{x^2+5x+4}\:\times \dfrac{x+1}{x+2}[/tex]
[tex]x^2-4 = (x + 2)(x-2) \;\;\;\; \text{Since $a^2-b^2 = (a +b)(a-b)$}[/tex]
[tex]x^2 + 5x + 4\;\;\; \text{can be factored as: }\\\\(x + 1)(x + 4)[/tex]
Therefore
[tex]\dfrac{x^2-4}{x^2+5x+4}\:\div \dfrac{x+2}{x+1}\\\\\\=\dfrac{x^2-4}{x^2+5x+4}\:\times \dfrac{x+1}{x+2}\\\\\\= \dfrac{(x+2)(x-2)}{(x+1)(x+4)} \times \dfrac{x+1}{x+2}\\\\\\\\[/tex]
The x+2 terms cancel out as do the x + 1 terms giving the result
[tex]\dfrac{x-2}{x+4}[/tex]
Answer:
[tex]\textsf{1. \quad $\dfrac{14x}{y}$}[/tex]
[tex]\textsf{2. \quad $\dfrac{16x^3}{9y^2}$}[/tex]
[tex]\textsf{3. \quad $10$}[/tex]
[tex]\textsf{4. \quad $\dfrac{x-2}{x+4}$}[/tex]
Step-by-step explanation:
When dividing two fractions, multiply the first fraction by the reciprocal of the second fraction.
The reciprocal of a fraction is its inverse (swap the numerator and denominator).
Question 1[tex]\boxed{\begin{minipage}{3 cm}\underline{Exponent Rules}\\\\$\dfrac{a^b}{a^c}=a^{b-c}$\\\\$a^{-n}=\dfrac{1}{a^n}$\\\end{minipage}}[/tex]
[tex]\begin{aligned}\dfrac{14x^5}{5y^2} \div \dfrac{2x^4}{10y} & = \dfrac{14x^5}{5y^2} \times \dfrac{10y}{2x^4}\\\\& = \dfrac{14x^5 \times 10y}{5y^2\times 2x^4}\\\\& = \dfrac{140x^5y}{10x^4y^2}\\\\& = \dfrac{14x^5y}{x^4y^2}\\\\& = 14x^{5-4}y^{1-2}\\\\& = 14x^1y^{-1}\\\\& = \dfrac{14x}{y}\end{aligned}[/tex]
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Question 2[tex]\begin{aligned}\left(\dfrac{x}{y} \right)\left(\dfrac{4y}{3x} \right) \div \dfrac{6y^2}{8x^3}&=\dfrac{4xy}{3xy} \div \dfrac{6y^2}{8x^3}\\\\&=\dfrac{4}{3} \div \dfrac{6y^2}{8x^3}\\\\&=\dfrac{4}{3} \times \dfrac{8x^3}{6y^2}\\\\&=\dfrac{4 \times 8x^3}{3 \times 6y^2}\\\\&=\dfrac{32x^3}{18y^2}\\\\&=\dfrac{16x^3}{9y^2}\end{aligned}[/tex]
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Question 3[tex]\begin{aligned}\left[\dfrac{5(y+6)}{2(xy-3)} \right] \div \left(\dfrac{y+6}{4(xy-3)}\right) & = \left[\dfrac{5(y+6)}{2(xy-3)} \right] \times \left(\dfrac{4(xy-3)}{y+6}\right)\\\\& = \dfrac{5(y+6) \times 4(xy-3)}{2(xy-3) \times (y+6)}\\\\& = \dfrac{20(y+6)(xy-3)}{2(xy-3)(y+6)}\\\\& = \dfrac{20}{2}\\\\&=10\end{aligned}[/tex]
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Question 4Factor the numerator and denominator of the first fraction.
Apply the Difference of Two Squares to the numerator:
[tex]\implies x^2-4=x^2-2^2=(x-2)(x+2)[/tex]
Factor the denominator:
[tex]\begin{aligned}\implies x^2+5x+4 & =x^2+x+4x+4\\ & =x(x+1)+4(x+1)\\ & =(x+1)(x+4)\end{aligned}[/tex]
Therefore:
[tex]\begin{aligned}\left(\dfrac{x^2-4}{x^2+5x+4}\right) \div \left( \dfrac{x+2}{x+1} \right) & = \left(\dfrac{x^2-4}{x^2+5x+4}\right) \times \left( \dfrac{x+1}{x+2} \right)\\\\& = \left( \dfrac{(x-2)(x+2)}{(x+1)(x+4)} \right) \times \left( \dfrac{x+1}{x+2} \right)\\\\& = \dfrac{(x-2)(x+2)(x+1)}{(x+1)(x+4)(x+2)}\\\\& = \dfrac{x-2}{x+4}\\\\\end{aligned}[/tex]
What is the correct notation for a ray with endpoint E
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