The statement is false. In the pair of linear equation, if a₁/a₂ ≠ b₁/b₂ then the equations have unique solution.
Given that,
The pair of linear equation
a₁x +b₁y +c₁=0 and
a₂x +b₂y +c₂=0
We have to prove if a₁/a₂ ≠ b₁/b₂ then the equations have no solution.
We know that,
In the pair of linear equation
a₁x +b₁y +c₁=0 and
a₂x +b₂y +c₂=0
If a₁/a₂ ≠ b₁/b₂ then System is called consistent and have a unique Solution
Lines intersect each other at unique point
a₁/a₂ = b₁/b₂ = c₁/c₂ then consistent (infinite solution) as both Equation represent same line
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ then inconsistent (no Solution) as Lines are parallel and never intersect each other
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How many roots does the equation x3 3x2 4 0 have?
The number of roots of the given equation x³ -3x² + 4 = 0 is equal to 3 which are -1, 2, 2.
As given in the question,
Given equation is written as:
x³ -3x² + 4 = 0
Number of roots of the given equation are as follow ;
Let f(x) = x³ -3x² + 4
For x = 1
f(1) = 1³ - 3(1)² + 4
= 2 ≠ 0
x = 1 is not the roots.
Now, x = -1 ,
f(-1) = (-1)³ - 3(-1)² + 4
= -1 -3 + 4
= 0
x = -1 is the roots of the given equation x³ -3x² + 4 = 0
x³ -3x² + 4 = 0
⇒x³ + x² -4x² -4x + 4x + 4 =0
⇒x² ( x + 1) - 4x( x + 1) + 4(x+1) =0
⇒(x+1) ( x² -4x +4 ) = 0
⇒(x+1)( x - 2 )² = 0
⇒ x = -1, 2, 2
Therefore, the number of roots of the given equation x³ -3x² + 4 = 0 are 3 and given by x = -1 , 2, 2.
The above question is incomplete , the complete question is :
How many roots does the equation x³ -3x² + 4 = 0 have ?
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Which is an asymptote of the graph of the function y tan 3 4x?
The graph of the function y tan 3 4x has a vertical asymptote at x = π/12, which means that as x approaches π/12 from either side, the output of the function increases without bound.
The graph of the function y tan 3 4x has a vertical asymptote at x = π/12, which means that as x approaches π/12 from either side, the output of the function increases without bound. This is because when x = π/12, the denominator of the tangent function (3/4x) is 0, which creates an undefined value. As a result, the output of the function increases without bound as x approaches π/12. This is why x = π/12 is an asymptote of the graph. Asymptotes can be thought of as boundary lines, beyond which the graph of the function cannot cross. In this case, the graph of y tan 3 4x will approach the vertical asymptote of x = π/12, but will never cross it.
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a. what is the area of the window seat when x=3
b. write a polynomial that helps represent the area of the window seats
a) The area of the window seat when x=3 is 8 ft².
b) Polynomial that represent area of the window is x² - 1
What is trapezium?
In American and Canadian English, a quadrilateral with at least one pair of parallel sides is referred to as a trapezoid. It is referred to as a trapezium in British and other varieties of English. In Euclidean geometry, a trapezoid is a convex quadrilateral by definition. The bases of the trapezoid are the parallel sides.Area of trapezium = 1/2 (a+b)*h
a= x ft
b = x+2 ft
c = x-1 ft
a ) When x=3;
a = 3 ft
b = 3 + 2 = 5 ft
c = 3-1 = 2 ft
Now, Area of window seat = 1/2 (3+5) 2
= 1/2 * 8 * 2
= 8 ft²
Hence, area of window seat is 8 ft².
b) Polynomial that helps represent the area of window seats
= 1/2(a+b)*h
= 1/2(x + x +2) * (x-1)
= 1/2 (2x+2)*(x-1)
= 1/2 * 2(x+1)* (x-1)
= (x+1) * (x-1)
= x² - 1
Hence, the Polynomial that represent area of the window is x² - 1.
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6 ÷ (–3) + [(4 – (–5)) × (–7)]
answer please
shuffle a standard deck of 52 playing cards. turn over the top card. put the card back in the deck and shuffle again. repeat this process until you get a queen. count the number of cards you had to turn over. which conditions for a binomial setting have been met for this scenario?
The conditions for a binomial setting have been met for this scenario are;
The probability of success, denoted p, remains the same from trial to trial.The n trials are independent.How to interpret Binomial Experiment?
The conditions for a binomial experiment to be true are;
The experiment consists of n identical trials.Each trial results in one of the two outcomes, called success and failure.The probability of success, denoted p, remains the same from trial to trial.The n trials are independent.Now, in this case we keep shuffling the standard deck of 52 playing cards and picking the first in each case and replacing until we get a queen. This means that each trial is independent and does not depend on the success or failure of another event.
Secondly, the probability of success remains the same in each trial because we can't predict when we will get a queen.
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How do you solve under root minus 16?
Answer:
4i
Step-by-step explanation:
Any negative numbers do not have any real square roots, only imaginary ones.
[tex]\sqrt{-16} = i\sqrt{16} = 4i[/tex]
hope this helps :)
Which is the same as 98.25%10
According to question,
We know that,
To determine the percentage,
we have to divide the value by the total value and then multiply the resultant by 100,
Percentage formula
Is /of=%?100
Or
Part/whole=%/100
With the help of question,
Given data,
Put the value of percentage formula,
98.25/100×10
=9.825
98.25%×10 is the same outcome of 9.825.
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What is the quadratic equation whose roots are 2 √ 3 and 2 − √ 3?
The quadratic equation whose roots are 2√3 and 2-√3 can be written in the form ax² + bx + c = 0.
To solve this equation, we can use the formula x = (-b ± √(b²-4ac))/2a.
Substituting the values of 2√3 and 2-√3 for x, we get:
2√3 = (-b + √(b²-4ac))/2a
2-√3 = (-b - √(b²-4ac))/2a
Subtracting the two equations, we get:
0 = √(b²-4ac)
Taking the square of both sides, we get:
b² - 4ac = 0
Rearranging the equation, we get:
ax² + bx + c = 0, where
a = 1
b = 0
c = -4ac
Therefore, the quadratic equation whose roots are 2√3 and 2-√3
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If f(x) = x² + 1, then f(f(3)) = ?
Answer:
f(f(3))=101
Step-by-step explanation:
To solve this problem, first solve for f(3):
[tex]f(3)=(3)^2+1\\f(3)=9+1\\f(3)=10[/tex]
Then, insert f(3) into f(2) by substituting f(3) in place of x:
[tex]f(f(3))=f(10)\\f(10)=(10)^2+1\\f(10)=101[/tex]
LCM [a,b] = 36 HCF [a,b] = 6 find the value of a=b
All the values that are possible here are (1, 36) (4,9) (9,4), and (36,1).
The possible values of and b are (1, 36) (4,9) (9,4), and (36,1).
Given: The LCM of a and b is 36 and HCF is 1
To find: The values of a and b?
Here is the solution:
The two numbers are a and b.
LCM of a and b is 36.
HCF of a and b is 1
We know that,
product of two numbers = LCM * HCF
= A*B=36*1
i.e = a, b=36,1
in a similar way:
The all-possible values of and b are (1, 36) (4,9) (9,4), and (36,1).
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There is another particular medicine that is modeled by the following:
f(t) = -15t² + 120t
1. Where f(t) is the amount of medication (in mg) in the bloodstream at time t (in hours).
2. Determine the maximum amount of medication in the bloodstream.
3.Using a function, determine the concentration of medication in the
bloodstream after 2 hours.
Answer:
Step-by-step explanation:
1. The function f(t) represents the concentration of medication in the bloodstream over time. The function has a parabolic shape, with the maximum value at t = 8 hours.
To find the maximum concentration of the medication, we can set the derivative of the function equal to 0 and solve for t. The derivative of f(t) is:
f'(t) = -30t + 120
Setting this equal to 0 and solving for t, we get:
-30t + 120 = 0
-30t = -120
t = 4
Therefore, the maximum concentration of the medication occurs at t = 4 hours. At this time, the concentration of the medication is:
f(4) = -15(4)^2 + 120(4) = 120 mg
It's important to note that this function represents the concentration of the medication in the bloodstream, not the total amount of medication that has been taken. The total amount of medication taken over a given time period can be found by integrating the function over that time period.
2. To determine the maximum amount of medication in the bloodstream, we need to find the maximum concentration of the medication and then multiply that concentration by the volume of the bloodstream.
We have already determined that the maximum concentration of the medication is 120 mg/L. The volume of the bloodstream in an adult is approximately 5 liters. Therefore, the maximum amount of medication in the bloodstream is:
120 mg/L * 5 L = 600 mg
This is the maximum amount of medication that can be present in the bloodstream at any given time. It's important to note that this amount may be affected by various factors, including the rate at which the medication is metabolized and eliminated from the body.
3. To determine the concentration of medication in the bloodstream after 2 hours, we can substitute 2 for t in the function f(t) and solve for the concentration. The function for the concentration of medication in the bloodstream is:
f(t) = -15t^2 + 120t
Substituting 2 for t, we get:
f(2) = -15(2)^2 + 120(2) = 60 mg/L
Therefore, the concentration of the medication in the bloodstream after 2 hours is 60 mg/L. It's important to note that this represents the concentration of the medication at a specific point in time, not the total amount of medication that has been taken. The total amount of medication taken over a given time period can be found by integrating the function over that time period.
In Angle abc and angle pqr ab:pq=4:5 and a(angle pqr)=125cm² then find a(angle abc)
In Angle abc and angle pqr ab:pq=4:5 and a(angle pqr)=125cm² then the measure of angle ABC is 100 cm².
What is the angle?Generally, To find the measure of angle ABC in the triangle ABC, we can use the fact that the ratios of the measures of the angles in a triangle sum to 180 degrees.
We are given that the ratio of the measures of angles ABC to PQR is 4:5. This means that if we let x represent the measure of angle ABC, then the measure of angle PQR is 5x/4.
The measure of angle PQR is also given to be 125 cm². We can use this to set up the equation:
5x/4 = 125
Solving for x, we find that the measure of angle ABC is:
x = 100
Therefore, the measure of angle ABC is 100 cm².
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What is the y-intercept of the line?
A. 1
B. 1/2
C. -2
D. 0
Answer:
A.1
Step-by-step explanation:
The line crosses when Y is one and X is 0 thus creating an intercept on the y line
Write an equation of the line with the given slope and y-intercept.
Undefined slope; (-8, 5)
A. x = -8
B. 5x - 8y = 0
C. y = 5
D. -8x + 5y = 0
Answer:
A. x = -8
Step-by-step explanation:
It can't be B because the slope of this equation is 5/8.
It can't be C because this equation slope is 0, not undefined.
It can't be D because the slope of this equation is 8/5
I say it is A because this is the only equation with an undefined slope!
What are examples of SAS congruence postulates?
The examples of SAS congruence postulates are
∆ ABC ≅ ∆DEF∆ABC ≅ ∆CEFThe SAS is stands for Side-Angle-Side congruence postulate which states that triangles are congruent if two sides and an included angle of a triangle are congruent with two sides and an included angle of a second triangle. Note that we need to include the angle between the two sides.
Example 1 : Let's say you have one triangle, ∆ABC with side lengths 5 and 10 and the angle included between those two sides is 30 degrees. If you have a second triangle, ∆DEF that also has side lengths 5 and 10 with a 30 degree angle in between, then by the SAS Postulate ∆ABC is similar to ∆DEF.
Example 2 : Point C is the midpoint of BF
so, AC = CE
Statement Reason
AC = CE C is mid point so, it
divides equal parts .
BC = CF C is mid-point
∠ACB =∠ECF Opposite angles
Thus, by the Side Angle Side postulate, the triangles are congruent i.e,∆ABC ≅ ∆CEF
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What is the mode of the following set of data 12 11 14 10 8 13 11 9?
The mode of the given data set is 11.
A mode is defined as the value that has a higher frequency in a given set of values. It is the value that appears the most number of times.
In statistics, the value that consistently appears in a particular collection is referred to as the mode. The mode or modal value in a data collection is sometimes referred to as the value or number that occurs most frequently in the data set. In addition, to mean and median, it is one of the three measures of central tendency.
The mode is the value that occurs the most often in a given set of data.
In the data set provided here, the mode is 11, hence 11 should occur the most number of times.
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INVERSE PROPORTION EXAMPLES
Answer:
Speed and Time are two examples of inverse proportion
How do you reverse a 2 digit number?
We are aware that to reverse a number, the TENS and ONE's positions must be reversed. As a result, its reverse "ba" usually takes the form (10 times "b" multiplied by "a")
What are 2-digit numbers?When a number has two digits, it is called a 2-digit number.
2-digit numbers have two digits and range from 10 to 99.
They are unable, to begin with, zero since it will then be regarded as a single-digit number.
Any number between 1 and 9 can be used as the digit in the tens place.
For instance, the numbers 45, 78, and 12 have two digits.
When it comes to two-digit numbers and their reversal, there are two common logics.
The two-digit number "ab" has the generic form (10 times a) Plus b.
We are aware that the TENS and ONE's positions should be switched in order to reverse a number.
As a result, the typical form of its reverse "ba" is (10 times "b" multiplied by "a")
Therefore, we are aware that to reverse a number, the TENS and ONE's positions must be reversed. As a result, its reverse "ba" usually takes the form (10 times "b" multiplied by "a")
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How do you solve using a graph?
Steps to graph-solve a problem:
Determine whether a graph can solve your issue. Get the graph data. Organize data to make graphing easier.Select the best graph for your data. Create the graph manually or using the software. Analyze the graph for data insights. Share your analysis. What is a graph?Generally, Identify the problem you are trying to solve and determine if a graph is an appropriate tool to help you solve it. Graphs are often used to represent relationships between different data points or to show changes in data over time.
Collect the data that you will need to create the graph. This may involve conducting research, conducting experiments, or gathering information from other sources.
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The complete Question was not found
We can solve using a graph by looking up the points on the graph.
How do you solve using a graph?We know that a graph can be regarded as a representation of facts and the graph is plotted in two axes. There is the vertical axis of the graph and there is the horizontal axis of the graph.
The vertical axis of the graph is called the y axis while the horizontal axis of the graph is called the x axis. The independent variable goes to the x axis while the dependent variable goes to the y axis.
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9 tens 9 ones 7 tenths 7 hundredths
9 ones, 9 tens 99.77 is equal to 7 tenths 7 hundredths.
Explain about the tenths and hundredths ?The place value of the digits that come after a decimal point is defined in mathematics as tenths and hundredths. When we count from right to left, the place values of the digits in a number are ones, tens, hundreds, and thousands.
The columns are labelled tenths, hundredths, thousandths, and so forth from left to right following the decimal point. The tenths column is the first digit following the decimal point. The column for hundredths is the second digit following the decimal point. The column for thousandths is the third digit following the decimal point.
The digit after the decimal point, or in the tens place, is known as the divisor. The first digit to the right of the decimal place is the tenths place digit.
9 tens = 9 x 10 = 90
9 ones =9 x 1 = 9
7 tenths = 7x 1/10 =0.7
7 hundredths = 7 x 1/100 = 0.07
= 90 + 9 +0.7 +0.07
= 99.77
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PLEASE HELP IM STUCK
Answer:
option b is the answer of the above
Match the size of a triangle to longest middle and shortest side
Answer:
AC is longest
BC is middle
AB is shortest
The table shows the height of a plant as it grows. Which equation in point-slope form gives the plant's height at any time?
A) y - 21 = 7/2 (x - 3)
B) y - 21 = 7 (x - 3)
C) y - 3 = 7/2 (x - 21)
D) The relationship is nonlinear.
The equation in point slope form is y - 21 = 7(x - 3).
What is point - slope form of the line?
A line's equation in point slope form is y - y1 = m. (x – x1). As a result,
y = mx and y - 0 = m(x - 0) are the equations for a line passing through the origin with a slope of m.
To find the equation in point slope form for the given table of height of a plant first we have to find the slope.
Here
Slope = change in height / Change in time
From given table,
Change in height = 14 and Change in time = 2
⇒ Slope = 14 / 2 = 7
Now to find the point slope form.
We know that the equation in point slope form for point [tex](x_1, y_1)[/tex] and slope m is written as,
[tex]y-y_1=m(x-x_1)[/tex]
Plug [tex](x_1, y_1) = (3, 21), m = 7[/tex]
So, the equation becomes
[tex]y-21 = 7(x-3)[/tex]
Hence, the equation in point slope form is y - 21 = 7(x - 3).
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I need answers for this questions
Answer:
a=7 b=2 c=0.2 d=9 e=0.064 f=8 g=16 h=0.03 i=0.714
Step-by-step explanation:
Translate the answer, t2 = a3, into words: the___of the orbital period, t, of a planet is equal to the____of the average distance, a, of the planet from the sun.
a. Squareb. Square Rootc. Cube
The answer is the square of the orbital period, t, of a planet is equal to the cube of the average distance, a, of the planet from the sun.
Translated into words, the equation t2 = a3 can be written as:
"The square of the orbital period, t, of a planet is equal to the cube of the average distance, a, of the planet from the sun."
The length of time it takes an astronomical object to complete one orbit around another object is known as the orbital period (also known as the revolution period). In astronomy, it often refers to bodies like planets or asteroids revolving about the Sun, as well as moons circling other stars, exoplanets, and double stars.
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Translated into words, the equation t2 = a3 can be written as:
"The square of the orbital period, t, of a planet is equal to the cube of the average distance, a, of the planet from the sun."
The orbital period is the amount of time it takes an astronomical object to complete an orbit around another one (also known as the revolution period). In astronomy, it frequently refers to objects revolving about the Sun, such as planets or asteroids, as well as moons orbiting around other stars, exoplanets, and double stars.
How do you find the horizontal asymptotes and vertical asymptotes of a function?
Horizontal asymptotes are found by looking at the degree of the numerator and denominator of a function. Vertical asymptotes are found by solving for all x-values where the denominator equals 0.
Horizontal asymptotes are horizontal lines that a graph of a function approaches as it goes towards positive or negative infinity. To find them, look at the degree of the numerator and denominator of a function. If the degree of the numerator is less than the degree of the denominator, the graph will approach the x-axis asymptote. If the degree of the numerator is equal to the degree of the denominator, the graph will approach the y-axis asymptote. Vertical asymptotes are vertical lines that a graph of a function approaches as it goes towards positive or negative infinity. To find them, solve for all x-values where the denominator equals 0. This will give you the x-values of the vertical asymptotes.
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Write the slope-intercept equation of the line through
the coordinates (4,5) and (-4,-3).
Answer:
y = 1/4x + 4
Step-by-step explanation:
Knowledge Needed
Slope: The amount the y-value changes on a constant line when the x-value increases by 1.
Slope Formula:
[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
y-intercept: When the line has a x-value of , or when it intersects the y-axis.
Slope-Intercept Form: y = mx+b
m is slope, b is y-intercept.
Leave y and x as variables so you can graph it.
Question
First, plug the points into the slope formula.
[tex]\frac{3-5}{-4-4}[/tex]
[tex]\frac{-2}{-8}[/tex]
1/4
y = 1/4x + b
Solve for b:
Use any of the 2 points to plug back in. I'll use (4,5).
We have: y = 1/4x + b
(5) = 1/4(4) + b
5 = 1 + b
5 - 1= 1 - 1 + b
b = 4
y = 1/4x + 4
find a vector that points in the same direction as the vector (i^+j^) and whose magnitude is 9.
A vector that points in the same direction as the vector (i^+j^) and whose magnitude is 9 is (9i^/ √2 + 9j^/ √2).
Therefore the answer is 9i^/ √2 + 9j^/ √2.
The vector direction is same as i^ + j^. We can obtain a unit vector in that direction by dividing the vector with magnitude of i^ + j^, which is √(1² + 1²) = √2.
Unit vector is (i^ + j^)/ √2.
Given that the required vector is of magnitude 9. We can determine the required vector thus by multiplying the magnitude with the unit vector in the direction, that is
9×(i^ + j^)/ √2
= 9i^/ √2 + 9j^/ √2
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How many positive four-digit integers have a remainder of 3 when divided by 4, a remainder of 3 when divided by 5, and a remainder of 5 when divided by 13?
Step-by-step explanation:
So this below is a solution I devised myself when I was around 11 yrs old doing math competitions with my friend so… (I don’t have a name for this solution, and can’t give any reference material)
alright so notice that when you subtract the remainder from the divisor in this question, all of their the values equate to 4:
6–2 = 4
9–5 = 4
11–7 = 4
I am going to use this property to devise a method from the problem.
The property above means
(n+ 4) % 6 = 0
(n+ 4) % 9 = 0
(n+ 4) % 11 = 0
where n is the dividend
notice that for n to be in accordance with the restriction of the question, n+4 must be a multiple of 6, 9 and 11 simultaneously (common multiple)
Since the question asks for the number of “n”s that are three-digit positives…
100 <= n <= 999 (which means)
104<= n+4 <= 1003
ok, so now we have to find the common multiples of 6,9,11 within the range 104~ 1003
the least common multiple of 6,9,11 is 198.
the smallest multiple of 198 that is larger than or equal to 104 is 198, which is 1*198… the largest multiple of 198 smaller than 1003 is 990, which is 5*198
so that means we have all the way from the first multiple of 198 to the fifth multiple of 198, inclusive, which is (5–1) + 1 = 5
and there we have it!!!
the answer is 5
P.S. we could also just count the multiples, but the (5–1)+1 is there because it may not always be countable, and the +1 is there to account for the first number that was subtracted
What are the 3 types of values?
Thousands, hundreds and tens are three types of values and there are many more.
What is value?Value in mathematics is a number that represents the outcome of a computation or function. In the aforementioned example, you may inform your teacher that 5 + 6 Equals 30 or that x + y = 9 if x = 6 and y = 3. A variable or constant can also be referred to as a value.
What is the value of 4?The number 4 is in this case in the tens column. As a result, the place value of the number four is tens or tens.
Thousands, hundreds and tens are three types of values and there are many more like ones etc. Moreover after solving any problem for unknown variable the resulting numerical solution can also be termed as value.
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