Answer: ab =6
have:
[tex]\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}=1\\\\=>\frac{b}{ab}+\frac{a}{ab}+\frac{1}{ab}=1\\\\<=>\frac{a+b+1}{ab}=1[/tex]
=> a + b + 1 = ab
⇔ a + b + 1 - ab = 0
⇔ b - 1 - a(b - 1) + 2 = 0
⇔ (b - 1)(1 - a) = -2
because a and b are postive integers => (b - 1) and (1 - a) also are integers
=> (b - 1) ∈ {-1; 1; 2; -2;}
(1 -a) ∈ {-1; 1; 2; -2;}
because (b -1).(1-a) = -2 => we have the table:
b - 1 -1 1 2 -2
1 - a 2 -2 -1 1
a -1 3 2 0
b 0 2 3 -1
a.b 0 6 6 0
because a and b are postive integers
=> (a;b) = (3;2) or (a;b) = (2;3)
=> ab = 6
Step-by-step explanation:
geometry of straight lines
Step-by-step explanation:
In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points or referred to using a single letter.
Types of Lines:Vertical lines: are straight up and down and perpendicular to horizontal lines.Horizontal lines: are straight up and down and perpendicular to vertical lines.Diagonal lines: are lines that straight in any direction except vertical or horizontal.The equation of a line is typically written as y=mx+b where m is the slope and b is the y-intercept. If you know two points that a line passes through, this page will show you how to find the equation of the line.
I hope it helps you!
Is this true or false? Pls help a dumb person out
Ishaan is 3 times as old as Christopher and is also 14 years older than Christopher.
How old is Ishaan?
Answer:
Ishaan is 21. Christopher is 7.
Answer: Ishann is 21 years old.
Step-by-step explanation:
In this situation, we will represent Ishaan age by the variable y and we will represent Christopher's age by x.
Ishaan is 3 times as old as Christopher and that can be represented by the equation, y = 3x
He is also 14 years older, and it can also be represented by the equation,
y = x + 14.
y= 3x
y = x + 14
Using both equations, substitute the value of y in one of the equations into the other.
x + 14 = 3x Solve for x .
-x -x
14 = 2x
x = 7
This means that Christopher is 7 years old.
To solve for Ishaan's age , input the value of x into one of the equations.
y = 3(7)
y = 21.
PlS HELP QUICKLY :One of the roots of the equation x2−bx+c=0 is equal to 5. Find c in terms of b.:
Guys pls help and find what C is equal to.
Answer:First suppose that the roots of the equation
x2−bx+c=0(1)
are real and positive. From the quadratic formula, we see that the roots of (1) are of the form
b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2.
For the root or roots to be real, we require that b2−4c≥0, that is, b2≥4c. In order for them to be positive, we require that
b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0.
This immediately tells us that b>0, but we can go further. We can rearrange this to get
b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,
which (assuming that b>0) is true if and only if
b2>b2−4c,
since both sides of the inequality are positive so we may square. But then
4c>0.
That is, if the roots are real and positive then b>0 and b2≥4c>0.
Now suppose that b>0 and b2≥4c>0.
Then the roots of (1) are real since b2−4c≥0, and b>0 guarantees that the root b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√ is positive.
So it remains to show that b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0. We have that
4c>0,
so that
b2>b2−4c,
then square rooting shows that
b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,
so the roots of (1) are real and positive, as required.
Approach 2
the curve y equals x squared minus b x plus c showing two positive roots for y equals zero
This is intended to be a proof without words! We have from the diagram that:
If c, b and b2−4c are all positive, there are two real positive roots for x2−bx+c=0 (if b2=4c, we have two real positive equal roots).
If there are two real positive roots for x2−bx+c=0, then c and b are positive and b2−4c is non-negative.
(Why is the distance between the roots b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√?)
Approach 3
When solving problems about the roots of polynomials, it is often useful to find expressions those roots must satisfy and see if this tells us anything new. If α and β denote the roots of the equation, then
x2−bx+c=(x−α)(x−β)=x2−(α+β)x+αβ
and so α+β=b and αβ=c.
We also know that the roots of a quadratic equation are real if and only if the discriminant is non-negative, that is, if and only if b2−4c≥0.
Using these facts, if α and β are both real and positive, then b=α+β>0, c=αβ>0 and b2≥4c, as above.
Conversely, if b>0 and b2≥4c>0, then we know the discriminant is positive and hence both roots are real. We also have that
αβ>0(2)
and
α+β>0.(3)
As α and β are both real, by (2), we know that α and β are either both positive or both negative. However, if α and β were both negative, then (3) could not possibly hold. Hence α and β are both positive, as required.
We now sketch on a graph the region where b>0, c>0 and b2≥4c:
The curve b squared = 4 c as a quadratic with the c-axis vertical and the b-axis horizontal. The region below it is shaded.
The region of the b-c plane for which b>0, c>0 and b2≥4c
Sketch the region of the b-c plane in which the roots of the equation are real and less than 1 in magnitude.
We know that in order for the roots to be real we need b2≥4c as in the first part. We now need to find the region where
−1<b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.(4)
We have b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2≤b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2 so we only need to consider the values for which both
−1<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2andb+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.
Firstly, we will consider the values for which
b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2.
Rearranging gives
b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2−b.
So b<2, as the square root is non-negative, and we can square both sides to get
b2−4c<4−4b+b2,
which we may rearrange to find c>b−1.
We will now consider the values for which
−2<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√.
Similarly, we can rearrange to get
b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<b+2.
So b>−2, and we can, as before, square to get
b2−4c<b2+4b+4,
and hence c>−b−1.
To sketch the graph, we start by considering the boundary curves b2=4c, c=b−1 and c=−b−1, and the points at which they intersect. We can see that the two lines only intersect when b=0 and c=−1, and the lines intersect the curve when
b2=4b−4andb2=−4b−4
which rearrange to
(b−2)2=0and(b+2)2=0.
This tells us that each line intersects with the curve in only one place and so these lines must be tangent.
Is there a way we could have deduced this directly from (4)?
What do the lines being tangent signify in terms of our equation x2−bx+c=0?
Is this a representation of a well-known property of these equations?
Sketching the graph then yields the following picture:
The graph with the previous curve and the lines 4 c = 4 b minus 4 and 4 c = minus 4 b - 4. Each line touches the curve once and the two lines intersect at (0, minus 1). The region between the three lines/curves is shaded.
The shaded region is where b2≥4c, c>b−1, c>−b−1 and −2<b<2
We might notice that when we derived the inequalities c>b−1 and c>−b−1, a lot of the work we did was very similar.
We might also notice that the graph above is symmetric about the c axis. Why is this?
Does the graph give us any ideas about how we might deduce one inequality from the other?
Step-by-step explanation:
Add a researcher compared the Heights and shoe sizes for 50 men, selected at random. The equation shown describes a line of best fit for the data, where X is the shoe size and why is the height, in inches (y=1.6x + 58)
Complete question :
a researcher compared the Heights and shoe sizes for 50 men, selected at random. The equation shown describes a line of best fit for the data, where X is the shoe size and y is the height, in inches (y=1.6x + 58)
Based on the equation, what is the approximate shoe size for a man having a height of 60 inches
Answer:
1.25
Step-by-step explanation:
Given the regression equation :
y=1.6x + 58
X = shoe size ; y = height
Using the equation, the approximate shoe size of a man with a height of 60 will be ;
60 = 1.6x + 58
60 - 58 = 1.6x
2 = 1.6x
2/1.6 = 1.6x / 1.6
x = 2 / 1.6
x = 1.25
Hence, approximate shoe size of a man with a height of 60 will be 1.25
At the annual dog show, Chantel noticed that there were three more Scotties than Schnauzers. She also
realized that the number of Wirehaired Terriers was five less than twice the number of Schnauzers. If there
were 78 dogs in all (counting Schnauzers, Scotties, and Wirehaired Terriers), how many Schnauzers were
there? Write and solve an equation.
9514 1404 393
Answer:
(s+3) +s +(2s-5) = 7820 SchnauzersStep-by-step explanation:
Each of the quantities is described in terms of the number of Schnauzers, and that is the number we are asked for. It is convenient to write an equation for the number of Schnauzers. We choose to use s for the variable representing that number.
s = # of Schnauzers
s+3 = # of Scotties, 3 more than the number of Schnauzers
2s-5 = # of Wire-haired Terriers, 5 less than twice the number of Schnauzers
The total number of dogs is then ...
(s+3) +(s) +(2s-5) = 78
4s -2 = 78 . . . . simplify
4s = 80 . . . . . . add 2
s = 20 . . . . . . . divide by 4
There were 20 Schnauzers at the dog show.
This is the problem that I need help with please. :)
Answer:
umm I think it 186
Step-by-step explanation:
The Ride-It Store rents bicycles.The cost is $8.50 for 1 hour, $13.65 for 2 hours,$18.80 for 3 hours, and $23.95 for 4 hours.If the pattern continues, how much will it cost Nate to rent a bike for 6 hours?
Answer:
in six hours it would be $34.25
Step-by-step explanation:
The computation of the total cost for 6 hours is shown below:
For one hour it would be $8.50
For two hours
= $13.65 - $8.50
= $5.15
For three hours
= $18.80 - $13.65
= $5.15
For four hours
= $23.95 - $18.80
= $5.15
For five hours
= $23.95 + $5.15
= $29.10
For six hours
= $29.10 + $5.15
= $34.25
Hence, in six hours it would be $34.25
Solve: 4-(9g-5)=-27 please help
Answer:
-2
Step-by-step explanation:
9g-4-5=-27
9g-9=-27
Add 9 to both sides
9g=-18
g=-2
Answer:
-18
if you need more info. just search up you question on Google and click the first link
Find the slope of the line. Explain or show your reasoning.
Write an equation for the line.
Find the values for and . Explain or show your reasoning.
what is the percent of change between 57000 people and 97000 people, and how do I solve it?
Answer:
58.76
Step-by-step explanation:
So, as you can tell 97,000 is the 100 percent of the people, in this case we don't know what 57,000 is. Just divide equation 1 by equation 2. Considering both have the same unit. I'm so sorry, if this doesn't explain the best way.
A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15?
A 3
B 4
C 5
D 6
E 8
Answer: B: 4
Step-by-step explanation:
A number is divisible by 15 if the number ends on a 0 or a 5, and the sum of its digits is divisible by 3.
We want to find a 5 digit flippy number, this can be: modeled with:
N = ababa
Where a and b are single digit numbers.
If we impose that N must end with a zero, then we will have:
N = 0b0b0 = b0b0
This is a 4-digit number, so we can discard all the options that end with a zero.
Then the only option that we have are the numbers like:
B = 5b5b5
This number will be only divisible by 15 if:
5 + b + 5 + b + 5 = K is divisible by 3.
Then let's try find the possible values of b.
K = 3*5 + 2*b
K has two terms, the left term is already divisible by 3, then K will be divisible by 3 only if the other term is also divisible by 3.
Then we want 2*b to be divisible by 3.
And 2 is a prime number, then b must be divisible by 3, and we know that b is a number between {0,1 , 2, 3, 4, 5, 6, 7, 8 ,9}
The options are:
2*0 = 0 is divisible by 3.
2*3 = 6 is divisible by 3.
2*6 = 12 is divisible by 3
2*9 = 12 is divisible by 3.
Then we have four values of b such that:
N = 5b5b5 is divisible by 15.
Then the correct option is: B: 4
Natalia and her friends held a bake sale to benefit a local charity, friends sold 15 cakes on the first dayand 22 cakes on the second day of the bake saleThey collected $60 on the first day $88 on the second day Write an equation to represent the amount R Natalia and her friends raised after selling c cakes .
Please i need this asap it’s due in 15 minutes!! I will give brainliest to whoever gets it right first!!
what is the x variable and y variable?
Answer:
x is the hours worked and y is the money earned
Step-by-step explanation:
Usually the x variable is the independent variable
The independent variable is the variable that does not rely on anything
The y variable is the dependent variable
The dependent variable is the variable that relies on the independent variable
In this case the money earned depends on how many hours you work for
I hope this helps :)
Answer:
I believe that it would be x is hours worked and y is dollars earned but I could be wrong I skipped a year of math because I completed it so quickly now I cant remember some stuff
Solve the following functions:
f(x) = 20x - 90 find f (x) = 10
Answer:
x=5
Step-by-step explanation:
Letters used in the word mathematics
Answer:
In this case, there are 11 letters in the word 'mathematics'; three letters appear twice: m, a, and t.
Step-by-step explanation:
I don't know why you want this but there you go
A circle is inscribed with 5 lines. The center of the circle is labeled point O. Four lines extend from point O. One connects to point W on the edge of the circle. The second line extends past the circle edge through point X on the edge. The third extends past the circle edge through point Z. The fourth extends past the circle edge through point Y. A line connects points W and X. The angle created by ray OX and ray OZ is angle . The two rays that create YOX are ray OX and ray . The center of the circle is point .
Answer:
XOZ
OY
O
Step-by-step explanation:
just took it on edge
subtract (7x^5-12x^2y+33y^2) from (5x^5-5x^2y+9y^2) simplify your results:
Answer:
- 2 [tex]x^{5}[/tex] + 7x²y - 24y²
Step-by-step explanation:
( 5 [tex]x^{5}[/tex] - 5x²y + 9y² ) - ( 7 [tex]x^{5}[/tex] - 12x²y + 33y² ) =
5 [tex]x^{5}[/tex] - 5x²y + 9y² - 7 [tex]x^{5}[/tex] + 12x²y - 33y² =
( 5 [tex]x^{5}[/tex] - 7 [tex]x^{5}[/tex] ) + ( - 5x²y + 12x²y ) + ( 9y² - 33y² ) =
- 2 [tex]x^{5}[/tex] + 7x²y - 24y²
Marching Bands: In how many ways can four marching bands and three floats line up for a paradeif two bands cannot march next to one another?
Answer:
There's 6 ways the boats can line up for the parade.
Step-by-step explanation:
Feel free to give brainliest.
Have an outstanding day!
Julie buys a new shirt. The original price of the shirt is $35.00, but it is on sale for 25% off. There is a 7% tax on the shirt. How much will Julie pay for the shirt?
Answer: $24.08
Step-by-step explanation: 25% off $35.00 is $22.50
and adding 7% tax which equals $1.50
so it adds up to 24.08
The function h(n) gives the number of person-hours it takes to assemble n engines in a factory. What is a reasonable domain for h(n)?
Answer:
For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
What is the best estimate of the measure of the angle shown?
*answer was incorrect please see below*
who good at math? , cuz i need help .
Answer:
what kind of math?
Step-by-step explanation:
Answer:
It depends your grade, but I might be able to help.
Step-by-step explanation:
What type of math?
pleaseeeee helpppppp
Answer:
BCE
Step-by-step explanation:
I hoped this helped and mark me as brainly pls
Answer:
a c b d
Step-by-step explanation:
What is the nature of the solution set to the following system of equations?
5x - y = 2
2x-0.5y=3
Answer:
u can use substitute or eliminate
I will give Brainlyest to the first person
A family gathered 12 seashells from the beach bringing the total number of seashells in their collection to 86
Which equation can be used to find, x, the number of seashells in the collection before the 12 were added?
Answer:
The first selection is the answer.
Step-by-step explanation:
x+12=86
The equation that can be used to find x is x + 12 = 86
A linear equation is of the form:
y = mx + b;
where y, x are variables, m is the slope of the line and b is the y intercept.
Let x represent the number of seashells in the collection before the 12 were added. Hence to find x, we use the equation:
x + 12 = 86
The equation that can be used to find x is x + 12 = 86
Find out more at: https://brainly.com/question/13911928
Graph the line
y=-2x=2
please hurry!
Answer:
-1,0
0,-2
slope is -2/1
Answer:
View the Screenshot that I have sent.
WILL GIVE BRAINLIEST. Will report if answer is absurd What is the trademark of a linear table?
Answer:
You can tell if a table is linear by looking at how X and Y change. If, as X increases by 1, Y increases by a constant rate, then a table is linear. You can find the constant rate by finding the first difference.
Step-by-step explanation:
Hope this helps!
[tex]\sqrt{8}+\sqrt{18}-\sqrt{32}[/tex]
Mikaila collected 27 pounds of glass bottles to bring to the recycling center. she also collected 400 ounces of newspapers and 160 ounces of aluminum cans. what is the total weight of the items Mikaila brought to the recycling center? write your answer in ounces. think you❤!