Answer:
No no don't click the link
Answer:
The biggest difference between exponential and linear functions is that linear functions change at a constant rate, while exponential functions change at a rate proportional to it's value, or exponent.
Basically, that's also what separates exponential functions from all others. It's the only function that changes at a rate proportional to its exponent.
Step-by-step explanation:
4(2a - 3) = 2(3a + 1)
What's the answer?
Answer:
a=7
Step-by-step explanation:
4(2a-3) = 2(3a+1)
8a-12 = 6a+2
2a-12=2
2a=14
a=7
What is the answer? Please help, I need this done today.
Answer:
27 is the answer
Step-by-step explanation:
Draw the diagram of LFSR with characteristic polynomial
x^6+x^5+x^3+x^2+1. What is the maximum period of the LFSR?
The maximum period of the LFSR with the given characteristic polynomial is 63.
To draw the diagram of a Linear Feedback Shift Register (LFSR) with a characteristic polynomial of [tex]x^6 + x^5 + x^3 + x^2 + 1,[/tex] we need to represent the shift register stages and the feedback connections.
The characteristic polynomial tells us the feedback taps in the LFSR. In this case, the feedback taps are at positions 6, 5, 3, 2, and 0 (the coefficients of the polynomial with non-zero exponents).
In the diagram, D1 represents the output of the first stage (bit), D2 represents the output of the second stage, and so on. The arrows represent the shift direction, with the feedback connections shown by the lines connecting the output of specific stages to the feedback taps.
Now, let's determine the maximum period of the LFSR with this characteristic polynomial. The maximum period of an LFSR is given by [tex]2^N - 1,[/tex] where N is the number of stages in the shift register.
In this case, there are 6 stages, so the maximum period is [tex]2^6 - 1 = 63.[/tex]
Therefore, the maximum period of the LFSR with the given characteristic polynomial is 63.
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pls help and show work i am screwed if i don’t do well on this
Answer:
x = - 2
Step-by-step explanation:
The axis of symmetry passes through the vertex, is a vertical line with equation equal to the x- coordinate of the vertex, that is
equation of axis of symmetry is x = 1
The zeros are equidistant from the axis of symmetry, on either side
x = 4 is a zero and is 3 units to the right of x = 1, so
3 units to the left of x = 1 is 1 - 3 = - 2
The other zero is therefore x = - 2
help me please it's due tonight
Answer:
π
Step-by-step explanation:
S = rФ
Arc length = radius x theta
S = (3)([tex]\frac{\pi }{3}[/tex]) = [tex]\pi[/tex]
Help please I’m a little confused
Answer:
answer is a . x²-8x+16=32
x²-8x=32-16
Español Tom opened a savings account with $600 and was paid simple interest at an annual rate of 2%. When Tom closed the account, he was paid $36 in interest. How long was the account open for, in years? If necessary, refer to the list of financial formulas. Dy years ?
The account was open for 3 years. This duration was determined by calculating the time using the formula for simple interest based on the initial principal, interest rate, and the amount of interest earned.
To determine the length of time the account was open, we can use the formula for simple interest:
Interest = Principal * Rate * Time
Given that Tom opened the account with $600, the annual interest rate was 2%, and he received $36 in interest, we can set up the equation:
36 = 600 * 0.02 * Time
Simplifying the equation:
36 = 12 * Time
Dividing both sides by 12:
Time = 3
Therefore, the account was open for 3 years.
In conclusion, Tom's savings account was open for 3 years, as calculated using the simple interest formula.
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What is the measure of the unknown segment?
Answer:6
Step-by-step explanation:
If 3 out of 4 people use a certain headache
medicine, how many in a city of 150,400
will use this medicine?
(A) 118, 200
(B) 37,600
(C) 50, 133
(D) 112,800
Answer:
(D) 112,800
Step-by-step explanation:
75%x150,400= 112,800
112,800 use the headache medicine.
Answer:
option D
Step-by-step explanation:
Out of 4 , 3 use headache medicine . So the fraction is 3/4 .
Total number of people in city = 150,400 .
So ,
⇒ n = 3/4 × Population
⇒ n = 3/4 × 150,400
⇒ n = 112,800
Hence option D is correct.(1 point) Are the following statements true or false? ? 1. If W = Span{V1, V2, V3 }, and if {V1, V2, V3 } is an orthogonal set in W, then {V1, V2, V3 } is an orthonormal basis for W. ? 2. If x is not in a subspace W, projw(x) is not zero. then x ?
3. In a QR factorization, say A = QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A.
1.An orthonormal basis for W is False.
2.If x is not in a subspace W, projw(x) is not zero then x True.
3.The QR factorization columns of Q form an orthonormal basis for the column space of A True.
An orthogonal set in a vector space necessarily mean that it is orthonormal are the vectors orthogonal to each other, but they unit length if {V1, V2, V3} is an orthogonal set in W, that the vectors are mutually orthogonal, but they may not have unit length {V1, V2, V3} assumed to be an orthonormal basis for W.
The projection of a vector x onto a subspace W, denoted as projW(x), is defined as the closest vector in W to x. If x is not in W, then the projection of x onto W will not be zero a nonzero vector in the subspace W that is closest to x.
In a QR factorization of a matrix A, where A has linearly independent columns, the matrix Q consists of orthonormal columns. The columns of Q form an orthonormal basis for the column space of A.
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Evaluate the expression x - 9, if x = 12.
1. 21
2. 17
3. 4
or 4. 3
what’s the answer what number?
Answer:
3
Step-by-step explanation:
I hope this answer has helped you
The value of expression x - 9, if x = 12 is 3, so the correct option is 4.
What is expression?A mathematical expression is made up of a statement, at least one arithmetic operation, and at least two integers or variables.
Given:
x - 9 and x = 12
Put the value of x in the expression as shown below,
x - 9 = 12 - 9
x - 9 = 3
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5th grade math. correct answer will be marked brainliest
Answer:
3
Step-by-step explanation:
Answer:
3
Step-by-step explanation:
9×[tex]\frac{1}{3}[/tex]
9×1/1×3=
[tex]\frac{9}{3}[/tex]÷3=
3!!!!
uppose V1 and V2 are both uniformly distributed between 0.2 and 0.8, and their probability distribution is modelled by using a Gaussian copula with a correlation coefficient of rho=0.5. Write down the joint probability Prob(V1<0.5, V2<0.3) in terms of the cumulative bivariate normal distribution function: M(U1
The joint probability, Prob(V1 < 0.5, V2 < 0.3), of two uniformly distributed variables V1 and V2, modelled using a Gaussian copula with a correlation coefficient of ρ = 0.5.
Given that V1 and V2 are uniformly distributed between 0.2 and 0.8, we need to transform these variables to standard normal variables before calculating the joint probability. The Gaussian copula is commonly used for this purpose.
The transformation from the uniform distribution to the standard normal distribution can be achieved using the inverse of the cumulative distribution function (CDF) of the standard normal distribution. Let Φ denote the CDF of the standard normal distribution. The transformed variables, denoted as U1 and U2, can be calculated as follows:
U1 = Φ^(-1)(V1)
U2 = Φ^(-1)(V2)
Since the correlation coefficient between U1 and U2 is ρ = 0.5, we can calculate the joint probability using the bivariate normal distribution function with mean 0, standard deviation 1, and correlation coefficient 0.5. Let Φ2 denote the cumulative bivariate normal distribution function.
[tex]Prob(V1 < 0.5, V2 < 0.3) = Prob(U1 < Φ^(-1)(0.5), U2 < Φ^(-1)(0.3))[/tex]
[tex]= Φ2(Φ^(-1)(0.5), Φ^(-1)(0.3); ρ = 0.5)[/tex]
By evaluating the bivariate normal distribution function at the given values, we can obtain the joint probability.
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To write down the joint probability Prob(V1 < 0.5, V2 < 0.3) in terms of the cumulative bivariate normal distribution function, we need to utilize the properties of the Gaussian copula and the correlation coefficient.
Which expression is equivalent to 13 22b
Jason wants to create a box with the same volume as the one shown below. He wants the length to be 4
inches. What would be the measurements of the width and height?
Length: 4 inches
Width: inches
Height: inches
Explain how you determined the width and height.
- cubic inch
Given : jason wants to create a box with the same volume as the one shown
To find : measurements of width and height .
Solution :
Volume of given box
=> 6 * 2 * 3
=36
length to be 4 inches .
lwh = 36
=> 4 wh = 36
=> wh = 9
9 = 1 * 9
9 = 3 * 3
Width and Height can be 3 inches each or 1 and 9 inches
Could someone please help me? Thank you and explain the work because I don’t get this
Answer:
5.59 times per second.
Step-by-step explanation:
Direct variation is in the form:
[tex]y=kx[/tex]
Where k is the constant of variation.
Inverse variation is in the form:
[tex]\displaystyle y=\frac{k}{x}[/tex]
In the given problem, the frequency of a vibrating guitar string varies inversely as its length. In other words, using f for frequency and l for length:
[tex]\displaystyle f=\frac{k}{\ell}[/tex]
We can solve for the constant of variation. We know that the frequency f is 4.3 when the length is 0.65 meters long. Thus:
[tex]\displaystyle 4.3=\frac{k}{0.65}[/tex]
Solve for k:
[tex]k=4.3(0.65)=2.795[/tex]
So, our equation becomes:
[tex]\displaystyle f=\frac{2.795}{\ell}[/tex]
Then when the length is 0.5 meters, the frequency will be:
[tex]\displaystyle f=\frac{2.795}{.5}=5.59\text{ times per second.}[/tex]
Is it possible for the rug to have an area of 71.25 square feet? If it is possible, give approximate dimensions of the rectangle and explain your work. If it is not possible, explain why not.
Answer:
Yes 2*35.625
Step-by-step explanation:
Just divide by two and now you have dimensions for a rug that has an area of 71.25
Yes, Is it possible for the rug to have an area of 71.25 square feet.
What is mean by Rectangle?A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.
We have to given that;
A rug weaver wants to create a large rectangular rug.
Now, From the figure;
Area (S) = (8 + x) (9 - x)
= - x² + x + 72
= - (x² - x + 1/4) + 1/4 + 72
= - (x - 1/2)² + 72.25
Hence,
S (max) = 72.25 square feet
And, 72.25 > 71.25
Thus, Is it possible for the rug to have an area of 71.25 square feet.
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Complete question is shown in figure.
The Leungs sold a valuable painting for $55,000. This price is $1,000
more than twice the amount they originally paid for it. How much
did they originally pay?
A. $25,000 B. $27,000 C. $27,500 D. $28,000
Answer:
B. $27,000
Step-by-step explanation:
So 55,000 = 2x + 1,000
Simply for x
55,000 = 2x + 1,000
54,000 = 2x
x = 27,000
Solve the following DE using Power series around x₁ = 0. Find the first eight nonzero terms of this DE. y" + xy' + 2y = 0.
To solve the differential equation y" + xy' + 2y = 0 using power series, we assume a power series representation for the solution and derive a recurrence relation for the coefficients. The first eight nonzero terms can be found by solving the recurrence relation.
To solve the differential equation y" + xy' + 2y = 0 using power series around x₁ = 0, we can assume a power series representation for the solution:
y(x) = ∑(n=0 to ∞) aₙxⁿ
Let's substitute this power series representation into the given differential equation and find the recurrence relation for the coefficients aₙ.
Differentiating y(x) with respect to x:
y'(x) = ∑(n=0 to ∞) aₙn xⁿ⁻¹
y''(x) = ∑(n=0 to ∞) aₙn(n-1) xⁿ⁻²
Substituting these expressions into the differential equation:
∑(n=0 to ∞) aₙn(n-1) xⁿ⁻² + x ∑(n=0 to ∞) aₙn xⁿ⁻¹ + 2∑(n=0 to ∞) aₙxⁿ = 0
Now, we can rearrange and collect like terms based on the powers of x:
∑(n=0 to ∞) [aₙn(n-1) xⁿ⁻² + aₙn xⁿ⁺¹ + 2aₙxⁿ] = 0
Since this equation must hold for all values of x, each coefficient of xⁿ must be zero. Therefore, we get the following recurrence relation for the coefficients:
aₙ(n-1)(n-2) + aₙ₋₁(n-1) + 2aₙ = 0
Simplifying the recurrence relation:
aₙ(n² - 3n + 2) + aₙ₋₁(n-1) = 0
Now, we can start finding the first few nonzero terms of the power series solution by using the recurrence relation.
First term (n=0):
a₀(0² - 3(0) + 2) + a₋₁(-1) = 0
a₀ + a₋₁ = 0
Second term (n=1):
a₁(1² - 3(1) + 2) + a₀(1-1) = 0
a₁ - a₀ = 0
From the first and second terms, we find a₀ = a₁ and a₋₁ = -a₀.
Third term (n=2):
a₂(2² - 3(2) + 2) + a₁(2-1) = 0
a₂ - 3a₁ = 0
a₂ - 3a₀ = 0
Fourth term (n=3):
a₃(3² - 3(3) + 2) + a₂(3-1) = 0
a₃ - 6a₂ = 0
a₃ - 6a₀ = 0
Continuing this process, we can find the values of a₄, a₅, a₆, and so on, using the recurrence relation.
By solving the recurrence relation for each term, we can determine the first eight nonzero terms of the power series solution to the differential equation y" + xy' + 2y = 0.
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Identify two ratios that are equivalent to 3:5
write the ratios sin m, cos m, and tan m. give the exact value and four decimal approximation. Please help.
Trigonometric functions are the ratio of different sides of a triangle. The ratios sin∠m, cos∠m, and Tan∠m are 0.758, 0.6522, and 1.1624.
What are Trigonometric functions?The trigonometric function gives the ratio of different sides of a right-angle triangle.
[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}[/tex]
where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.
As it is given that the base of the triangle for the ∠m is Mk(15 units), the perpendicular is KL(4√19), and the hypotenuse is 23. Now, the trigonometric ratios can be written as,
Sine
[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\\rm Sin (\angle m)=\dfrac{KL}{ML}\\\\\\\rm Sin (\angle m)=\dfrac{4\sqrt{19}}{23}\\\\\\\rm Sin (\angle m)=0.758069\approx 0.758[/tex]
Cosine
[tex]\rm Cos\theta=\dfrac{Base}{Hypotenuse}\\\\\\\rm Cos(\angle m)=\dfrac{MK}{ML}\\\\\\\rm Cos(\angle m)=\dfrac{15}{23}\\\\\\\rm Cos (\angle m)=0.65217\approx 0.6522[/tex]
Tangent
[tex]\rm Tan\theta=\dfrac{Perpendicular}{Base}\\\\\\\rm Tan(\angle m)=\dfrac{KL}{MK}\\\\\\\rm Tan(\angle m)=\dfrac{4\sqrt{19}}{15}\\\\\\\rm Tan(\angle m)=1.16237\approx 1.1624[/tex]
Hence, the ratios sin m, cos m, and tan m are 0.758, 0.6522, and 1.1624.
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help pleaseee i don’t understand thisss
Answer:
The answer is;
box A:3/2
box B:2/3
This is a scale drawing of a flag. The scale factor of a drawing to the actual flag is represented by the ratio 1:18. What is the area in square inches of the actual flag.
Answer:
2:36
Step-by-step explanation:
i don't know for my answer
Find the centre of mass of the 2D shape bounded by the lines y = ±1.3x between x = 0 to 2.1. Assume the density is uniform with the value: 3.5kg. m-2 Also find the centre of mass of the 3D volume created by rotating the same lines about the x-axis. The density is uniform with the value: 1.9kg. m (Give all your answers rounded to 3 significant figures.) a) Enter the mass (kg) of the 20 plate: Enter the Moment (kg.m) of the 2D plate about the y-axis: Enter the x-coordinate (m) of the centre of mass of the 2D plate: Submit part ed Enter the mass (kg) of the 3D body: Enter the Moment (kg.m) of the 3D body about the y-axis: Enter the x-coordinate (m) of the centre of mass of the 3D body:
The mass of the 2D plate is 20.067 kg, with a moment of 5.742 kg.m about the y-axis. The x-coordinate of the center of mass of the 2D plate is 0.286 m. The mass of the 3D body is 62.137 kg, with a moment of 39.748 kg.m about the y-axis. The x-coordinate of the center of mass of the 3D body is 0.640 m.
To determine the center of mass of the 2D shape bounded by the lines y = ±1.3x between x = 0 to 2.1, we need to calculate the mass, moment, and x-coordinate of the center of mass.
1) Mass of the 2D plate:
The area of the 2D shape can be calculated by finding the difference in the areas under the two lines y = ±1.3x between x = 0 and x = 2.1.
Area = ∫(1.3x)dx - ∫(-1.3x)dx
= ∫1.3xdx + ∫1.3xdx
= 2 * ∫1.3xdx
= 2 * [0.65x²] between x = 0 and x = 2.1
= 2 * (0.65 * (2.1)²)
= 2 * (0.65 * 4.41)
= 2 * 2.8665
= 5.733
Mass = Area * Density
= 5.733 * 3.5
≈ 20.067 kg
Therefore, the mass of the 2D plate is approximately 20.067 kg.
2) Moment of the 2D plate about the y-axis:
The moment of the 2D shape about the y-axis is given by the integral of the product of the y-coordinate and the area element.
Moment = ∫(y * dA)
= ∫(±1.3x * dA)
= 2 * ∫(1.3x * dA) between x = 0 and x = 2.1
= 2 * 1.3 * ∫(x * dA) between x = 0 and x = 2.1
= 2 * 1.3 * ∫(x * dx)
= 2 * 1.3 * [0.5x²] between x = 0 and x = 2.1
= 2 * 1.3 * (0.5 * (2.1)²)
= 2 * 1.3 * (0.5 * 4.41)
= 2 * 1.3 * 2.205
= 5.742 kg.m
Therefore, the moment of the 2D plate about the y-axis is 5.742 kg.m.
3) x-coordinate of the center of mass of the 2D plate:
The x-coordinate of the center of mass of the 2D shape can be calculated using the formula:
x-coordinate = Moment / Mass
x-coordinate = 5.742 kg.m / 20.067 kg
≈ 0.286 m
Therefore, the x-coordinate of the center of mass of the 2D plate is approximately 0.286 m.
For the 3D body created by rotating the same lines about the x-axis:
1) Mass of the 3D body:
The volume of the 3D body can be calculated by finding the difference in the volumes between the two shapes obtained by rotating y = ±1.3x about the x-axis between x = 0 and x = 2.1.
Volume = π * ∫(1.3x)^2 dx - π * ∫(-1.3x)^2 dx
= π * ∫1.69x^2 dx - π * ∫1.69x^2 dx
=
2 * π * ∫1.69x² dx
= 2 * π * [0.5633x³] between x = 0 and x = 2.1
= 2 * π * (0.5633 * (2.1)³)
= 2 * π * (0.5633 * 9.261)
= 2 * π * 5.2167
≈ 32.703 m³
Mass = Volume * Density
= 32.703 * 1.9
≈ 62.137 kg
Therefore, the mass of the 3D body is approximately 62.137 kg.
2) Moment of the 3D body about the y-axis:
The moment of the 3D body about the y-axis can be calculated similarly to the 2D plate but considering the additional dimension.
Moment = ∫(x * dV)
= π * ∫(x * (1.3x)² dx) - π * ∫(x * (-1.3x)^2 dx)
= 2 * π * ∫(1.3x³ dx)
= 2 * π * [0.325x⁴] between x = 0 and x = 2.1
= 2 * π * (0.325 * (2.1)⁴)
= 2 * π * (0.325 * 19.4481)
= 2 * π * 6.3252
≈ 39.748 kg.m
Therefore, the moment of the 3D body about the y-axis is approximately 39.748 kg.m.
3) x-coordinate of the center of mass of the 3D body:
The x-coordinate of the center of mass of the 3D body can be calculated using the formula:
x-coordinate = Moment / Mass
x-coordinate = 39.748 kg.m / 62.137 kg
≈ 0.640 m
Therefore, the x-coordinate of the center of mass of the 3D body is approximately 0.640 m.
To summarize the answers:
a) Mass of the 2D plate: 20.067 kg
b) Moment of the 2D plate about the y-axis: 5.742 kg.m
c) x-coordinate of the center of mass of the 2D plate: 0.286 m
d) Mass of the 3D body: 62.137 kg
e) Moment of the 3D body about the y-axis: 39.748 kg.m
f) x-coordinate of the center of mass of the 3D body: 0.640 m
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The volume of a sphere is 36π cubic inches. What is the radius of the sphere?
Answer 2.05
Step-by-step explanation: hope this helps
Find the area of the triangle with vertices: Q(2,-1,1), R(3,-2,-2), S(5,1,-1).
The area of the triangle with vertices: Q(2,-1,1), R(3,-2,-2), S(5,1,-1) is 6.5 square units.
To find the area of the triangle with vertices: Q(2,-1,1), R(3,-2,-2), S(5,1,-1), we can use the formula Area of triangle = 1/2 * |(x2-x1) (y3-y1)-(x3-x1)(y2-y1)|
where, x1, y1, x2, y2, x3 and y3 are the coordinates of the given triangle Q(2,-1,1)
corresponds to x1=2, y1=-1R(3,-2,-2)
corresponds to x2=3, y2=-2S(5,1,-1)
corresponds to x3=5, y3=1
We can substitute these values in the above formula to get Area of triangle = 1/2 * |(x2-x1) (y3-y1)-(x3-x1)(y2-y1)|= 1/2 * |(3-2)(1-(-1)) - (5-2)(-2-(-1))| = 1/2 * |-4 - 9| = 1/2 * 13 = 6.5
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estimate each product 49.2×10.3
A local charity receives 1/3 of funds raised during a craft fair and a bake sale. The total amount given to the charity was $137.45. How much did the bake sale raise?
(NEED ANSWER ASAP!!)
Answer:
$412.35
Step-by-step explanation:
Answer: $412.35
Step-by-step explanation:
$137.45 x 3 = $412.35 for the total that the bake sale raised.
What is equivalent ratio of 5:8:4 and explain how u got it
this is example
So, first we need to write the given ratio as fraction,
= 8/18
= (8 × 2)/(18 × 2)
= 16/36
= 16 : 36 (one equivalent ratio),
So, 16 : 36 is an equivalent ratio of 8 : 18.
Now we will find another equivalent ratio of 8 : 18 by using division.
Similarly, first we need to write the given ratio as fraction,
= 8/18
= (8 ÷ 2)/(18 ÷ 2)
= 4/9
= 4 : 9 (another equivalent ratio)
So, 4 : 9 is an equivalent ratio of 8 : 18.
Therefore, the two equivalent ratios of 8 : 18 are 16 : 36 and 4 : 9.
2. Frame two equivalent ratios of 4 : 5.
Solution:
To find two equivalent ratios of 4 : 5 we need to apply multiplication method only to get the answer in integer form.
First we need to write the given ratio as fraction,
= 4/5
= (4 × 2)/(5 × 2)
= 8/10
= 8 : 10 is one equivalent ratio,
Similarly again, we need to write the given ratio 4 : 5 as fraction to get another equivalent ratio;
= 4/5
= (4 × 3)/(5 × 3)
= 12/15 is another equivalent ratio
Therefore, the two equivalent ratios of 4 : 5 are 8 : 10 and 12 : 15.
Note: In this question we can’t apply division method to get the answer in integer form because the G.C.F. of 4 and 5 is 1. That means, 4 and 5 cannot be divisible by any other number except 1.
The two equivalent ratios of 5:8:4 are 8: 10 and 12: 15.
What is a fraction?The fraction is defined as the division of the whole part into an equal number of parts. In mathematics, ratios are used to determine the relationship between two numbers it indicates how many times is one number to another number.
First, we need to write the given ratio as a fraction,
F= 8/18
F= (8 × 2)/(18 × 2)
F= 16/36
F= 16 : 36 (one equivalent ratio),
16: 36 is an equivalent ratio of 8: 18.
Find another equivalent ratio of 8: 18 by using division. Similarly, first, we need to write the given ratio as a fraction,
F= 8/18
F= (8 ÷ 2)/(18 ÷ 2)
F= 4/9
F= 4 : 9 (another equivalent ratio)
4: 9 is an equivalent ratio of 8: 18.
Therefore, the two equivalent ratios of 5:8:4 are 16: 36 and 4: 9.
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Find the intersections of these pairs of linear equations. Al) 4x-3y=6 -2x+(3/2)y-3 A2) x-4y=-5 3x-2y.15 A3) 4x+y=9 2x-3y22 A4) -6x+9y = 9 2x-3y=6 Note on answers: If the answer is a point, write It as an ordered pair, (a,b). No spaces. Include the parentheses. If there is no solution, enter none, If they are the same line and there is an Infinite number of solutions, enter
A1) To find the intersection of the two equations, we can solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
4x - 3y = 6
Solving for y:
-3y = -4x + 6
y = (4/3)x - 2
Now let's look at the second equation:
-2x + (3/2)y - 3 = 0
Solving for y:
(3/2)y = 2x + 3
y = (4/3)x + 2
Now we have two expressions for y:
y = (4/3)x - 2 and y = (4/3)x + 2
These lines have different y-intercepts and the same slope, so they are not parallel and must intersect at some point.
Setting the two expressions equal to each other:
(4/3)x - 2 = (4/3)x + 2
Subtracting (4/3)x from both sides:
-2 = 2
This is a contradiction, so there is no solution.
Answer: none
A2)
To find the intersection of the two equations, we can again solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
x - 4y = -5
Solving for y:
-4y = -x - 5
y = (1/4)x + (5/4)
Now let's look at the second equation:
3x - 2y = 15
Solving for y:
-2y = -3x + 15
y = (3/2)x - 7.5
Now we have two expressions for y:
y = (1/4)x + (5/4) and y = (3/2)x - 7.5
Setting the two expressions equal to each other:
(1/4)x + (5/4) = (3/2)x - 7.5
Subtracting (1/4)x and adding 7.5 to both sides:
(11/4) = (5/2)x
Multiplying both sides by 2/5:
x = 22/20 = 11/10
Now we can substitute this value of x into either equation to find y:
y = (1/4)(11/10) + (5/4) = (11/40) + (50/40) = 61/40
Answer: (11/10, 61/40)
A3)
Starting with the first equation:
4x + y = 9
Solving for y:
y = -4x + 9
Now let's look at the second equation:
2x - 3y = 22
Solving for y:
-3y = -2x + 22
y = (2/3)x - (22/3)
Now we have two expressions for y:
y = -4x + 9 and y = (2/3)x - (22/3)
Setting the two expressions equal to each other:
-4x + 9 = (2/3)x - (22/3)
Adding 4x and (22/3) to both sides:
33/3 = (14/3)x
Multiplying both sides by 3/14:
x = 9/14
Now we can substitute this value of x into either equation to find y:
y = -4(9/14) + 9 = -18/7
Answer: (9/14, -18/7)
A4)
Starting with the first equation:
-6x + 9y = 9
Solving for y:
9y = 6x + 9
y = (2/3)x + 1
Now let's look at the second equation:
2x - 3y = 6
Solving for y:
-3y = -2x + 6
y = (2/3)x - 2
Now we have two expressions for y:
y = (2/3)x + 1 and y = (2/3)x - 2
Setting the two expressions equal to each other:
(2/3)x + 1 = (2/3)x - 2
Adding -2/3x and -1 to both sides:
0 = -3A1)
To find the intersection of the two equations, we can solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
4x - 3y = 6
Solving for y:
-3y = -4x + 6
y = (4/3)x - 2
Now let's look at the second equation:
-2x + (3/2)y - 3 = 0
Solving for y:
(3/2)y = 2x + 3
y = (4/3)x + 2
Now we have two expressions for y:
y = (4/3)x - 2 and y = (4/3)x + 2
These lines have different y-intercepts and the same slope, so they are not parallel and must intersect at some point.
Setting the two expressions equal to each other:
(4/3)x - 2 = (4/3)x + 2
Subtracting (4/3)x from both sides:
-2 = 2
This is a contradiction, so there is no solution.
Answer: none
A2)
To find the intersection of the two equations, we can again solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
x - 4y = -5
Solving for y:
-4y = -x - 5
y = (1/4)x + (5/4)
Now let's look at the second equation:
3x - 2y = 15
Solving for y:
-2y = -3x + 15
y = (3/2)x - 7.5
Now we have two expressions for y:
y = (1/4)x + (5/4) and y = (3/2)x - 7.5
Setting the two expressions equal to each other:
(1/4)x + (5/4) = (3/2)x - 7.5
Subtracting (1/4)x and adding 7.5 to both sides:
(11/4) = (5/2)x
Multiplying both sides by 2/5:
x = 22/20 = 11/10
Now we can substitute this value of x into either equation to find y:
y = (1/4)(11/10) + (5/4) = (11/40) + (50/40) = 61/40
Answer: (11/10, 61/40)
A3)
Starting with the first equation:
4x + y = 9
Solving for y:
y = -4x + 9
Now let's look at the second equation:
2x - 3y = 22
Solving for y:
-3y = -2x + 22
y = (2/3)x - (22/3)
Now we have two expressions for y:
y = -4x + 9 and y = (2/3)x - (22/3)
Setting the two expressions equal to each other:
-4x + 9 = (2/3)x - (22/3)
Adding 4x and (22/3) to both sides:
33/3 = (14/3)x
Multiplying both sides by 3/14:
x = 9/14
Now we can substitute this value of x into either equation to find y:
y = -4(9/14) + 9 = -18/7
Answer: (9/14, -18/7)
A4)
Starting with the first equation:
-6x + 9y = 9
Solving for y:
9y = 6x + 9
y = (2/3)x + 1
Now let's look at the second equation:
2x - 3y = 6
Solving for y:
-3y = -2x + 6
y = (2/3)x - 2
Now we have two expressions for y:
y = (2/3)x + 1 and y = (2/3)x - 2
Setting the two expressions equal to each other:
(2/3)x + 1 = (2/3)x - 2
Adding -2/3x and -1 to both sides: 0 = -3
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