Point P cannot be located on either the x-axis or y-axis, as neither a nor b is 0.
The coordinate system's two axes, the x-axis and the y-axis, constitute a region called a quadrant.
If both a and b have the same sign, point P could be located in one of the four quadrants:
For First quadrant (a > 0, b > 0)
For Second quadrant (a < 0, b > 0)
For Third quadrant (a < 0, b < 0)
For Fourth quadrant (a > 0, b < 0)
Hence, Point P cannot be located on either the x-axis or y-axis, as neither a nor b is 0.
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the diagonals of a trapezoid are perpendicular and have lengths $3$ and $4$. find the length of the median of the trapezoid.
The correct answer is 5/2. The median trapezoid is the line segment that bisects the area of the trapezoid and has its end points on the 2 bases of the trapezoid.
The median length is √( (3/2)^2 + (4/2)^2 ) = √(9/4 + 16/4) = √(25/4) = 5/2. So, the length of the median of the trapezoid is 5/2.
Since the diagonals of a median trapezoid bisect each other, the median of the trapezoid is the line segment that connects the midpoints of the diagonals. To find the median length, we can use the distance formula, which is the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates of the 2 points. In this case, the median length is the square root of the sum of the squares of 1/2 length of the diagonals.
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The length of the median of the trapezoid will be 5/2 units.
Given that the median trapezoid is the line segment that divides the trapezoid's area and has its ends on its two bases,
The median length will be:
√( (3/2)^2 + (4/2)^2 )
= √(9/4 + 16/4)
= √(25/4)
= 5/ units
So, the length of the median of the trapezoid is 5/2.
The median of a median trapezoid is the line segment that links the midpoints of the diagonals since the diagonals of a median trapezoid bisect one another. We may use the distance formula, which is the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates of the two points, to get the length of the median. The median length in this instance is equal to the square root of the sum of the squares of the diagonals' half-lengths.
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what is the value of the expression below? (15^1/5)^5
Answer:
15
Step-by-step explanation:
Given the expression:
[tex]\displaystyle{\left(15^{\frac{1}{5}}\right)^5}[/tex]
We can apply the law of exponent of:
[tex]\displaystyle{\left(a^m\right)^n = a^{mn}}[/tex]
Therefore, we will have:
[tex]\displaystyle{\left(15^{\frac{1}{5}}\right)^5 = 15^{\frac{5}{5}}}\\\\\displaystyle{\left(15^{\frac{1}{5}}\right)^5 = 15^{1}}\\\\\displaystyle{\left(15^{\frac{1}{5}}\right)^5 = 15}[/tex]
Hence, the value of expression is 15.
The relationship between time in months and height in inches for Kylie's plant is represented by the
equation y = 12x. The relationship for Glen's plant is represented by the equation 2y = 24x. Is there a
time when the two plants are the same height?
Both equations are equivalent, which means that the two plants always have the same height.
Is there a time when the two plants are the same height?We know that the height in inches for Kylie's plant is represented by the
equation y = 12x, where x is the time in months.
And the height of Glen's plant is represented by the equation 2y = 24x
Notice that if we take the second equation and we divide both sides by 2, we will get:
2y = 24x
2y/2 = 24x/2
y = 12x
This is the same equation than the one that represents the height of Kylie's plant.
Then the two plantes have the same height always.
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chooe all of the following expreion which are even for all value of n
2n8
5n10
2n3
n2
4n-14
All of the expressions 2n8, 5n10, 2n3, n2, and 4n-14 are even for all values of n because all of the coefficients of n are even numbers.
2n8, 5n10, n2, 4n-14
2n8 : This expression is even for all values of n because the coefficient of n is an even number (2).
5n10 : This expression is even for all values of n because the coefficient of n is an even number (5).
2n3 : This expression is even for all values of n because the coefficient of n is an even number (2).
n2 : This expression is even for all values of n because the coefficient of n is an even number (1).
4n-14 : This expression is even for all values of n because the coefficient of n is an even number (4).
All of the expressions 2n8, 5n10, 2n3, n2, and 4n-14 are even for all values of n because all of the coefficients of n are even numbers.
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An octagon with vertices at (-4, 2), (-2,5), (1,5), (3, 2), (3, -6), (1, -3), (-2, -3),
and (4, -6) was dilated by a scale factor of 3 and with a center of dilation at the
origin. What are the coordinates of the vertices of the dilated octagon
Coordinates of the vertices of the octagon after dilation by scale factor 3 are (-12, 6), (-6,15), (3,15), (9, 6), (9, -18), (3, -9), (-6, -9),and (12, -18).
Original coordinates of the vertices of an octagon are :
(-4, 2), (-2,5), (1,5), (3, 2), (3, -6), (1, -3), (-2, -3), and (4, -6).
Scale factor is equal to 3
Figure transformed using dilation of scale factor of 3 with center of dilation at the origin.
Coordinates of the new vertices after dilation of the octagon.
(-4×3, 2×3) = ( -12, 6)
(-2 ×3,5×3) = ( -6 , 15)
(1 ×3,5×3) = ( 3 ,15)
(3×3, 2×3) = ( 9,6)
(3×3, -6×3) = ( 9, -18)
(1 ×3, -3×3) = ( 3 , -9)
(-2×3, -3×3) = ( -6 , -9)
(4×3, -6×3) = (12, -18)
Therefore, the new coordinates of the vertices after dilation are (-12, 6), (-6,15), (3,15), (9, 6), (9, -18), (3, -9), (-6, -9),and (12, -18).
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a certain corner of a room is selected as the origin of a rectangular coordinate system. if a fly is crawling on an adjacent wall at a point having coordinates (1.7, 0.6), where the units are meters, what is the distance of the fly from the corner of the room?
A certain corner of a room is selected as the origin of a rectangular coordinate system. A fly has coordinate (1.7, 0.6). The distance of the fly from the corner of the room is 1.8 meters.
Suppose we have a point having coordinate (x,y). Then the horizontal axes, vertical axis, and the distance from the origin form a right triangle which its base is x and y.
Hence, we can find the distance of the point (x,y) from the origin using the Pythagorean Theorem.
In this case, the origin is a certain corner of the room. The coordinate of the fly is (1.7, 0.6) meters.
Using the Pythagorean Theorem:
distance = sqrt (1.7² + 0.6²)
distance = sqrt (2.89 + 0.36)
distance = sqrt (3.25)
distance = 1.8 meters
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A new long-life tire has a tread depth of 3/8 inches, instead of the more typical 7/32 inches. How much deeper is the new tire?
Using the subtraction operation, the new long-life tire with 3/8 inches depth is deeper than the more typical 7/32 inches by 5/32 inches.
What is a subtraction operation?Subtraction operations involve the minuend, the subtrahend, and the result called the difference.
It is one of the four basic mathematical operations, including addition, multiplication, and division.
The depth of the new long-life tire = 3/8 inches
The depth of the old tire = 7/32 inches
The difference = 5/32 inches (3/8 - 7/32)
Thus, the depth difference between the new long-life tire and the typical tire is 5/8 inches.
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The average birth weight of a Boxer breed dog is 1.25 kilograms. What is the average birth weight in grams?The average birth weight of a Boxer breed dog is 1.25 kilograms. What is the average birth weight in grams?
Answer:
1250g
Step-by-step explanation:
1.25 kg × 1000 = 1250g
What is the solution to this system of
equations?
s 6x – 2y = 8
1-3x + y = -4
According to the elimination method, the solution of the system of equation has infinite number of solutions.
In math the term elimination method is the process of eliminating one of the variables in the system of linear equations using the addition or subtraction methods in conjunction with multiplication or division of coefficients of the variables
Here we know that the equations are
=> 6x - 2y = 8 ---------------(1)
=> -3x + y = -4 ------------------(2)
when we simplify the equation (1) by dividing the terms by 2, then we get
=> 3x - y = 4
Now, we have to add the two equation then we get the result as,
=> (-3x + 3x) +( -y + y) = (-4 + 4)
=> 0 + 0 = 0
Hence the system has an infinite number of solutions.
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evaluate h(x)= -10x when x= -3,0, and 4
In order to replace the value in the explicit expression for h, one must evaluate a function h(x) for a particular value of the independent variable x. h(-3)=30, h(0)=0, h(4)=-40
How to find the calculation?A function is evaluated by determining the value of f(x) =... or y =... that corresponds to a certain value of x. Simply substitute whatever x has been assigned for all of the x variables to do this.Meaning: Simplify and replace x with 5 Parentheses should be used to enclose the value that is being substituted.A variable is replaced with a provided number or expression when a function is evaluated. In order to replace the value in the explicit expression for h, one must evaluate a function h(x) for a particular value of the independent variable x.The task is as follows:h(x)=-10x
Let's determine the values below:
h(0)= -10 * (0) = 0 and h(4)= -10 * (4) = -40, respectively.
h(-3)=30, h(0)=0, h(4)=-40.
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Chris opens up a new visa credit card and credit line with U. S. Bank. He is paying a 15% annual percentage rate (APR) on all new purchases. If he has a current balance of $2,800 on this credit card, how much does he owe in interest to the back next month?
Chris owes $35 in interest to the back next month.
We know that a credit card's interest rate is nothing but the price one pay for borrowing money. This is called the annual percentage rate (APR).
The Interest paid is calculated by the formula:
I = PTR ÷ 100
where P = Principal amount ,
R= rate of interest in percentage ,
and T= period.
Here, P = $2800,
R = 15%
T= 1 year
Using above formula,
I = (2800 x 15 x 1) ÷ 100
I = 420.
He will pay an interest of 420 ÷ 12= $35 per month.
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The price of a new car was $22,800 in April. However, the price of the car was reduced by 5% in May. Find the price in May.
In May, the price of the car was ? % as compared to
the price of the car in April.
Answer: To find the price of the car in May, we can use the formula:
New price = Original price - (Original price x Discount rate)
In this case, the original price is $22,800, and the discount rate is 5%, which can be written as 0.05.
So, the new price in May is:
New price = 22700 - (22700 x 0.05)
New price = 22700 - 1135
New price = $21,565
So, the price of the car in May was $21,565.
To find the percentage of the price of the car in May as compared to the price of the car in April, we can use the formula:
(New price / Original price) x 100
In this case, the new price is $21,565 and the original price is $22,800
(21565 / 22700) x 100 = 0.95 x 100 = 95%
Therefore, in May, the price of the car was 95% as compared to the price of the car in April.
Step-by-step explanation:
The difference between the water levels for high and low tide was 3. 6 feet. Write and solve an equation to find the water level at high tide
The difference between the water levels for the high and low tide was 3. 6 feet. the equation to find the water level at high tide x = y + 3.6, where High Tide Water Level = x and Low Tide Water Level = y.
In simple form it will be High Tide Water Level = Low Tide Water Level + 3.6. To explain this equation, the water level at high tide is equal to the water level at low tide plus 3.6 feet. The variable 'x' represents the water level at high tide, while the variable 'y' represents the water level at low tide. By adding 3.6 feet to the water level at low tide, we can find the water level at high tide.
For example, if the water level at low tide is 4 feet, the water level at high tide would be 4 + 3.6 = 7.6 feet. To find the water level at high tide for any given low tide water level, one can simply add 3.6 feet to the known low tide water level. This equation can be used to accurately calculate the water level at high tide for any given low tide water level.
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5b-2c + 3)-d, a +2b-c+d, 2-d + 3c + a
The sum of the given expression will be 2a+7b+-d.
What are expressions?
Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division. For example, x + y is an expression, where x and y are terms having an addition operator in between. In math, there are two types of expressions, numerical expressions - that contain only numbers; and algebraic expressions- that contain both numbers and variables.
e.g. A number is 6 more than half the other number, and the other number is x. This statement is written as x/2 + 6 in a mathematical expression. Mathematical expressions are used to solve complicated puzzles.
Now
Given expressions are 1.5b-2c+3-d 2.a+2b-c+d 3.2-d+3c+a
By adding these
=a+a+5b+2b-2c-c+3c-d+d-d
=2a+7b-d
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an integer is called parity-monotonic if its decimal representation $a 1 a 2 a 3 \dots a k$ satisfies $a i < a {i 1}$ if $a i$ is odd, and $a i > a {i 1}$ if $a i$ is even. how many four-digit parity-monotonic integers are there?
The total number of four-digit parity-monotonic integers is 432*1 = 24.
Parity-monotonic four-digit integers countWe can solve this problem by counting the number of parity-monotonic integers in each of the four digits individually and then multiplying the counts together.
For the first digit, we must have [tex]$1 \leq a_1 \leq 4$[/tex] as the integer must be four-digit.For the second digit, we must have [tex]$0 \leq a_2 < a_1$[/tex] if [tex]$a_1$[/tex] is odd and [tex]$a_1 < a_2 \leq 9$[/tex] if [tex]$a_1$[/tex] is even.For the third digit, we must have [tex]$0 \leq a_3 < a_2$[/tex] if [tex]$a_2$[/tex] is odd and [tex]$a_2 < a_3 \leq 9$[/tex] if [tex]$a_2$[/tex] is even.For the fourth digit, we must have [tex]$0 \leq a_4 < a_3$[/tex] if [tex]$a_3$[/tex] is odd and [tex]$a_3 < a_4 \leq 9$[/tex] if [tex]$a_3$[/tex]is even.By counting the possible values of [tex]$a_1$[/tex], [tex]$a_2$[/tex], [tex]$a_3$[/tex] and [tex]$a_4$[/tex] independently, we have:
[tex]$a_1$[/tex] has 4 possibilities[tex]$a_2$[/tex] has 3,2,1 possibilities depending on [tex]$a_1$[/tex] is even or odd[tex]$a_3$[/tex] has 2,1,2,1 possibilities depending on [tex]$a_2$[/tex] is even or odd[tex]$a_4$[/tex] has 1,0,1,0 possibilities depending on [tex]$a_3$[/tex] is even or oddThus, the total number of four-digit parity-monotonic integers is 432*1 = 24.
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(4)(-0.5)(8)(-2/4) a positive product?
Answer: +8
Step-by-step explanation:
yes, it is going to be a positive product, since two negative signs multiply to give a positive sign.
⇒(4)(-0.5)(8)(-2/4)
⇒+8 ..Answer
hope that helps...
Use the figure shown for Items 1–4. In the figure, m∠IJL=0.5 m∠JIK.
Triangle I J K with side I J congruent to side J K. Segment J L is drawn from vertex K perpendicular to base I K such that it bisects the base. Segment I L has length 3 x and is congruent to segment L K which has length 2 x plus 5.
Select all the statements that must be true.
A. KL=15
B. JL bisects IK
C. ΔIJK is equilateral.
D. JL is the perpendicular bisector of IK
E. JL=2LK
The true statements are:
A. KL=15
B. JL bisects IK
D. JL is the perpendicular bisector of IK
What is perpendicular?Two geometric objects are perpendicular in simple geometry if they intersect at a right angle. The perpendicular sign, ⊥ , can be used to visually depict the state of perpendicularity. It can be defined between two planes, two lines, or two planes and another line.
Given:
We have a triangle IJK.
m∠IJL=0.5 m∠JIK.
Side IJ ≅ side JK.
JL ⊥ IK.
It bisects the base.
That means,
JL bisects IK and JL is the perpendicular bisector of IK.
So, IL = LK.
3x = 2x + 5
x = 5
KL=15.
Therefore, A, B and D are the true statements.
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Sara says if you subtract 19 from my number and multiply the difference by -6 the result is -96,
what is saras number
PLEASE PLAE PLASE PALSE PLASEP APLSASEP PLASEP PLEASE PLAESESEEPZ.
Write the decimal as a fraction. 0.83 = F 10 x 0.83 = F x 10 8.33= [?]F
The decimal 8. 33 written as a fraction is 8 3/10
What is a fraction?
A fraction can be defined as the part of a whole number, variable, element,
The several types of fractions are;
Proper fractionsMixed fractionsImproper fractionsSimple fractionsComplex fractionsExamples of mixed fractions: 2 4/5, 7 1/2
Examples of simple fractions: 1/3, 1/4
Examples of improper fractions: 6/4, 7/5
Example of proper fractions: 2/3, 4/5
From the information given, we have that;
8.33
This is represented as;
8 3/10
Hence. the value is 8 3/10
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The art teacher has 48 paintbrushes. She puts 8 paintbrushes on each table in her classroom. How many tables are in her classroom?
Answer:
6 Tables
Step-by-step explanation:
48 tables divided by 8 paintbrushes equals 6 tables.
(05.01 MC)
Currently, the temperature of an ice cream is 81°C below room temperature. The temperature of the ice cream decreases based on the
function f(t)=81(0.83) where t represents time elapsed in minutes. Where does the graph of f(t) cross the y-axis, and what does this represent
in the context of the problem?
The graph of f(t) crosses the y-axis at t = 0, which represents the initial temperature of the ice cream (81°C below room temperature).
What does this represent in the context of the problem?The graph of f(t) crosses the y-axis at y=81, which represents the initial temperature of the ice cream, 81°C below room temperature. This is the starting point of the graph, and all subsequent points on the graph represent the temperature of the ice cream as time progresses. As time passes, the temperature of the ice cream decreases, and the graph of f(t) illustrates this change.
The y-intercept of a graph measures the initial value of the function. In this case, the initial temperature of the ice cream is 81°C below room temperature. As the time passes, the temperature of the ice cream decreases at a rate of 0.83°C per minute. This rate of decrease is represented by the slope of the graph, which is 0.83.
The point where the graph of f(t) crosses the y-axis illustrates the initial temperature of the ice cream, and all subsequent points on the graph represent the temperature of the ice cream as time progresses. This is a helpful visual representation of the rate at which the temperature of the ice cream is decreasing. The graph of f(t) is a useful tool for understanding how the temperature of the ice cream changes over time.
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Use the Bisection Method up to five iterations and find the root to 3 decimal places for the following:
f(x) = x2 − 3x + 1 in the interval [0, 1]
A.
0.375
B.
0.438
C.
0.406
D.
0.391
Answer:
D. 0.391
Step-by-step explanation:
You want the approximate solution to f(x) = x² -3x +1 = 0 on the interval [0, 1] using 5 iterations of the Bisection Method.
IterationThe Bisection Method makes one iteration by finding the function value at the midpoint of the interval. The midpoint replaces the end of the interval whose function value has the same sign. At the end of one iteration, the midpoint of the halved interval is calculated.
Start:
f(0) = 1, f(1) = -1. Interval: [0, 1]; midpoint: 1/2
First iteration:
f(1/2) < 0. Interval: [0, 1/2]; midpoint: 1/4
Second iteration:
f(1/4) > 0. Interval: [1/4, 1/2]; midpoint: 3/8
Third iteration:
f(3/8) > 0. Interval: [3/8, 1/2]; midpoint: 7/16
Fourth iteration:
f(7/16) < 0. Interval: [3/8, 7/16]; midpoint: 13/32
Fifth iteration:
f(13/32) < 0. Interval: [3/8, 13/32]; midpoint: 25/64
SolutionThe approximate solution after 5 iterations is x ≈ 25/64 ≈ 0.391.
__
Additional comments
The approximate solution in the interval to full calculator precision is 0.38196601125. The exact solution is 1.5-√1.25.
You will notice that the function values for the ends of the interval [0, 1] are [positive, negative]. So, when the function value at the midpoint is negative, that point replaces the right end of the interval.
Of course, the midpoint is the average of the interval end values.
On average, it takes about 3.3 iterations to improve the accuracy of the solution by 1 decimal place.
The nature of the roots for the equation
If D > 0 and not an excellent square, then the roots of the quadratic equation are real, unequal and irrational.
What is Quadratic Equation ?
Quadratic equation can be defined as equation in which it is in the form of ax^2+bx+c = 0 .
Nature of Roots depending upon Discriminant
in step with the value of discriminant, we shall discuss the subsequent cases about the character of roots.
Case 1: D = 0
If the discriminant is same to 0 (b2 – 4ac = zero), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax2 + bx + c = zero, are real and equal. In this case, the roots are x = -b/2a. The graph of the equation touches the X axis at a single point.
Case 2: D > 0
If the discriminant is greater than zero (b2 – 4ac > 0), a, b, c are actual numbers, a≠0, then the roots of the quadratic equation ax2 + bx + c = 0, are actual and unequal. The graph of the equation touches the X-axis at two unique points.
Case 3: D < 0
If the discriminant is less than 0 (b2 – 4ac < 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax2 + bx + c = 0, are imaginary and unequal. The roots exist in conjugate pairs. The graph of the equation does not touch the X-axis.
Case 4: D > zero and perfect square
If D > zero and an excellent square, then the roots of the quadratic equation are actual, unequal and rational.
Case five: D > 0 and now not an excellent square
If D > 0 and not an excellent square, then the roots of the quadratic equation are real, unequal and irrational.
Therefore, If D > 0 and not an excellent square, then the roots of the quadratic equation are real, unequal and irrational.
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Which property is being demonstrated?
3^9/3^4=3^9-4=3^5
quotient of powers property
power of a product property
product of powers property
power of a quotient property
From the given example we can say that , the given expression is in the form of quotient of a powers property.
What is Algebraic expression ?
Algebraic expression can be defined as the combination of variables and constants.
Given expression,
3^9 / 3^4
= 3^(9-4)
= 3^(5)
So,
if a^m / a^n = a^(m-n)
Its in the form of quotient of powers property.
And from above we can say that, The given expression also in the form of quotient of powers property.
Hence, From the given example we can say that , the given expression is in the form of quotient of a powers property.
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in a negatively skewed distribution of exam scores, bender scored at the mean, fry scored at the median, and lela scored at the mode. who had the highest score?
Lela had the highest score.
Now, According to the question:
A skewed distribution is neither symmetric nor normal because the data values trail off more sharply on one side than on the other. In business, you often find skewness in data sets that represent sizes using positive numbers.
In a negatively skewed distribution, higher values are more frequent than lower values causing the distribution to present a longer left-tail. This implies in a median higher than the mean and in a mode higher than the median:
mean < median < mode.
Therefore, since Lela scored at the mode, Lela had the highest score.
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Simplify (4s+7 -2S) +4 (3s-5^2) 
Answer:
14S -93
Step-by-step explanation:
(4s+7 -2S) +4 (3s-5^2)
= 2S +7 +12S -4*25
=14S -93
Find the slope of the line through (-2,6) and (-5,9)
-1
Step-by-step explanation:Slope describes the rise over run for a graph.
Slope Formula
Slope is the change in y over the change in x. To find the slope of any line from two points we can use the slope formula. The slope formula is [tex]\displaystyle\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]. We can plug the points we are given into this formula.
[tex]\frac{9-6}{-5-(-2)}[/tex]This equals [tex]\frac{3}{-3}[/tex] which is simply -1. The slope of the line is -1.
Counting the Change
The formula above just represents the changes in the y-values divided by the changes in the x-values. So, we can count that the y-values increase by 3. Then, the x-values decrease by 3. This creates the slope [tex]\frac{3}{-3}[/tex] (remember that increases are positive and decreases are negative). Once again this gives the slope of -1.
In a survey of 50 first graders, 30% said their favorite food is hot dog. Which box is correctly set up to show this?
The probability that the number of first graders who have liked the hot dog is 15
In math the term probability is defined as a number that reflects the chance or likelihood that a particular event will occur
Here we have know that in a survey of 50 first graders, and then 30% said their favorite food is hot dog.
Here we have to find the probability that the number of first graders who have liked the hot dog
As we know that the number of first graders is 50.
And here we also know that 30% of them are liked the hot dog.
Then it can be calculated as,
=> 50 x 30%
=> 50 x 30/100
=> 15.
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Find the 67th term of the arithmetic sequence -17,-23,-29
The 67th term of the arithmetic sequence -17,-23,-29 is -413.
What is the arithmetic sequence?
An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is always the same. This difference is called the common difference, and it can be represented by the variable d.
Given the sequence -17,-23,-29, we can see that the common difference is d = -23 - (-17) = -6
The nth term of an arithmetic sequence can be found using the formula:
a_n = a_1 + (n-1)d
where a_1 is the first term of the sequence, n is the term number and d is a common difference.
For the 67th term, a_1 = -17, n = 67 and d = -6
So the 67th term is:
a_67 = -17 + (67-1) * -6 = -17 + 66 * -6 = -17 - 396 = -413
Hence, the 67th term of the arithmetic sequence -17,-23,-29 is -413.
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Answer: -413
Step-by-step explanation:
Function rule or explanation of why it is not possible to find a function rule