Final Answer:
The derived relationship between p and q that is independent of y is: q = 1/2 * p
Explanation:
To find the relation between 'p' and 'q' that is independent of 'y,' we will combine the two given equations and eliminate 'y'.
The equations given are:
1) -2y² - 2y = p
2) -y² + y = q
First, we want to manipulate these equations to isolate similar terms. Notice that the first equation has -2y² and the second has -y². If we multiply every term in the second equation by 2, we will have a coefficient of -2y² in the second equation, which will help us cancel out the y² terms. Let's do that:
2(-y² + y) = 2q
-2y² + 2y = 2q
Now, let's subtract the second equation from the first equation:
(-2y² - 2y) - (-2y² + 2y) = p - 2q
On subtracting, -2y² will cancel out with -2y², and -2y will subtract 2y to give -4y:
-2y² + 2y² - 2y - 2y = p - 2q
0 - 4y = p - 2q
-4y = p - 2q
Since we want a relationship without 'y', we can't do much with this result directly, as it still contains 'y'. But let's look at the equations we've been given once more.
The goal is not to solve for 'y' but to find a relationship between 'p' and 'q'. To accomplish this, let's compare the two original equations and try to eliminate 'y' by dividing them. Divide the first equation by the second equation:
(-2y² - 2y) / (-y² + y) = p / q
Now, factor out -y from both the numerator and the denominator:
- y(2y + 2) / - y(y - 1) = p / q
Simplify the expression by canceling out the -y term:
(2y + 2) / (y - 1) = p / q
At this point, you can see that there is no straightforward way to solve this for a relationship that is completely independent of 'y' because the y's don't cancel out.
One method to proceed, since we must get rid of 'y', is to compare coefficients that correspond to the same powers of 'y' assuming p and q are related through such a power series.
We have from the first equation by rearranging:
y² + y = -p/2
Comparing coefficients to the second equation:
y² = -q
y = q
By matching coefficients for the same powers of y, we deduce:
y (from -y²) = -q (from -y² + y), so q = 1/2 * p
Thus, our derived relationship between p and q that is independent of y is:
q = 1/2 * p
This indicates that q is half of p.
The longer base of an isosceles trapezoid measures 18 ft. The nonparallel sides measure 8 ft, and the base angles measure 75 degrees.
a) Find the length of a diagonal.
b) Find the area.
Answer:
a) The length of the diagonal is 17.71 feet
b) The area of the trapezoid is 123.14 feet²
Step-by-step explanation:
* Lets explain how to solve the problem
- Look to the attached figure
- ABCD is an isosceles trapezoid
∵ DC is the longer base with length 18 feet
∵ AD and BC are the two non-parallel sides with length 8 feet
∵ ∠ ADC and ∠ BCD are the bases angles with measure 75°
- AE and BF are ⊥ DC
# In Δ BFC
∵ m∠BFC = 90° ⇒ BF ⊥ CD
∵ m∠C = 75°
∵ BC = 8
∵ sin∠C = BF/BC
∴ sin(75) = BF/8 ⇒ multiply both sides by 8
∴ BF = 8 × sin(75) = 7.73
∵ cos∠C = CF/BC
∴ cos(75) = CF/8 ⇒ multiply both sides by 8
∴ CF = 8 × cos(75) = 2.07
# In Δ BFD
∵ m∠BFD = 90°
∵ DF = CD - CF
∴ DF = 18 - 2.07 = 15.93
∵ BD = √[(DF)² + (BF)²] ⇒ Pythagoras Theorem
∴ BD = √[(15.93)² + (7.73)²] = 17.71
a)
∵ BD is the diagonal of the trapezoid
* The length of the diagonal is 17.71 feet
b)
- The area of any trapezoid is A = 1/2 (b1 + b2) × h, where b1 and b2
are the barallel bases and h is the height between the two bases
∵ b1 is CD
∴ b1 = 18
∵ b2 is AB
∵ AB = CD - (CF + DE)
∵ ABCD is an isosceles trapezoid
∴ CF = DE
∴ AB = 18 - (2.07 + 2.07) = 13.86
- BF is the perpendicular between AB and CD
∴ BF = h
∴ h = 7.73
∵ A = 1/2 (18 + 13.86) × 7.73 = 123.14
* The area of the trapezoid is 123.14 feet²
What is the inverse of the function f(x) = 2x + 17
To find the inverse of a function switch the place of y (aka f(x) ) with x. Then solve for y.
Original equation:
y = 2x + 17
Switched:
x = 2y + 17
Solve for y by isolating it:
x - 17 = 2y + 17 - 17
x - 17 = 2y
(x - 17)/2 = 2y/2
[tex]\frac{1}{2}x-\frac{17}{2}= y[/tex]
Hope this helped!
~Just a girl in love with Shawn Mendes
Simplify the expression.
-81 = (-9)
Answer:
it must be 9
Step-by-step explanation:
it is a simple division.
What is the solution to the system of equations graphed below?
А.(6, 0)
B.(1, 5)
С.(0.3)
D.(0,6)
Answer:
B
Step-by-step explanation:
The solution to a system of equations given graphically is at the point of intersection of the 2 lines, that is
Solution = (1, 5 ) → B
[tex]\huge{\boxed{\text{(1, 5)}}}[/tex]
All you need to do is find where the intersection of the lines is located.
Count how many units to the right. [tex]1[/tex] This is our [tex]x[/tex] value.
Count how many units up. [tex]5[/tex] This is our [tex]y[/tex] value.
About 95% of sixth-grade students will have heights between ______ inches and ______inches.
Answer:
53.4 and 62.6
Step-by-step explanation:
Answer:
53.4 and 62.6
Step-by-step explanation:
Got it right :/
sin y +cos y + tan y sin y = sec y +cos y tan y. Verify the Identity. Show all Steps!
[tex]\bf sin(y)+cos(y)+tan(y)sin(y)=sec(y)+cos(y)tan(y) \\\\[-0.35em] ~\dotfill\\\\ sin(y)+cos(y)+tan(y)sin(y)\implies sin(y)+cos(y)+\cfrac{sin(y)}{cos(y)}\cdot sin(y) \\\\\\ sin(y)+cos(y)+\cfrac{sin^2(y)}{cos(y)}\implies \stackrel{\textit{using the LCD of cos(y)}}{\cfrac{sin(y)cos(y)+cos^2(y)+sin^2(y)}{cos(y)}} \\\\\\ \cfrac{sin(y)cos(y)+\stackrel{cos^2(y)+sin^2(y)}{1}}{cos(y)}\implies \cfrac{sin(y)cos(y)+1}{cos(y)} \\\\\\ \cfrac{sin(y)}{cos(y)}\cdot cos(y)+\cfrac{1}{cos(y)}\implies tan(y)cos(y)+sec(y)[/tex]
r=2sec(theta) converted into a cartesian equation
[tex]\bf r=2sec(\theta )\qquad \begin{cases} x=rcos(\theta )\\ \frac{x}{r}=cos(\theta ) \end{cases}\qquad \implies r=2\cdot \cfrac{1}{cos(\theta )}\implies r=\cfrac{2}{~~\frac{x}{r}~~} \\\\\\ r=\cfrac{\frac{2}{1}}{~~\frac{x}{r}~~}\implies r=\cfrac{2r}{x}\implies x=\cfrac{2~~\begin{matrix} r \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{~~\begin{matrix} r \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies x=2[/tex]
a^3b^-2c^-1d if a=2 b=4 c=10 d=15 express as a reduced fraction
[tex]\bf a^3b^{-2}c^{-1}d\implies \cfrac{a^3d}{b^2c}\qquad \begin{cases} a=2\\ b=4\\ c=10\\ d=15 \end{cases}\implies \cfrac{2^3\cdot 15}{4^2\cdot 10}\implies \cfrac{120}{160}\implies \cfrac{3}{4}[/tex]
Given the system of equations, match the following items.
x + 3y = 5
x - 3y = -1
To solve the system of equations x + 3y = 5 and x - 3y = -1, we add both equations to get 2x = 4, solve for x to find x = 2, and then substitute x back into one of the equations to find y = 1, resulting in the solution (2, 1).
The solution of the system of linear equations given by x + 3y = 5 and x - 3y = -1 involves manipulating the equations to find the values of x and y that satisfy both equations simultaneously. One common method to solve these is to add or subtract the equations, which eliminates one variable, making it possible to solve for the other.
Starting with the addition method, we align the equations and add them together:
x + 3y = 5
x - 3y = -1
Adding these equations, we get 2x = 4, and solving for x gives us x = 2. We can substitute x = 2 back into one of the original equations to find the value of y, yielding y = 1.
The solution to the system is the intersection point of the two lines represented by the equations, which is the point (2, 1).
the area of this rectangle is 4x^2.what does the coefficient 4 mean in terms of the problem?
Answer:
If the width of the rectangle is x than the length is 4x because 4x*x is 4x^2
Step-by-step explanation:
All rhombuses are. Parallelograms square rectangules quadrilaterals
Step-by-step explanation:
Look at the picture.
All rhombuses are
parallelograms
quadrilaterals
Identify the restrictions on the domain of f(x) = quantity x plus 5 over quantity x minus 2.
The restriction on the domain of the function [tex]f(x) = {x + 5}/{x - 2}.[/tex] is that x cannot be equal to 2, since it would make the denominator zero, which is undefined in real numbers.
The student is asking to identify the restrictions on the domain of the function [tex]f(x) = {x + 5}/{x - 2}.[/tex] The domain of a function includes all the values that x can take for which the function is defined. In the case of a rational function, any values that make the denominator zero must be excluded from the domain since division by zero is undefined.
In this function, the denominator is x - 2. Therefore, the value that makes the denominator zero is x = 2. To identify the restrictions on the domain of [tex]f(x) = {x + 5}/{x - 2}.[/tex] we set the denominator equal to zero and solve for x:
x - 2 = 0x = 2Hence, the only restriction on the domain of this function is that x cannot be 2. So the domain of f(x) is all real numbers except x = 2.
Solve for x. Write the smaller solution first, and the larger solution second. (x-10)^2-1=0
Answer:
[tex]x_1 = 9[/tex] and [tex]x_2 = 11[/tex].
Step-by-step explanation:
Start by adding 1 to both sides of this equation.
[tex](x - 10)^{2} = 1[/tex].
The square of what number or numbers will lead to the number "1"? It turns out that not only [tex]1^{2} = 1[/tex], but [tex](-1)^{2}= 1[/tex] as well. In other words, the value [tex](x - 10)[/tex] can be either 1 or -1. Either way, the equation is still going to hold. That's the reason why there are two solutions to this equation.
Consider the case when [tex]x - 10 = 1[/tex]. Add 10 to both sides of the equation. [tex]x = 11[/tex].
Now, consider the case when [tex]x - 10 = -1[/tex]. Again, add 10 to both sides of the equation, [tex]x = 9[/tex].
Order the two solutions in an increasing order:
[tex]x_1 = 9[/tex],[tex]x_2 = 11[/tex].Find the final amount for a $750 investment at 5.25% interest compound semiannually for 25 years
Answer:
=$2739.81
Step-by-step explanation:
To find the total amount if the interest is compounded, we use the compound interest formula.
A=P(1+R/100)ⁿ
A is the amount, P- principal, is the invested amount R is the % interest rate, n is the number if periods.
If compounded semi-annually, it means we have two periods in 1 year
The rate is also divided by 2
Thus 25 years have (25×2) = 50 periods.
A= 750(1+5.25/200)⁵⁰
=750(1.02625)⁵⁰
=$2739.81
a line passes through (3,-2) and (6,2). write an equation in point-slope form. rewrite the equation in standard form
again, bearing in mind that standard form for a linear equation means
• all coefficients must be integers, no fractions
• only the constant on the right-hand-side
• all variables on the left-hand-side, sorted
• "x" must not have a negative coefficient
[tex]\bf (\stackrel{x_1}{3}~,~\stackrel{y_1}{-2})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{2}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{2-(-2)}{6-3}\implies \cfrac{2+2}{6-3}\implies \cfrac{4}{3} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-2)=\cfrac{4}{3}(x-3)\implies y+2=\cfrac{4}{3}x-4 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf y=\cfrac{4}{3}x-6\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{3}}{3(y)=3\left( \cfrac{4}{3}x-6 \right)}\implies 3y=4x-18 \\\\\\ -4x+3y=-18\implies \stackrel{\textit{standard form}}{4x-3y=18}[/tex]
Find the equation in slope-intercept form that describes a line through (4, –2) with slope –3
Answer:
y = - 3x + 10
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here slope m = - 3, hence
y = - 3x + c ← is the partial equation
To find c substitute (4, - 2) into the partial equation
- 2 = - 12 + c ⇒ c = - 2 + 12 = 10
y = - 3x + 10 ← equation of line
Answer: [tex]y=-3x+10[/tex]
Step-by-step explanation:
The equation of a line in intercept form: [tex]y=mx+c[/tex]
The equation of a line passing through (a,b) and has slope m is given by :_
[tex](y-b)=m(x-a)[/tex]
Similarly, the equation in slope-intercept form that describes a line through (4, -2) with slope -3 will be :_
[tex](y-(-2))=-3(x-4)\\\\\Rightarrow\ y+2=-3x+12\\\\\Rightarrow\ y=-3x+12-2\\\\\Rightarrow\ y=-3x+10\ \ \text{In intercept form}[/tex]
Hence, the equation in slope-intercept form that describes a line through (4, -2) with slope -3 = [tex]y=-3x+10[/tex]
rectangle with a side length of 11" and a diagonal of 14" what is the perimeter
Answer:
10sqrt3+22
Step-by-step explanation:
Ok, let us imagine it as a sort of rectangle split upon its diagonal.
Using that, we can Pythag it out,
11^2+b^2=14^2
121+b^2=196
b^2=75
b=sqrt75
b=5sqrt3
Ok, using this info, we find the perimeter,
5sqrt3+5sqrt3+11+11
10sqrt3+22
The answer is 10sqrt3+22
The answer is:
The perimeter of the rectangle is equal to 39.32".
[tex]Perimeter=39.32in[/tex]
Why?Since we are working with a rectangle, we can use the Pythagorean theorem to find the missing side of the rectangle and calculate its perimeter. We must remember that we can divide a rectangle into two equal right triangles.
According to the Pythagorean Theorem, we have:
[tex]a^{2}=b^{2}+c^{2}[/tex]
Where:
a, represents the hypotenuse of the triangle which is equal to the diagonal of the given rectangle (14")
b and c are the other sides of the triangle.
Now, let be "a" 14" and "b" 11"
So, solving we have:
[tex]a^{2}=b^{2}+c^{2}[/tex]
[tex]14^{2}=11^{2}+c^{2}[/tex]
[tex]14^{2}-11^{2}=c^{2}[/tex]
[tex]14^{2}-11^{2}=c^{2}\\\\c=\sqrt{14^{2} -11^{2} }=\sqrt{196-121}=\sqrt{75}=8.66in[/tex]
Now, that we already know the the missing side of the rectangle, we can calculate the perimeter using the following formula:
[tex]Perimeter=2base+2length\\\\Perimeter=2*11in+2*8.66in=22in+17.32in=39.32n[/tex]
Hence, we have that the perimeter of the rectangle is equal to 39.32".
Have a nice day!
Help!!
Which of the following options is the cheapest per month over all? Assume a month has 30 days
A. rent 11.95 a day
B Lease 149.00 a month 3180 due at signing
C. Buying 16,000.00
D Finance 389.00 /month
Answer:
The correct option is A.
Step-by-step explanation:
We need to find the cheapest per month over all.
Assume a month has 30 days.
In option A:
Rent = 11.95 a day
Monthly rent = 11.95 × 30 = 358.5
Total renting amount is 358.5.
In option B:
Lease = 149.00 a month 3180 due at signing
Total amount = 149 + 3180 = 3329
Total leasing amount is 3329.
In option C:
Buying = 16,000
In option D:
Finance = 389.00 /month
The cheapest amount for a month is 358.5 .Therefore the correct option is A.
Answer: renting a car
Step-by-step explanation:
Find the equation of the line that
is perpendicular to y =1/6 x + 3
and contains the point (-3,23).
Answer:
y = - 6x + 5
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = [tex]\frac{1}{6}[/tex] x + 3 ← is in slope- intercept form
with slope m = [tex]\frac{1}{6}[/tex]
Given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{1}{6} }[/tex] = - 6, hence
y = - 6x + c ← is the partial equation of the perpendicular line.
To find c substitute (- 3, 23) into the partial equation
23 = 18 + c ⇒ c = 23 - 18 = 5
y = - 6x + 5 ← equation of perpendicular line
Kelly drinks 0.5 liters of coffee and 0.3 liters of yogurt drink at breakfast. How much did she drink in total in milliliters?
Answer:
800 milliliters
Step-by-step explanation:
we know that
To find out the total amount Kelly drank, add the amount of coffee and the amount of yogurt and convert the result to milliliters.
so
0.5+0.3=0.8 liters
Remember that
1 liter= 1,000 milliliters
so
0.8 liters=0.8*1,000=800 milliliters
Answer:
800 mL
Step-by-step explanation:
Because we know that
1 liter equals 1000 milliliters
So 0.5+0.3=0.8
0.8 Liters=0.8*1,000 ML
(50 Points)
Drag each description to the correct location on the table. Each description can be used more than once.
Some systems of equations and their graphs are given in the table. For each system, place the description(s) in the box that correctly describe the type of system shown.
Please helppppp :((((
Answer:3x+y=3 is the red line.
6x+y=-4 is the blue line.
Step-by-step explanation:I answer it on the test it is right..
Sin2x=______
A.2sinxcosx
B.1/2(cos(a-b)-cos(a+b))
C.1-2sin^2x
D.2sinx+2cosx
Answer:
A.2sinxcosx
Step-by-step explanation:
We know the trig identity
Sin (2a) = 2 sin a cos a
sin (2x) = 2 sin x cos x
Answer:
2sinxcosx
Step-by-step explanation:
A P E X
classify XYZ.
A. Scalene triangle
B. Right triangle
C. Isosceles triangle
D. Equilateral triangle
Answer:
Scalene Triangle
Step-by-step explanation:
By definition, scalene triangles have 3 sides of unequal length.
FYI,
Right Triangle : triangle with one of the angles = 90°
Isosceles Triangles: Triangle with 2 sides of the same length.
Equilateral triangle: Triangle with 3 sides of the same length.
What is the volume of the triangular prism shown below? PLEASE HELP 10 points
Answer:
270
Step-by-step explanation:
base area=18
18*15=270
The volume of the triangular prism is equal to [tex]270[/tex] cu. units.
What is volume?
" Volume is defined as the total space occupied by a three-dimensional object."
Formula used
Volume of a triangular prism = Area of the base × height
Area of the base [tex]= \frac{1}{2} \times base \times height[/tex]
According to the question,
Given dimensions,
Base of triangle [tex]= 9 units[/tex]
Height of the triangle [tex]=4 units[/tex]
Height of the triangular prism [tex]= 15 units[/tex]
Substitute the value in the formula to get the area of the base we have,
Area of the base [tex]= \frac{1}{2}\times 9\times 4[/tex]
[tex]= 18 square units[/tex]
Volume of a triangular prism [tex]= 18 \times 15[/tex]
[tex]= 270 cu.units[/tex]
Hence, the volume of the triangular prism is equal to [tex]270[/tex] cu. units.
Learn more about volume here
https://brainly.com/question/1578538
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Which equation shows the variable terms isolated on one side and the constant terms isolated on the other side for the equation 3x-5=-2+10
To isolate the variable terms on one side and the constant terms on the other side of the equation 3x - 5 = -2 + 10, add 2 to both sides, simplify to 3x - 3 = 10, then add 3 to both sides to get the final simplified equation 3x = 13.
Explanation:The equation 3x - 5 = -2 + 10 needs to be rearranged to isolate the variable terms on one side and the constant terms on the other. To do this, follow these steps:
Add 2 to both sides to move the constant term from the right to the left side: 3x - 5 + 2 = 10.Simplify both sides: 3x - 3 = 10.Add 3 to both sides to completely isolate the constant terms on one side: 3x = 10 + 3.Simplify the equation: 3x = 13.Now, we have successfully isolated the variable terms (3x) on one side of the equation and the constant terms (13) on the other side.
Solve: ( Brainliest ) -- TIME LIMIT: 8:00 minutes
2/3·z=10/9
Answer in proper and improper
Answer:
z = 5/3 or 1 2/3
Step-by-step explanation:
2/3·z=10/9
Multiply each side by 3/2
3/2*2/3·z=10/9*3/2
z = 30/18
We can simplify by dividing the top and bottom by 6
z = 5/3
Changing to a mixed number
z =1 2/3
Answer:
1⅔ [OR 5⁄3]
Step-by-step explanation:
2 × ? = 10
---------------
3 × ? = 9
That would be 1⅔.
I am joyous to assist you anytime.
Which numbers are irrational? Check all that apply
Irrational numbers cannot be expressed as a fraction or ratio of two integers and have decimal representations that go on forever without repeating.
Explanation:Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers. They are decimal numbers that go on forever without repeating. Examples of irrational numbers include π, √2, and √3. These numbers cannot be expressed as a simple fraction or as a terminating or repeating decimal.
Determine if parallel, perpendicular, or neither.
3y+4x=12
-6y=8x+1
What does h(40)=1820 mean in terms of the problem ? Help please
Final answer:
The notation h(40)=1820 means that the function h produces an output of 1820 when the input is 40, although additional context is needed to determine what h represents specifically in this scenario.
Explanation:
The expression h(40)=1820 typically means that a function h is being evaluated at the input value of 40, and the output is 1820. This could represent a variety of contexts, such as the height of a rocket in meters at 40 seconds after launch, the amount of money saved after 40 weeks, or any other situation described by a function where the variable h depends on the number 40. Without additional context, it's impossible to say precisely what 1820 refers to, but it is the result of the function h when the input is 40.
3
[tex]( - x + 12) - ( - 4x + 2)[/tex]
Answer:
3x+10
Step-by-step explanation:
( - x + 12) - ( - 4x + 2)
Distribute the minus sign
( - x + 12) + 4x - 2
Combine like terms
3x +10