Instructions: For the following quadratic functions, write the function in factored form and then find the -intercepts, axis of symmetry, vertex, and domain and range. Round to one decimal place, if necessary.
Answers
Factored form: y = (x+1)(x-8)
x-intercept: (-1, 0) and (8, 0)
Axis of symmetry: x = 7/2
Vertex: (7/2, -81/4)
Domain: All real numbers
Range: y ≥ -81/4
Explanations:Given the quadratic equation expressed as:
[tex]y=x^2-7x-8[/tex]Factorize
[tex]\begin{gathered} y=x^2-8x+x-8 \\ y=x(x-8)+1(x-8) \\ y=(x+1)(x-8)\text{ Factored form} \end{gathered}[/tex]The x-intercept is the point where y= 0. Substitute y = 0 into the factored form
[tex]\begin{gathered} (x+1)(x-8)=0 \\ x=-1\text{ }and\text{ }8 \\ The\text{ x-intercept are \lparen-1, 0\rparen and \lparen8, 0\rparen} \end{gathered}[/tex]The axis of symmetry of the equation is given as x = -b/2a where:
a = 1
b = -7
Substitute:
[tex]\begin{gathered} axis\text{ of symmetry:}x=\frac{-(-7)}{2(1)} \\ axis\text{ of symmetry: }x=\frac{7}{2} \end{gathered}[/tex]The vertex form of the equation is in the form (x-h)^2+k where (h, k) is the vertex. Rewrite in vertex form:
[tex]\begin{gathered} y=x^2-7x-8 \\ y=x^2-7x+(-\frac{7}{2})^2-(-\frac{7}{2})^2-8 \\ y=(x-\frac{7}{2})^2-\frac{49}{4}-8 \\ y=(x-\frac{7}{2})^2-\frac{81}{4} \end{gathered}[/tex]The vertex of the function will be (7/2, -81/4)
The domain are the independent values of the function for which it exists. The domain of the given quadratic function exists on all real number that is:
[tex]Domain:(-\infty,\infty)[/tex]The range of the function are the dependent value for which it exist. For the given function, the range is given as:
[tex]Range:[-\frac{81}{4},\infty)[/tex]Related Questions
Suppose that the functions and s are defined for all real numbers x as follows r(x) = 4x; s(x) = 3x ^ 2 Write the expressions for (s + r)(x) and (sr)(x) and evaluate (s - r)(2) .
Answers
Solution
Given
[tex]\begin{gathered} r(x)=4x \\ \\ s(x)=3x^2 \end{gathered}[/tex]Then
[tex]\begin{gathered} (s+r)(x)=s(x)+r(x)=4x+3x^2 \\ \\ \Rightarrow(s+r)(x)=4x+3x^2 \end{gathered}[/tex][tex](s\cdot r)(x)=s(r(x))=s(4x)=3(4x)^2=3\times16x^2=48x^2[/tex][tex](s-r)(2)=s(2)-r(2)=3(2)^2-4(2)=12-8=4[/tex](-25) + (42) + (-62) + (20) =
Answers
Answer
The answer = -25
Explanation
As long as we note that
(+) × (-) = (-)
(-) × (+) = (-)
We can easily solve this,
(-25) + (42) + (-62) + (20)
= -25 + 42 - 62 + 20
= 17 - 62 + 20
= -45 + 20
= -25
Hope this Helps!!!
Write the first six terms of each arithmetic sequence:a,=200d=20
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Recall that the nth term of an arithmetic sequence is as follows:
[tex]\begin{gathered} a_n=a_1+d(n-1), \\ where\text{ }a_1\text{ is the first element and d is the common difference between terms.} \end{gathered}[/tex]We know that:
[tex]\begin{gathered} a_1=200, \\ d=20. \end{gathered}[/tex]Therefore:
1) The second term of the given arithmetic sequence is:
[tex]a_2=200+20(2-1),[/tex]simplifying the above result we get:
[tex]a_2=200+20(1)=220.[/tex]2) The third term of the given arithmetic sequence is:
[tex]a_3=200+20(3-1)=200+20(2)=240.[/tex]3) The fourth therm is:
[tex]a_4=200+20(4-1)=200+20(3)=260.[/tex]4) The fifth term is:
[tex]a_5=200+20(5-1)=200+20(4)=280.[/tex]5) The sixth term is:
[tex]a_6=200+20(6-1)=200+20(5)=300.[/tex]Answer: The first six terms of the given sequence are:
[tex]200,\text{ }220,\text{ }240,\text{ }260,\text{ }280,\text{ }300.[/tex]A woman wants to measure the height of a nearby building. She places a 9ft pole in the shadow of the building so that the shadow of the pole is exactly covered by the shadow of the building. The total length of the building shadow is 117ft, and the pole casts a shadow that is 6.5 ft long. How tall is the building? Round to the nearest foot.
Answers
ANSWER
[tex]162ft[/tex]EXPLANATION
Let us make a sketch of the problem:
Let the height of the building be H.
The triangles formed by the shadows of the building and the pole are similar triangles.
In similar triangles, the ratios of the corresponding sides of the triangles are equivalent.
This implies that the ratio of the length of the shadow of the pole to the pole's height is equal to the ratio of the length of the shadow of the building to the building's height.
Hence:
[tex]\frac{6.5}{9}=\frac{117}{H}[/tex]Solve for H by cross-multiplying:
[tex]\begin{gathered} H=\frac{117\cdot9}{6.5} \\ H=162ft \end{gathered}[/tex]That is the height of the building.
Could you please help me with questions 36 & 37?
Answers
We have to determine if the functions are linear or not.
We can do this by rearranging the equations in this form:
[tex]y=mx+b[/tex]where m and b are constants.
NOTE: There are many ways to prove that a function is linear, but this is the easiest for this question.
36.
[tex]\begin{gathered} x+\frac{1}{y}=7 \\ \frac{1}{y}=-x+7 \end{gathered}[/tex]As this function can not be written in the form y=mx+b, then it is not linear.
37.
[tex]\begin{gathered} \frac{x}{2}=10+\frac{2y}{3} \\ \frac{x}{2}-10=\frac{2y}{3} \\ \frac{2y}{3}=\frac{1}{2}x-10 \\ y=\frac{3}{2}(\frac{1}{2}x-10) \\ y=\frac{3}{4}x-15 \end{gathered}[/tex]This function is now in the form y=mx+b, where m=3/4 and b=-15. Then, this function is a linear function.
Answer:
36. Non-linear.
37. Linear.
find bounds on the real zeros of the polynomial functionf(x)= 17x^4 + 17x^3 - x^2 - 68x - 68
Answers
In order to identify bounds on the real zeros of this polynomial, first we need to find tendencies about the signal of f(x)
We know that the two term with highest degree is being multiplied by a positive coefficient. Therefore, we can initially conclude the f(x) tends to positive infinite as x grows either positive or negative.
We can check that, for x = -2, the first term is 272, and the remaining thermis togheter are given by:
[tex]17\cdot(-2)^3-(-2)^2-68\cdot(-2)-68=-72[/tex]Then for x < -2, and for also for x > 2, we can state for sure that f(x) remains always positive.
Then, any possible roots must lies in the intervel (-2,2)
e can also chegck that, for x = 1, f(x) = -103, and, for x = -1, f(x) = -1.
Therefore, f(x) must have a root between
asymptoteg(x) = -3*2^x+5
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determine if the statement is true or false if it is false explain why must be changing the statements and make it true if it's true explain why you believe to be true the common difference for a geometric sequence given 5:1 -1 negative 3 negative 5 negative 7 is 2
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The common difference can be determined by subtracting the first term with the second term, second term with the third term, and so forth.
In this case we have:
[tex]1-(-1)=1+1=2[/tex][tex]-1-(-3)=-1+3=2[/tex][tex]-3-(-5)=-3+5=2[/tex]In this way we can determine that the common difference of the sequence is 2, so the statement is true.
The probably of selecting a blue pin is 18/25. The chance of selecting a blue pin is _________A.) likely B.) unlikelyC.) impossible
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We have the following:
The probability is as follows
[tex]p=\frac{18}{25}=0.72[/tex]That is, we can say that the probability of selecting selecting a blue pin occurs 72% of the time or 18 times out of 25 attempts, therefore we can conclude that it is likely to happen.
The answer is A) likely
Give the following numberin Base 10.1215 = [ ? ]10
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To convert a number in base five to base ten, we shall use the expanded notation with the place value of each of the base five numbers.
The procedure is shown below;
[tex]\begin{gathered} 121_5 \\ \text{Assign place values starting from the right to the left} \\ \text{That is 0, 1 and 2.} \\ We\text{ now have}; \\ (1\times5^2)+(2\times5^1)+(1\times5^0) \end{gathered}[/tex]We can now simplify this as follows;
[tex]\begin{gathered} (1\times25)+(2\times5)+(1\times1) \\ =25+10+1 \\ =36 \end{gathered}[/tex]ANSWER:
[tex]121_5=36_{10}[/tex]what is x? how would i find the value of x?
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Given:
Find-:
The value of "x."
Explanation-:
Use a trigonometric is:
[tex]\tan\theta=\frac{Perpendicular}{\text{ Base}}[/tex]In a triangle:
[tex]\begin{gathered} \text{ Angle}=x \\ \\ \text{ Base }=3 \\ \\ \text{ Pespendicular }=4 \end{gathered}[/tex]The value of "x" is:
[tex]\begin{gathered} \tan\theta=\frac{\text{ Perpendicular}}{\text{ Base}} \\ \\ \tan x=\frac{4}{3} \\ \\ x=\tan^{-1}(\frac{4}{3}) \\ \\ x=53.13 \end{gathered}[/tex]So, the angle is 53 degree
Complete the table using the equation y = 7x +4. NO -1 0 1 2
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We will have the following:
*x = -1 => y = 7(-1) + 4 = -3
*x = 0 => y = 7(0) + 4 = 4
*x = 1 => y = 7(1) + 4 = 11
*x = 2 => y = 7(2) + 4 = 18
*x = 3 => y = 7(3) + 4 = 25
I think you have to create a tree diagram PLS HELP
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Given a coin and spinner
We need to flip the coin once and spin the spinner once
so, the tree diagram will be as following :
So, the all possible outcomes will be :
H1 , H2 , H3 , H4 , H5 , T1 , T2 , T3 , T4 , T5
Which of the following could be the end behavior of f(x) = -x6 + 3x4 + 8x3 – 4x2 – 6? f(x) → ∞ as x → ±∞f(x) → -∞ as x → ±∞f(x) → -∞ as x → -∞ and f(x) → ∞ as x → ∞f(x) → ∞ as x → -∞ and f(x) → -∞ as x → ∞
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The given function f is given by:
[tex]f(x)=-x^6+3x^4+8x^3-4x^2-6[/tex]Therefore, as:
[tex]x\to\infty,f(x)\to-\infty[/tex]and
[tex]\begin{gathered} \text{ as } \\ x\to-\infty,f(x)\to-\infty \end{gathered}[/tex]Therefore, the correct answer is:
f(x) → -∞ as x → ±∞
The points (2,-2) and (-4, 13) lie on the graph of a linear equation. What isthe linear equation? *
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Answer:
[tex]y=-\frac{5}{2}x+3[/tex]Explanation:
Given the two points on the graph to be (2, -2) and (-4, 13), we can use the point-slope form of the equation of a line below to write the required linear equation;
[tex]y-y_1=m(x-x_1)[/tex]where m = slope of the line
x1 and y1 = coordinates of one of the points
Let's go ahead and determine the slope of the line;
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{13-(-2)}{-4-2}=\frac{13+2}{-6}=-\frac{15}{6}=-\frac{5}{2}[/tex]Let's go ahead and substitute the value of the slope into our point-slope equation using x1 = 2 and y1 = -2;
[tex]\begin{gathered} y-(-2)=-\frac{5}{2}(x-2) \\ y+2=-\frac{5}{2}x+5 \\ y=-\frac{5}{2}x+5-2 \\ y=-\frac{5}{2}x+3 \end{gathered}[/tex]consider the discrete random variable x given in the table below calculate the mean variance and standard deviation of eggs also calculate the expected value of x around solution to three decimal places if necessary
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The mean and the expected value are computed as follows:
[tex]\mu=\sum ^{}_{}x_i\cdot P(x_i)[/tex]Substituting with data:
[tex]\begin{gathered} \mu=1\cdot0.07+6\cdot0.07+11\cdot0.08+15\cdot0.09+18\cdot0.69 \\ \mu=0.07+0.42+0.88+1.35+12.42 \\ \mu=15.14 \\ E(x)=15.14 \end{gathered}[/tex]The variance is calculated as follows:
[tex]\sigma^2=(x_i-\mu)^2\cdot P(x_i_{})[/tex]Substituting with data:
[tex]\begin{gathered} \sigma^2=(1-15.14)^2\cdot0.07+(6-15.14)^2\cdot0.07+(11-15.14)^2\cdot0.08+(15-15.14)^2\cdot0.09+(18-15.14)^2\cdot0.69 \\ \sigma^2=(-14.14)^2\cdot0.07+(-9.14)^2\cdot0.07+(-4.14)^2\cdot0.08+(-0.14)^2\cdot0.09+2.86^2\cdot0.69 \\ \sigma^2=199.9396\cdot0.07+83.5396\cdot0.07+17.1396\cdot0.08+0.0196\cdot0.09+8.1796\cdot0.69 \\ \sigma^2=26.860 \end{gathered}[/tex]And the standard deviation is the square root of the variance, that is:
[tex]\begin{gathered} \sigma=\sqrt[]{\sigma^2} \\ \sigma=\sqrt[]{26.8604} \\ \sigma=5.183 \end{gathered}[/tex]The function [tex]f(t) = 1600(0.93) ^{10t} [/tex]represents the change in a quantity over t decades. What does the constant 0.93 reveal about the rate of change of the quantity?
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Given the function;
[tex]f(t)=1600\cdot(0.93)^{10t}[/tex]The function shows exponential decay with a rate of 0.93
So, the quantity will decrease each year with the rate of 0.93
A skydiver jumps out of a plane form a certain height. The graph below shows their height h in meters after t seconds. What is the skydiver’s initial height?
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The x-intercept and the slope Set y=0 giving 0=10x+3000 x=300010=300 seconds, or precisely 5 minutes.
How to you interpret the x-intercept and the slope?Y is described as being "above ground" in height. He will get into problems if he descends below ground!
In regard to time x, the right hand side (RHS) has negative 10x, which decreases 3000. Consequently, the starting height must be 3000 feet. Moreover, it is the y-intercept.
The number of feet that the parachutist will descend in x seconds must be the -10x. It is also the graph's gradient or slope.
The graph's intersection with the x-axis indicates that y=0 and y is the height above the earth.
The point when the item is about to strike the ground is therefore the x-intercept.
The rate of descent, or slope, remains constant as time (x) passes.
The rate of fall is actually a extremely rounded value In metric form, I believe the speed to be 9.81 meters per second per second, to two decimal places.
It is approximately 32.2 feet per second per second in imperial form.
to determine how long it will take to reach the ground.
Set y=0 giving 0=10x+3000 x=300010=300 seconds, or precisely 5 minutes.
The complete question is : A skydiver parachutes to the ground. The height y (in feet) of the skydiver after x seconds is y=−10x+3000, how do you graph and Interpret the x-intercept and the slope?
To learn more about x intercept refer to:
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Answer 5 mins
becasue i solved in. my phone and my teacher said its right
Question 13 of 19 Use the elimination method to solve the system of equations. Choose the correct ordered pair x+y=9 x-y=7 O A. (8,1) B. (12,-3) C. (16, -7) D. (7.0) SUBMIT
Answers
The solution for the system of equations is x = 8, y = 1
Explanation:Given the pair of equations:
x + y = 9 .....................................................(1)
x - y = 7 ......................................................(2)
To solve by elimintation, add (1) and (2) to eliminate y
2x = 16
x = 16/2 = 8 ............................................(3)
Subtract (2) from (1) to eliminate x
2y = 2
y = 2/2 = 1 ................................................(4)
(3) and (4) form the solution of the equation.
x = 8, y = 1
On Thursday Tyler‘s math teacher helped him write the expression T equals -2 parentheses 3+ age parentheses to represent the temperature change for that day indicate all the expressions below the equivalent to T equals negative age parentheses 3+ H parentheses
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t = - 2 (3 + h )
Step 1: Expand the parenthesis so that -2 multiplies all the terms in the bracket
t = -2 x 3 - 2 x h
t = - 6 - 2h
Comparing the answer to the options provided
Option C is the best option
If the area of a trapezoid is 46 sq. cm. and its height is 4 cm. Find the shorter base if itslonger base is 15 cm.
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We are given the following information about a trapezoid.
Area = 46 sq cm
Height = 4 cm
Longer base = 15 cm
We are asked to find the shorter base of the trapezoid.
Recall that the area of a trapezoid is given by
[tex]A=\frac{a+b}{2}\cdot h[/tex]Let us substitute the given values and solve for the shorter side of the trapezoid.
[tex]\begin{gathered} 46=(\frac{a+15}{2})\cdot4 \\ 2\cdot46=(a+15)\cdot4 \\ 92=(a+15)\cdot4 \\ \frac{92}{4}=a+15 \\ 23=a+15 \\ a=23-15 \\ a=8\; cm \end{gathered}[/tex]Therefore, the shorter base of the trapezoid is 8 cm
Hi I’m I don’t understand a variable of what this is saying.
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given expression to simplify,
[tex]q^{-3}r^0s^{-1}\cdot\: q^3r^{-9}s^0[/tex][tex]\begin{gathered} q^{-3}r^0s^{-1}\cdot\: q^3r^{-9}s^0 \\ q^{-3}\cdot q^3=1 \\ \: =r^0s^{-1}r^{-9}s^0 \\ =1\cdot\frac{1}{r^9}s^{-1}s^0 \\ =1\cdot\frac{1}{r^9}\cdot\frac{1}{s} \\ =\frac{1}{r^9}\cdot\frac{1}{s} \\ =\frac{1}{r^9s} \\ =r^{-9}s^{-1} \end{gathered}[/tex]EXERCISE Carlos is jogging at a constant speed. He starts a timer when he is 12 feet from his starting position. After 3 seconds, Carlos is 21 feet from his starting position. Write a linear equation to represent the distance d of Carlos from his starting position t seconds after starting the timer.
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The linear equation that represent the distance d of Carlos from his starting position is d=3t+12 where d denotes the distance and t denotes the time.
What is the meaning of speed?
The speed at which an object's location changes in any direction. The distance travelled in relation to the time it took to travel that distance is how speed is defined.
Given that when Carlos is 12 feet from his starting position, starts a timer.
He is 21 feet from his starting position after 3 s.
He covers (21 - 12) = 9 feet in 3 second.
The speed of an object is the distance that covers in unit time.
The Carlos's speed is 9/3 = 3 feet/s.
After t seconds, he covers (3×t) = 3t feet.
The distance of his from his starting position after t seconds is (3t + 12) feet.
The linear equation is d = 3t + 12.
To learn more about linear inequality, click on below link:
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From the diagram below, if side AB is 48 cm., side DE would be ______.
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The triangle ABC is similar to the triangle DCE.
Hence, we need to find a proportion to find side DE:
If side AB = 48, it will represent the double value of the side DE.
Hence, DE = 48/2 = 24
The correct answer is option b.
i need help on this question
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a) Based on the naming of the triangle, the line segment WR is congruent with the line segment PL.
[tex]\bar{WR}\cong\bar{PL}[/tex]b) Based on the naming of the triangle, the third point for triangle BGT is T. The third point for the other triangle, DSN, is N. Hence, the angle for T is congruent with angle N.
[tex]\angle T\cong\angle N[/tex]c) The name of the triangle RHK is rearranged as KRH. This means that name of the triangle WVO can also be rearranged as OWV and is congruent with the triangle KRH.
[tex]\Delta KRH\cong\Delta OWV[/tex]Dominick weighs 30 pounds more than his sister. Together they weigh 330 pounds. How much does Dominick weigh?
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Domik weights 30 pounds more than his sister
Together they weigh 330 pounds
Let "x" represent the weight of Dominik's sister, then his weight can be expressed as "x+30"
And their total weight can be calculated as:
[tex]x+(x+30)=330[/tex]From this expression you can calculate the value of x:
[tex]\begin{gathered} 2x+30=330 \\ 2x+30-30=330-30 \\ 2x=300 \\ \frac{2x}{2}=\frac{300}{2} \\ x=150 \end{gathered}[/tex]x=150 pounds
x+30=150+30=180 pounds
Dominik weights 180pounds and his sister 150 pounds
If everyone had the same body proportion your weight in pounds would vary directly with the cube root of your height in feet according to Wikipedia the most recent statics available in 2009 indicated that the average height and weight for an adult male in the United States is 5 feet 9.4 inches in 191 pounds
Answers
Given
Height
5 ft 9.4 inches
Weight
191 pounds
Procedure
Let's calculate the equation to define the weight of a person. The structure of the equation would be as follows:
[tex]w=kh^3[/tex]Replacing the values to calculate k
[tex]\begin{gathered} 191=k(5.7833)^3 \\ k=\frac{191}{5.78^3} \\ k=0.98 \end{gathered}[/tex]The equation would be
[tex]w=0.98h^3[/tex]The median for the set of six ordered data values is 28.5.7,12,23,_,41,49What is the missing value
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Given data:
• 7,12,23,,X, ,41,49
• Median : 28.5
• X=?
→When we divide the data set into three equal parts , as in (7;12) (23,X) and (41;49)
→ We get that the median is (23+x )/2= 28.5
23+x = (28.5*2 )
x = 57 -23
x =34Check : 34+23 = 57/2 = 28.5 This means that x = 34 is correct.What Is 3×10³=3×10×10×10=
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3 x 10 ^3 = 3 x 10 x 10 x10 = 3 x 100 = 300
100 x 3 = 300 = 3 x 10 x 10 x 10
300
A stick is 10 1/5 inches in length. A carpenter will cut it into shorter pieces, each 1 2/15 inches in length. How many pieces will the stick be cut into?
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Answer:
9 pieces
Explanation:
From the question, we're told that the length of the stick is 10 1/5 inches, let's convert the mixed fraction into an improper fraction;
[tex]10\frac{1}{5}=\frac{51}{5}[/tex]Also, we're told that the stick was cut into shorter pieces of length 1 2/15 inches each. Converting 1 2/15 into an improper fraction, we'll have;
[tex]1\frac{2}{15}=\frac{17}{15}[/tex]To determine the number of pieces that the stick will be cut into, we'll need to divide 51/5 by 17/15;
[tex]\frac{\frac{51}{5}}{\frac{17}{15}}=\frac{51}{5}\ast\frac{15}{17}=\frac{15}{1}\ast\frac{3}{17}=\frac{153}{17}=9[/tex]