This is equivalent to finding the solutions of the equation x1 + x2 + x3 + x4 + x5 = 21 and with an application of Theorem 2: r = 21, n = 5, so we have C(21+5-1, 21) = C(25, 21) = 12,650 solutions
Three Types of Solutions of a System of Linear EquationsConsider the pair of linear equations in two variables x and y.
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Here a1, b1, c1, a2, b2, c2 are all real numbers.
Note that, a12 + b12 ≠ 0, a22 + b22 ≠ 0
1. If (a1/a2) ≠ (b1/b2), then there will be a unique solution. If we plot the graph, the lines will intersect. This type of equation is called a consistent pair of linear equations.
2. If (a1/a2) = (b1/b2) = (c1/c2), then there will be infinitely many solutions. The lines will coincide. This type of equation is called a dependent pair of linear equations in two variables
3. If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. If we plot the graph, the lines will be parallel. This type of equation is called an inconsistent pair of linear equations.
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What is the equation of a horizontal line passing through the point (- 3 8?
The equation of a horizontal line passing through the point (- 3 8) is y = 8
How to determine the equation of the lineFrom the question, we have the following parameters that can be used in our computation:
Point = (-3, 8)
This means that
(x, y) = (-3, 8)
Horizontal lines that do not pass through the origin do not intersect with the x-axis
This means that
Line = y
So, we have
y = 8
Hence, the equation is y = 8
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What are the measures of â X and â Y?
The measure of angle x and angle y is x = 75° and y = 135°, respectively.
We know that,
If the angles of a triangle are θ₁, θ₂ and θ₃, respectively. Then the sum of θ₁, θ₂ and θ₃ is equal to 180°.
In triangle ABC,
⇒ 45° + 60° + x = 180°
⇒x = 180° − 105°
⇒x = 75°
Now, If the exterior angle of a triangle is θ and the two opposite interior angles are θ₁ and θ₂, then the sum of θ₁ and θ₂ will be equal to the θ.
⇒y = 60° + x
⇒y = 60° + 75°
⇒y = 135°
Therefore, x =75° and y = 135°
--The given question is incorrect, the correct question is
"What are the measures of angle x and angle y?"--
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A newspaper advertisement offers a $10 000 used car for nothing down and 36 monthly payments of $450. What is the interest rate used? How much interest will you have to pay if you purchase this car?
The interest paid if you purchase this car is $4,860.
What exactly is a monthly payment?A monthly payment is a payment made every month to pay off loans or advances. It's similar to EMI.
The interest rate used can be found using the formula: Interest rate = (monthly payment x number of payments) / principal.
Interest rate = (450 x 36) / 10000 = 16.2%
The interest rate used is 16.2%
The interest paid can be calculated using the formula: Interest = Principal x Interest rate x Number of payments / 12
Interest = 10000 x 0.162 x 36/12
Apply the multiplication operation, and we get
Interest = 4,860
Thus, the interest paid if you buy this car is $4,860
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1) A cleaning service charges $45 per hour and $95 for supplies. Create an equation that shows the cost, c, of hiring the service for h hours.
When two expressions are joined by an equal sign, a mathematical statement is called an equation.
What is meant by equation?When two expressions are joined by an equal sign, a mathematical statement is called an equation. An equation is something like 3x - 5 = 16. By solving for x, we discover that x equals 7, which is the value for the variable. Expressions that equal one another are the building blocks of an equation. There is an equation: 4x2+3x = 22. A formula is an equation that has at least two variables and depicts the relationship between them. Molecular, structural, projection formulas, as well as empirical forms, are frequently used. Symbols for the components that make up a compound, like Na for sodium and Cl for chlorine, are used in empirical formulas, together with subscripts that show how many atoms are present in each constituent element.Let h be the number of hours for which the two costs would be the same.
PP charges $2220 + ($45/hr)*h
D&S charges $1850 + ($65/hr)*h
Note that the ($/hr)*hours gives $, because hours cancel out
Set the expression equal and solve for h (this is called the "break even point"):
2200 + 45h = 1850 + 65h
370 = 20h (subtract 1850 from both sides)
18,5 = h
At 18.5 hours, the companies would charge the same amount.
18 1/2 hours is 18 hours 30 minutes.
The complete question is,
Write and solve an equation to determine the number of hours for which the two costs would be the same.
Ms. Silver wants to hire a painter to paint her house. Painters Plus charges $2,220 plus $45 per hours. Davis & Sons charges $1,850 plus $65 per hour. Write and solve an equation to determine the number of hours for which the two costs would be the same.
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For the school play, 205 students buy tickets and 232 adults buy tickets. All tickets cost $16. How much did the school make in ticket revenue
Answer: $6,992
Step-by-step explanation:
First you add 205 and 232 then multiply your answer by 16
Write y=-1/2|x+2|+5 as a piecewise function
The piecewise function of the given function is
y= -x/2+4 if x ≥ -2
y= x/2+6 if x < -2
The solution is obtained using the modulus function.
What is modulus function?
The magnitude of a given number is indicated by the modulus of the number. By definition, a modulus function is a real-valued function, such as f: R -> R, where y = |x| for any x R OR f(x) = |x|.
F(x) is equal to the positive value of x for every non-negative value of x. However, f(x) is equal to the negative value of x for negative x values.
We are given y=-1/2|x+2|+5
So, first we need to solve for x+2≥0
We get x≥-2
Now, we get two equations,
y=-1/2(x+2)+5 if x ≥ -2
y= 1/2(x+2)+5 if x < -2
On solving the equations, we get
y=-1/2(x+2)+5 if x ≥ -2
y= -x/2-1+5
y= -x/2+4
y= 1/2(x+2)+5 if x < -2
y= x/2+1+5
y= x/2+6
Hence, the final answer as a piecewise function is
y= -x/2+4 if x ≥ -2
y= x/2+6 if x < -2
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Is 22 divisible by 2 Yes or no?
What is acute angle triangle example?
Acute angle triangle examples are Equilateral Acute Triangle, Isosceles Acute Triangle, and Scalene Acute Triangle.
The definition of an acute triangle states that it is a type of triangle in which all three interior angles are acute angles or less than 90°. The sides of an acute-angled triangle can be similar or unequal depending on whether the triangle is equilateral, isosceles, or scalene.
As we know that triangles can be categorised on the basis of sides and angles, an acute triangle can also be further categorised as:
Equilateral Acute Triangle: In an equilateral acute triangle, all three angles are equal to 60° and all sides are equal.
Isosceles Acute Triangle: In an isosceles acute triangle, two sides and two angles are equal, and all interior angles are less than 90°.
Scalene Acute Triangle: In a scalene acute triangle, all the 3 sides are of dissimilar lengths and the 3 interior angles are of dissimilar measures but all the interior angles measure less than 90°.
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At what point on the curve y slope is 1?
The point at which the slope of the curve y = [tex]e^x[/tex] is 1 is (0,1)
Now, According to the question:
Slope and Derivative:
The slope of a curve is the rate at which y is changing with respect to x at a particular point on the curve. The slope at a given point on a curve is also equal to the derivative of the curve at that point. We can use this fact to solve various problems and applications involving slope.
We know that the slope of the curve y = [tex]e^x[/tex] will be equal to 1 when the derivative of the curve is equal to 1. The derivative of y = [tex]e^x[/tex] is y ' = [tex]e^x[/tex] , so we can set this equal to 1, and solve for x to determine the x-value where the slope will be equal to 1.
[tex]e^x[/tex] = 1
Take the natural log of each side.
ln([tex]e^x[/tex]) = ln(1)
Simplify each side.
x = 0
We get that the slope of y = [tex]e^x[/tex] will be 1 when x = 0. We can find the corresponding y-value of the point by plugging 0 in for x into the original equation, and solving for y.
y = [tex]e^0[/tex]
Simplify the right-hand side.
y = 1
We get that the point where the slope of the curve is y = [tex]e^x[/tex] is (0,1).
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The complete question is this :
At what point on the curve y = [tex]e^x[/tex] slope is 1?
WILL GIVE BRAINLIEST FOR SOLVE: To write 4^-4 with a positive exponent, ______________.
A. Use the reciprocal of the base, then change the exponent to a positive number and
place it with the number in the denominator.
B. Simply change the exponent to a positive number.
C. Write it as 4/4
D. Change the base to a negative number and make the exponent positive.
Writing 4⁻⁴ to have a positive exponent should be of the form 1/4⁴
What is exponent?In math, the term exponent refers to the number of times a certain number multiplies it self.
The number that is being multiplies is referred to as the base number while the number of times it multiplies itself is the exponents.
In the law of exponents, numbers with negative exponents will have positive when written in reciprocal form.
Because of this law, the number as in the problem is 4⁻⁴ the reciprocal form of this number is 1/4⁴ and the negative exponent vanishes
The positive exponent of the number is 4⁻⁴
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Answer:
1/(4^4)
Step-by-step explanation:
A. Use the reciprocal of the base, then change the exponent to a positive number and place it with the number in the denominator.
Gavin was thinking of a number. Gavin doubles it, then adds 15 to get an answer of 44.2. What was the original number?
Answer: 29.2!
Step-by-step explanation: You must subtract 15 from 44.2 which would get you to 29.2!
Algebra 1 class work
Answer: Categorical bar graph
Step-by-step explanation:
Question: What are 3 things you notice about the parabola below?
The 3 things, I noticed about parabola is given below.
What is parabola?A parabola is a curve drawn in a plane.
Where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix)
We have a parabola.
From the graph of parabola:
1). The vertex of the parabola = (2, -5)
2). And the y-intercept = (0, 7)
3). The curve passes through (1, 2) and (3, 2).
Therefore, all the three points are given above.
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Meah bought a dress for $550 at 20% discount. What was the original price?
Let's take this problem in steps:
--> original price of the dress
What does this question give us:
--> a 20% discount of the original price is $550
That means:
==> (100 - 20)% of the original price is $550
==> 80% of the original price is $550
In equation form:
[tex]\dfrac{80}{100} * \text{ original price}=550\\\\\\\dfrac{80* \text{ original price}}{100} =550\\\\\\80* \text{ original price}=550*100=55000\\\\\\\text{original price}=\dfrac{55000}{80} = 687.5[/tex]
Thus the original price is $687.50
Answer: $687.50
The original price of Meah bought a dress is $687.50.
When using trade discounts and discount rates, how do you determine the initial price?The percent discount should be divided by 100% in order to be converted to a decimal. Create the equation P=(1d)x P = (1 d) x to determine the item's original price, where P is the sale price, d is the discount expressed as a decimal, and x is the item's original price.
Explanation:
To solve this issue, let's do it piecemeal:
the dress's initial purchase price
This inquiry provides us with:
--> The amount after a 20% reduction equals $550.
that implies:
(100 - 20)% of the purchase price is $550.
The amount is $550, or 80% of the purchase price.
written as an equation
Consequently, $687.50 is the starting price.
In response, $687.50.
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Find the Average Rate of Change on the table from 2 < x < 5:
The Average Rate of Change on the table from 2 ≤ x ≤ 5 is 3
How to find the average rate from the table?The average rate of change of a function is a measure of how the output value of the function (y) changes with respect to the input value (x) as x changes. Mathematically:
Average rate of change = Δy/Δx
To find the Average Rate of Change on the table from 2 ≤ x ≤ 5:
when x =2, h(x) = 0 and when x = 5, h(x) = 9
Average rate of change = Δh/Δx
Average rate of change = (9 - 0) / (5 - 2)
Average rate of change = 9/3
Average rate of change = 3
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Solve the following system of equations
5x+2y=-6
3x-2y=22
Answer:
y = 8
Step-by-step explanation:
5x + 2y = -6 .... 5x = -6 - 2y ...... x = -6/5 -2y/5
3 × ( -6/5 -2y/5 ) - 2y = 22
-18/5 - 6y/5 - 2y = 22 ... (×5)
-18 - 6y - 10y = 110
-16y = 128 .... (× -1)
16y = -128 .... (÷16)
y = 8
Do the segments connecting points A, B, and C form a right triangle? Show work that leads to your answer. Round to the nearest tenths place if necessary.
Show your work!
No, the segments connecting points A, B, and C does not to form a right-angled triangle.
What is Pythagorean theorem?Mathematically, Pythagorean's theorem is given by this mathematical expression:
a² + b² = c²
Where:
a, b, and c represents the length of sides of a right-angled triangle.
In order for the segments connecting points A, B, and C to form a right-angled triangle, its side lengths must satisfy the Pythagorean's theorem:
AB² = (xB - xA)² + (yB - yA)²
AB² = (6 - (-2))² + (2 - 2)²
AB² = 8² + 0²
AB² = 8²
AB = 8
AC² = (xC - xA)² + (yC - A)²
AC² = (0 - (-2))² + (6 - 2)²
AC² = 2² + 4²
AC² = 4 + 16
AC = √20
BC² = (0 - 6)² + (6 - 2)²
BC² = (-6)² + 4²
BC² = 36 + 16
BC = √52
From Pythagorean's theorem, we have:
a² + b² = c²
√20² + √52² = 8²
20 + 52 = 64
72 ≠ 64 (False, not a right-angled triangle).
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what is the probability that a 5-card straight will be drawn (a straight is 5 cards in a row of any suit) from a standard deck of 52 cards if the first three cards drawn are the 2 of spades, the 4 of diamonds, and the 6 of diamonds?
The probability that a 5-card straight will be drawn from a standard deck of 52 cards if the first three cards drawn are the 2 of spades, the 4 of diamonds, and the 6 of diamonds is 0.005.
A standard deck has 52 cards.
In this case, three cards are already drawn from the deck, which are the 2 of spades, the 4 of diamonds, and the 6 of diamonds. Thus, remaining number of cards= 49.
Probability of getting first card: 1/4
Probability of getting the second card from same suit= 1/13+ 1/13+ 1/12+ 1/11= 563/1716
Probability of getting the third card from same suit= 1/12+1/12+ 1/11+ 1/10= 472/ 1320
Probability of getting the fourth card from same suit= 1/11+ 1/11+ 1/10+ 1/9= 389/ 990
Probability of getting the fifth card from same suit= 1/10+ 1/10+ 1/9+ 1/8= 314/ 720
Therefore, total probability of the event that a 5-card straight will be drawn,
= (1/4)* (563/1716)* (472/ 1320)* (389/990)* (314/720)
= 0.005
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An alloy is a mixture of different kinds of metal a company is going to produce an alloy that is 80% nickel and 20% iron it is using 160 pounds of nickel to make the alloy how many pounds of iron should it add
If an alloy is a mixture of different kinds of metal a company is going to produce an alloy that is 80% nickel . The number of pounds of iron that should be added is 640 grams.
How to find the number of pounds of iron?Since 80% of the alloy is made of Iron and 20% is Nickel so,
Let x represent the total weight
160 grams = 20% × x
x = 160 / 20%
x = 800 grams
Iron should be 80% of the full weight of the alloy:
Pounds of iron = 80% x weight of Alloy x
Pounds of iron = 80% x 800 grams
Pounds of iron = 640 grams
Therefore 640 grams of Iron should be added.
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Graph the line with slope -1/3 passing through the point (2,-5) .
Considering the definition of a line, the equation of the line that passes through the point (2, -5) and has a slope of -1/3 is y= -1/3x - 13/3 and the graph is attached.
What is a Linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin and always coincides with the value of y corresponding to the value of x= 0, that is represents the coordinate of the point where the line crosses the y axis.Line in this caseIn this case, you know:
The line has a slope of -1/3.The line passes through the point (2, -5).Substituting the value of the slope and the value of the point in the expression y = mx + b, the value of the ordinate to the origin b can be obtained:
-5= -1/3×2 + b
-5= -2/3 + b
-5 + 2/3= b
-13/3= b
Finally, the equation of the line is y= -1/3x - 13/3.
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A solution containing 32 % insecticide is to be mixed with a solution containing
55% insecticide to make 100 L of a solution containing 44% insecticide. How much
of each solution should be used?
40L of each solution should be used
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Let x=amount of 32% insecticide needed
Then 100-x=amount of 55% insecticide needed
Now we know that the amount of pure insecticide in the 32% solution(0.32x) plus the amount of pure insecticide in the 55% solution(0.55(100-x)) has to equal the amount of pure insecticide in the final mixture (0.44×100).
So our equation to solve is:
0.32x+0.52(100-x)=0.44×100 get rid of parens
0.32x+52-0.52x=44
52-44-0.2x=0
8=0.2x
x=40L
Hence, 40L of each solution should be used
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1 out of 8 students wants pancakes and the rest want waffles. What is the ratio of the number of students who want waffles to the total number of students?
The ratio of students who want waffles to the total number of students is 7/8.
What is the ratio?
The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.
Out of 8 students, 1 student wants pancakes and 8 - 1 = 7 students want waffles.
To find the ratio of the number of students who want waffles to the total number of students, we can divide the number of students who want waffles by the total number of students and express the result as a fraction.
So the ratio is:
(Number of students who want waffles) / (Total number of students) = 7/8
This fraction can be simplified by dividing both the numerator and denominator by the greatest common factor, which in this case is 1, so it remains the same.
Hence, the ratio of students who want waffles to the total number of students is 7/8.
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question 2 please answer
Based on the function f(x) = 9x - 27, the input value when f(x) = 18 is equal to 5.
What is a function?In Mathematics, a function can be defined as a mathematical expression which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair in tables or relations.
What is a domain?In Mathematics, a domain is sometimes referred to as input value and it can be defined as the set of all real numbers for which a particular function is defined.
When the range (output value) of the given function f(x) = 9x - 27, the input value (domain) can be calculated as follows;
f(x) = 9x - 27
18 = 9x - 27
9x = 18 + 27
9x = 45
Input value, x = 45/9
Input value, x = 5
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Can someone help me with these questions?
Also, you have to multiply the fractions/mixed numbers.
a) The fraction of tank filled with gasoline is given by the equation A = 3/8
b) The solution to the system of equations is
xz - y = 11/6
yz + x = 13
3yz = 29
What is a Fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.
Given data ,
a)
Let the fraction be represented as A
Now , After first journey the amount of fuel left was = 1 - ( 7/16 )
On simplifying the equation , we get
The amount of fuel left = 9/16
( 1/3 ) of the fuel was used , so 2/3 of the fuel is left in the tank
Substituting the values in the equation , we get
The amount of fuel left in the tank A = ( 2/3 ) of ( 9/16 )
The amount of fuel left in the tank A = 3/8
Hence , the fraction is A = 3/8
b)
The value of x = 3 1/3 = 10/3
The value of y = 4 5/6 = 29/6
The value of z = 2
Let the equation be B = xz - y
Substituting the values in the equation , we get
B = ( 10/3 ) 2 - ( 29/6 )
B = 20/3 - 29/6
B = ( 40 - 29 ) / 6
B = 11/6
Let the equation be C = yz + x
Substituting the values in the equation , we get
C = ( 29/6 ) 2 + ( 10/3 )
C = 58/6 + 10/3
C = ( 58 + 20 ) / 6
C = 13
Let the equation be D = 3yz
Substituting the values in the equation , we get
D = 3 ( 29/6 ) 2
D = ( 29/6 ) x 6
D = 29
Hence , the equations are solved
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Multiple Choice: Which interval on the graph below is positive?
Answer:c
Step-by-step explanation:
it easy
How do you classify a triangle having the sides 3cm 4cm and 5cm?
It is possible to construct a triangle whose sides are 3 cm 4 cm and 5 cm because the sum of two sides are greater than three sides.
In the given question, we have to classify a triangle having the sides 3cm 4cm and 5cm.
The sum of the two sides must always be greater than the third side in order to determine whether or not the given sides may form a triangle.
To check this we firstly add the 10.2 and 5.8
3 + 4 > 5
7 > 5
Now we add 3 and 5
3 + 5 > 4
8 > 4
Now we add 4 and 5
4 + 5 > 3
9 > 3
It is possible to construct a triangle whose sides are 3 cm 4 cm and 5 cm because the sum of two sides are greater than three sides.
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Triangle JKL, with vertices J(2,3), K(7,4), and L(3,9), is drawn inside a rectangle, as
shown below.
What is the area, in square units, of triangle JKL?
Answer:
14.5
Step-by-step explanation:
The area of the rectangle is [tex](5)(6)=30[/tex].
The areas of the three right triangles in the rectangle but not in [tex]\triangle JKL[/tex] are 2.5, 10, and 3.
[tex]\implies [JKL]=30-2.5-10-3=14.5[/tex]
Which expressions are equivalent to the given expression?
√252
Tonya is 9 years younger than twice Marty's age Daniella is 2 years older than Tanya's age the sum of the ages of the three siblings can be expressed as 3m + 8 where m represents Marty's age how old is Marty
The age of Morty is 12 years.
What is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
given:
let the Age of Marty be m.
Then, Tonya's Age = 2m- 9
and, Daniella's age = 2m- 9 + 2= 2m - 7
Now, sum of their ages = 3m+ 8
So, m + 2m-9 + 2m -7 = 3m+ 8
5m - 16= 3m + 8
5m - 3m = 8 +16
2m = 24
m = 12
So, Marty's age is 12 years.
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What is 12x^3-4x^2 while being fully factorised?
Answer:187
Step-by-step explanation: