We are to determine the equation of line by interpreting tabulated results between an independent variable ( x ) and a dependent variable ( y ).
A function is usually expressed as follows:
[tex]y\text{ = f ( x )}[/tex]The above notation gives us the output ( y ) which is a function of input variable ( x ). This means that whatever relationship these two variables have the value of output ( y ) is related to the imput variable ( x ).
We are given a table/list of values of output ( y ) corresponding to each value of input variable ( x ) as follows:
Input ( x ) Output ( y )
3 -5
6 -4
9 -3
There are a series of steps that we must take to arrive at the equation that relates two variables.
Step 1: Determine the type of relationship between two variables by intuition
The first step in the process is the hardest of all. We have to critically analyze each input value ( x ) and its corresponding output value ( y ) with successive pair of values.
There are many types of relationships possible ( polynomial order, exponential, logarithmic, trigonometric, radical, etc .. ).
We can conjure up a way by comparing outputs of successive values to determine the type of relationship possible.
So looking at the first value:
[tex]y\text{ = f ( 3 ) = -5}[/tex]The successive value:
[tex]y\text{ = f ( 6 ) = -4}[/tex]The next successive value:
[tex]y\text{ = f ( 9 ) = -3}[/tex]Here if scrutinize between each successive value of input variable ( x ) we see that there is a "3 unit step-up" in each pair of values i.e ( 3 -> 6 -> 9 ).
Next we compare each output values ( y ) for successive pairs. We see that with every step increase of 3 units in ( x ) value there is an increase of ( 1 ) unit in output value i.e ( -5 -> -4 -> -3 ).
Conclusion: Combing the result of above analysis we see that with each 3 step increase in input value ( x ) there is an increase in output value ( y ) by 1 unit.
This gives us the idea that the two variables are linearly related to one another.
Therefore, the type of relationship is:
[tex]\text{straight line }\text{ }[/tex]Step 2: Recall the equation for the type of relationship between two vairbales x and y
Once we have determined the type of relationship between two variables. We will have to resort to our equation bank and pluck out the corresponding equation that expresses a LINEAR relationship i.e equation of a straight line.
The slope-intercept form of a straight line is:
[tex]y\text{ = m}\cdot x\text{ + c}[/tex]Step 3: Determine the complete equation of function by defining the arbitrary constants.
The above equation is valid for all straight lines that express a linear relationship. However, we seek to find a unique straight line for the given set of points.
Every unique straight line equation would have either of the constants different. The constants defined in a striaght line equation are:
[tex]\begin{gathered} m\colon\text{ The slope( gradient ) of the line} \\ c\colon\text{ The y-intercept} \end{gathered}[/tex]To determine these constants we will use the given pairs of coordinates of input and output variables, x and y respectively.
To determine the slope (m) of an equation:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]The above expression relates the change in output value ( y ) with respect to change in input variable ( x ).
To determine the constant ( m ) we will use the conclusion from Step 1:
"3 step increase in input of ( x ) value there is an increase in output value ( y ) by 1 unit."
Therefore,
[tex]m\text{ = }\frac{+1}{+3}\text{ = }\frac{1}{3}[/tex]To determine the value of y-intercept ( c ). We will plug in the value of ( m ) into the general equation of a straight line written in step 2:
[tex]y\text{ = }\frac{1}{3}x\text{ + c}[/tex]Now, we will use any pair of input and output value.
[tex]x\text{ = 3 , y = -5}[/tex]Substitute the pair of values into the derived equation expressed above and solve for constant ( c ):
[tex]\begin{gathered} -5\text{ = }\frac{1}{3}\cdot(3)\text{ + c} \\ -5\text{ = 1 + c} \\ c\text{ = -6} \end{gathered}[/tex]Note: The above step implies that following equation must satisfy each and every data pair of point given to us ( table ). Or each and every value must lie on the line. For that each value must satisfy the equation of line.
Step 4: Write the complete equation of the relationship
Once we have evaluated the values of equation defining constants ( m and c ). We can simply plug in the values into the general equation relationship ( Linear - slope intercept form ) as follows:
[tex]m\text{ = }\frac{1}{3}\text{ , c = -6}[/tex]Therefore, the equation for the set of values given to us is:
[tex]\begin{gathered} \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{\frac{1}{3}\cdot x}\text{\textcolor{#FF7968}{ - 6}} \\ OR \\ y\text{ = }\frac{x\text{ - 18}}{3} \\ \textcolor{#FF7968}{3y}\text{\textcolor{#FF7968}{ = x - 18}} \end{gathered}[/tex]What is the solution set for: 4x - 3>29 *X> 4OX >8OX8OX > 3
Let us first solve the inequality
[tex]\begin{gathered} 4x-3>29\rightarrow \\ 4x>29+3=32 \\ 4x>32 \\ x>\frac{32}{4}=8 \\ x>8 \end{gathered}[/tex]So the answer is x>8
please help me with this calculate the volume of the cone and round it to the nearest hundredth
ANSWER:
The volumen of the cone is 667.07 cubic meters
The volume has the following formula
[tex]V=\frac{1}{3}\pi\cdot r^2\cdot h[/tex]replacing:
[tex]\begin{gathered} V=\frac{1}{3}\cdot3.1416\cdot7^2\cdot13 \\ V=667.07 \end{gathered}[/tex]Question 1] Find the mean for each set of data6,9,2,4,3,6,5
Given:
The given set of data is
[tex]6,9,2,4,3,6,5[/tex]To find: The mean for the given set of data?
Explanation:
We can calculate the mean of a given set of data by using the formula
[tex]Mean=\frac{Sum\text{ of the terms}}{Number\text{ of terms}}[/tex]Here,
We can calculate the sum of terms as
The sum of terms = 6+9+2+4+3+6+5
The sum of terms =35.
and
The number of terms = 7.
Now, by substituting the value in the above formula
Therefore,
[tex]\begin{gathered} Mean=\frac{35}{7} \\ \\ Mean=5 \end{gathered}[/tex]Hence, the mean = 5.
Answer: The mean for a given set of data is 5.
Determine whether the graph represents a linear, quadratic, or exponential function.o Exponentialo Linearo Quadratic
When we graph a linear equation we draw a line. When we want to graph a quadratic equation we draw a parabola, a parabola is a U-shaped curve. The graph of an exponential function grows or decays and has asymptotic values.
In this case, the first graph is a line, so it represents a linear equation.
The second graph is a U-shaped curve, then it represents a quadratic equation.
The third graph decays, then it represents an exponential equation.
Two numbers are called relatively prime if their greatest common divisor is $1$.
Grogg's favorite number is $10!$, the product of the integers from $1$ to $10$. (He pronounces it TEN!)
What is the smallest integer greater than $800$ that is relatively prime to Grogg's favorite number?
Answer:
Two numbers are called relatively prime if their greatest common divisor is $1$. Grogg's favorite number is the product of the integers from $1$ to $10$
Answer: The answer is 803. :) lol
Step-by-step explanation:
You gotta keep trying numbers. But the numbers cant be even because if it's even, it can be divided by 2! Try the odd numbers starting at 800. If the number is not divisible by any of the numbers from 1-10 then it's the correct number. Got it on my second try. LOL! I'm from AOPS like you. :) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Which number line represents the solution set for the inequality 2x - 6 2 6(x - 2) + 8?-1.5-1-0.500.51.5-1.5-1-0.500.51.55-0.500.51.5-1-0.500.51.5
Solve the inequality:
2x - 6 ≥ 6(x - 2) + 8
Operating the parentheses:
2x - 6 ≥ 6x - 12 + 8
Subtracting 6x:
2x - 6x - 6 ≥ - 12 + 8
Adding 6:
2x - 6x ≥ - 12 + 8 + 6
Simplifying:
- 4x ≥ 2
Dividing by -4 and changing the symbol:
x ≤ 2/(-4)
x ≤ -0.5
The solution in the number line starts at -0.5 and goes to the left with an arrow:
A car travels about 32 miles on 1 gallon of gas. While a truck drives about 260 miles on 9.75 gallons of gas. Which gets better gas mileage?
In order to determine which vehicle gets better gas mileage, we need to determine which of the two vehicles consumes less gas for each mile travelled. The car consumes 1 gallon of gas for every 32 miles. We shall now determine how many miles the truck covers to consume 1 gallon of gas.
[tex]\begin{gathered} \text{Truck} \\ 9.75gl=260\text{miles} \\ 1gl=\frac{260}{9.75}\text{miles} \\ 1gl=26.67miles \end{gathered}[/tex][tex]\begin{gathered} \text{Car} \\ 1gl=32\text{miles} \end{gathered}[/tex]Therefore, the results shows that the truck gets better gas mileage
I need help with this question but no one is helping me! Can someone please help me.
4) Say there are 500,000 people in a county, of which approximately 300,000 are registered voters. According to the pir chart below, what is an estimate to the nearest person of how many people in the county are registered independents? (10 points) Approximately people in this county are registered independents. Registered County Voters Other 64 Independent 40% - Democrat u Republican independent Other
Since the chart has the name "Registered Country Voters", it only includes the registered voters, that is the approximately 300,000 people. Since 16% are independet, we can take 16% of 300,000 to get the registered independets. that is, multiply 0.16 (16%) by 300,000:
[tex]0.16\cdot300,000=48,000[/tex]So, the answer is 48,000.
1. In a factory, a machine takes 5 minutes to stitch a book. How many machines are needed to stitch
50 books in 10 minutes?
Answer: just multiply 5 by 2
Step-by-step explanation:
Below, the two-way table is given for a classof students.FreshmenSophomore Juniors Seniors TotalMale4622Female 3463TotalIf a male student is selected at random, what is theprobability the student is a freshman.
29%
Explanations:Probability is the likelihood or chance that ab event will occur. Mathematically;
Probability = Expected/Total
From the given table, the total number of male student is the total outcome as shown:
Total outcome = 4 +6+ 2 + 2
Total outcome = 14
If a male freshmen is selected at random, the expected outcome will be 4.
Determine the probability
[tex]\begin{gathered} Pr(Freshman\text{ \mid Male})=\frac{Male\text{ freshmen}}{Total\text{ male studnt}} \\ Pr(Freshman\text{ \mid Male})=\frac{4}{14} \\ Pr(Freshman\text{ \mid Male})=0.2857\times100 \\ Pr(Freshman\text{ \mid Male})\approx29\% \\ \end{gathered}[/tex]Hence the probability that the male student is freshman is approximately 29%
n(n+1)(n+5) is multiple of?
n(n+1)(n+5) can be a multiple of 3.
Utilizing the mathematical induction formula for any n > N, we are going to demonstrate it.
P(n) = n(n+1) n(n+5) = 3d, where d N
P(1)=1(2)(6)=12, that is cleavable by three, for n=1.
Suppose P(k) is true.
Where m N k three +6k a pair of +5k=3m k, P(k)=k(k+1)(k+5)=3m.
3\s =−6k \s2\s −5k+3m
We will currently demonstrate that P(k+1) is true.
P(k+1)=K+1, K+2, K+6, K 3 +9 K 2, K twenty K + twelve
When we plug the worth of k three into the equation on top of, we tend to get 3m6k a pair of 5k)+9k a pair of +20k+12 =3.
m+3k \s2\s +15k+12\s=3
(m+k \s2\s +5k+4)
in which r=m+k a pair of + 5k + four
P(k+1) is often true since P(k) is often true.
Therefore, in step with the induction principle, P(n) is cleavable by three for all n N.
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1ptWhat value of c would make x2 – 14.c +ca perfect square?C=What is the factored form of this expression for this value of c?(.)?
The value of c would be 49 and the factored form will be;
[tex](x-7)^2[/tex]Here, we want to get the value of c that will make the expression a perfect square
We proceed as follows;
Divide the coefficient of x by 2 ;
That would be; -14/2 = -7
Now, square this value; that would be (-7)^2 = 49
The value 49 would make the expression a perfect square
Thus, we have the factored form as follows;
[tex](x^2-14x+49)\text{ = (x-7)(x-7)}[/tex]Carry your intermediate competations at least four decimal places were on your answer to 2 decimal places
Given that Z follows the standard normal distribution, and that the left-tailed probability for a z-score c is:
[tex]P(Z\leqslant c)=0.8531[/tex]Then, we can see that the value of c, round to two decimal places, is:
[tex]c=1.05[/tex]Answer the question right and i will give u brainiest
Answer:
21 is -2 away, 22 is- 1 away 28 is 5 away, and 29 is 6 away
Step-by-step explanation:
Answer:
-2
-2
-2
-1
-1
-1
-1
-1
5
6
a. Draw the figure and its reflection in the x-axis.
A(2,0), B(1,5), C(4,3)
Answer:
A: 2,0 B: 1,-5 C: 4,-3
Step-by-step explanation:
w
Solve for x usingcross multiplication.3x + 6 = 4x - 364=35x = [?]Enter
Solving for x,
[tex]\begin{gathered} \frac{3x+6}{4}=\frac{4x-3}{5} \\ \\ \rightarrow5(3x+6)=4(4x-3) \\ \rightarrow15x+30=16x-12 \\ \rightarrow30+12=16x-15x \\ \rightarrow42=x \\ \\ \Rightarrow x=42 \end{gathered}[/tex]There were 20 baby devils 4 born in 2013 on Maria Island. In 2014, the number born was 200% of the 2013 devils. How many were born in 2014?
There are 20 baby devils in 2013 on Maria Island
in 2014 200 % where born
the number of baby devils born in 2014 can be gotten by
20 -------- 100
x -----------200
by proper analysis x = (200 x 20)/100 = 4000/100 = 40
40 baby devils were born in 2014
Two fields of equal soil types and slope both had corn planted last year. It was the first year that the newly installed center pivot irrigation systems were used - one system per field. However, these systems were made by two different manufacturers.
Suspecting that one of the systems in particular is oversupplying water to the corn, the farmer set up sensors on the edge of the field adjacent to a drainage ditch to measure the total runoff in the two fields. The data from these two sensors is displayed below:
Water runoff (gallons)
Irrigation System #1
11.4 12.4 12.1 11.8 11.8 11.7 12.3 12.0 11.8
Irrigation System #2
13.0 12.4 12.4 12.1 12.3 12.1 12.5 12.0 12.1
The question of interest is whether these two systems produce significantly different runoff. Find the value of the test statistic for the this test (hint: this is a kind of t-test - read the problem carefully to infer exactly which kind).
More surprising was a recent report of severe erosion brought on by rain in a well-kept, drip-irrigated subtropical fruit orchard in Limpopo
It is widely known and has been extensively researched for many years how soil erosion occurs in rangelands and locations where crops are watered by rain. However, it appears that knowledge of and research on the severe soil erosion that occurs in irrigated regions is lacking. When center pivot irrigation was used on highly crusting, extremely unstable soils, such as those from the Eastern Cape and the Lowveld of Mpumalanga, major gully (donga) erosion ensued, according to information the author has previously and more recently received. After the irrigation was put in place, they grew quite fast. More surprising was a recent report of severe erosion brought on by rain in a well-kept, drip-irrigated subtropical fruit orchard in Limpopo.
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Xiao Mei covers a table with 8 rows of square unit tiles. There are 7 tiles in each row. What is the area that Xiao Mei covers in square units?
The area that Xiao Mei covers in square tiles is 56 sq. units.
Given,
In the question:
Xiao Mei covers a table with 8 rows of square unit tiles.
and, There are 7 tiles in each row.
To find the area that Xiao Mei covers in square tiles.
Now, According to the question:
Tables covers = 8 rows of sq. unit
In each row = 7 tiles
All you would have to do is multiply 8 times 7
it means, 8 x 7 = 56
Hence, The area that Xiao Mei covers in square tiles is 56 sq. units.
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The midpoint of overline PQ is M = (- 2, 1) . One endpoint is P = (- 5, - 1) Find the coordinates of the other endpoint , Q.
Answer;
The coordiantes of the point Q is;
[tex](1,3)[/tex]Explanation;
Here, we want to get the coordinates of the other endpoint
Mathematically, we can find the midpoint of two endpoints using the formula below;
[tex](x,y)\text{ = (}\frac{(x_1+x_2)}{2},\frac{(y_1+y_2)}{2})[/tex]The notation 1 and 2 represents the coordinates of the two end points
Now, by replacing the values given in the question, we have it that;
(x,y) is the coordinate of the point m which is (-2,1)
Now the first end point has the coordinates (-5,-1)
The second endpoint Q has the endpoint (x2,y2) which we want to calculate
We proceed to make the substitutions as follows;
[tex](-2,1)\text{ =}\frac{(-5+x_2)}{2},\frac{(-1+y_2)}{2}[/tex]We proceed to susbtitute the individual branches as follows;
[tex]\begin{gathered} -2\text{ = }\frac{-5+x_2}{2} \\ 2(-2)=-5+x_2 \\ -4=-5+x_2 \\ x_2=-4+5 \\ x_2\text{ = 1} \end{gathered}[/tex]Finally, we have;
[tex]\begin{gathered} 1=\frac{-1+y_2}{2} \\ 2(1)=-1+y_2 \\ 2+1=y_2 \\ y_2\text{ = }3 \end{gathered}[/tex]A group of friends wants to go to the amusement park tey have no more than $115 to spend on parking and admission parking is $15 and the tickets cst $25 per person including tax wrote and solve an inequality which can be used to determine x the number of people who can go to the amuement park
An inequality which can be used to determine x the number of people who can go to the amusement park is: 15 + 25x ≤ 115
Also, approximately 4 people can go to the amusement park.
In the given question,
cost for admission parking = $15
Tickets = $25 per person
Let x be the number of people those who go to the amusement park.
On parking and admission spent not more than $115
So we get an inequality,
15 + 25x ≤ 115
We solve above inequality.
25x ≤ 115 - 15
25x ≤ 100
25x / 25 ≤ 100/25
x ≤ 4
This means that approximately 4people can go to the amusement park.
Therefore, an inequality which can be used to determine x the number of people who can go to the amusement park is: 15 + 25x ≤ 115
Also, approximately 4 people can go to the amusement park.
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You buy five equally priced gifts for a total of $75 what was the price of each candle
Answer:
$15
Explanation:
To find the price of each candle, we need to divide $75 by 5, so the price of each candle is:
$75 ÷ 5 = $15
So, the answer is $15
Mrs. Jenkins is catering a school dance and wants to know how manystudents prefer tacos and how many prefer hot wings.296 students preferred tacos63% of students preferred hot wings.How many students in total were attending the dance?
Let x represent the total number of students that attended the dance. This means that
number of students that prefer tacos + number of students that prefer hot wings = x
Percentage is expressed in terms of 100. If 63% of the students preferred hot wings, then the number of students that preferred hotwings is
63/100 * x = 0.63x
If 296 students preferred tacos, it means that
293 + 0.63x = x
x - 0.63x = 296
0.37x = 296
x = 296/0.37
x = 800
Thus, a total of 800 students attended the dance
A 2070 kg space station orbits Earth at an altitude of 457 km. Find the magnitude of the force with which the space station attracts Earth. The mass and mean radius of Earth are 5.98×1024 kg and 6370 km, respectively.
The magnitude of the force with which the space station attracts Earth.
17714.8532 N
This is further explained below.
The magnitude of the force :An influence that has the potential to alter the motion of an item is referred to as a force.
An object having mass may experience a change in its velocity, often known as acceleration, when subjected to a force.
Generally, the equation for the magnitude of the force is
F = G.M.m/r2
Where
G=gravitational constant
M and m mass
r=radius
F=force of attraction
Therefore
6.67*10^{-11} * 5.98*10^{24} * 2070/((6370+457)2*10^6)
= 17714.8532 N
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The table shows how much money Tearra has after buying some games and represents a linear function.
Number of games 0 1 2 3
Amount of money $106.50 $71 $35.50 $0
What is the yintercept of the function shown in the table, and what does it represent?
A. The y-intercept is (0,106,50). Tearra has $106.50 before buying any games.
B. The y-intercept is (0,106.50). Tearra spends $106.50 per game.
C. The y-intercept is (3,0), and the 3 represents the number of games Tearra buys.
D. The y intercept is (3,0), and the 3 represents the amount of money Tearra spends per game.
Answer:
C. The y-intercept is (0, 150). Diego has $150 before buying any
Step-by-step explanation:
Brainlylist?
Step-by-step explanation:
Answer: The answer would be A.
Step-by-step explanation:
PLEASE HELP ITS DUE SOON! I DONT GET ANY OF THIS! HELP WOULD BE MUCH APPRECIATED! NEED THIS DONE BEEN STUCK ON THIS FOR WAY TO LONG!
YOU WILL GET 100 POINTS IF YOU HELP! QUESTION DOWN BELOW!!!!!
Answer:
1. given
2. isoceles triangle
3. PN = QN
4. Angle 3 = 4
5. MP = OQ
6. common
7. MP + PQ = OQ + PQ
8. not sure about this one sorry!
9. Angle Side Angle
Step-by-step explanation:
Simplify 6x(9-2)-6 divided by 2
i don't have any idea but ok JQJAJDJDJKSBA
Malcolm's Bakery Shop sells gourment cupcakes. The shop's cost for making 3 cupcakes is $6.75, and the cost of making 5 cupcakes is $9.75 . What is the shop's cost per minute?
A. $1.50
B. $1.95
C. $2.25
D. $3.00
17. Identify all the angles that are congruent to
Answer:
ad, bc, eh, bf, ae, jk, mp, pn, ln, lm