find the general solution of the indicated differential equation. If possible, find an explicit solution. 1. y =xy 2. xy =2y 3. y =e
x−y 4. y =(1+y 2)e x5. y =xy+y 6. y =ye x −2e x +y−2 7. y =x/(y+2) 8. y =xy/(x−1) 9. x 2 y =ylny−y 10. xy −y=2x 2y 11. y 3y =x+2y 12. y =(2xy+2x)/(x 2−1)
Answers
The equations you mentioned:
1. y = xy:
To solve this differential equation, we can separate the variables and integrate both sides.
dy/y = x dx
Integrating both sides gives:
ln|y| = (1/2)x² + C
Exponentiating both sides gives the general solution:
|y| = \(e^{((1/2)x^2 + C)\)
Taking the positive and negative values of y, we get two branches of solutions:
y = \(Ae^{(1/2)x^2\) and y = \(-Ae^{(1/2)x^2\), where A is an arbitrary constant.
2. xy = 2y:
Rearranging the equation, we get:
xy - 2y = 0
Factoring out y, we have:
y(x - 2) = 0
This equation has two solutions:
y = 0 and x - 2 = 0, which leads to x = 2.
3. y = \(e^x\) - y:
Rearranging the equation, we get:
y + y = \(e^x\)
Combining like terms, we have:
2y = \(e^x\)
Dividing both sides by 2, we get:
y = (1/2)\(e^x\)
4. y = (1 + y²)\(e^x\):
Rearranging the equation, we get:
y - y² = \(e^x\)
Factoring out y, we have:
y(1 - y) = \(e^x\)
This equation has two solutions:
y = 0 and 1 - y = \(e^x\), which leads to y = 1 - \(e^x\).
5. y = xy + y:
Rearranging the equation, we get:
y - xy - y = 0
Combining like terms, we have:
-xy = 0
This equation has two solutions:
x = 0 and y = 0.
6. y = y\(e^x\) - 2\(e^x\) + y - 2:
Rearranging the equation, we get:
y\(e^x\) - y = 2\(e^x\)- 2
Factoring out y, we have:
y(\(e^x\) - 1) = 2(\(e^x\) - 1)
Dividing both sides by (\(e^x\) - 1), we get:
y = 2
7. y = x/(y + 2):
Rearranging the equation, we get:
y(y + 2) = x
Expanding the equation, we have:
y² + 2y - x = 0
This equation doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
8. y = xy/(x - 1):
Rearranging the equation, we get:
(x - 1)y = xy
Dividing both sides by y and rearranging, we have:
x - 1 = x/y
Solving for y, we get:
y = x/(x - 1)
9. x²y = ylny - y:
This is a nonlinear differential equation that doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
10. xy - y = 2x²y:
Rearranging the equation, we get:
xy - 2x²y - y = 0
Factoring out y, we have:
y(x - 2x² - 1) = 0
This equation has two solutions:
y = 0 and x - 2x² - 1 = 0, which leads to x = (1 ± √3)/2.
11. y - 3y² = x + 2y:
Rearranging the equation, we get:
-3y² + y + 2y - x = 0
Combining like terms, we have:
-3y² + 3y - x = 0
This equation doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
12. y = (2xy + 2x)/(x² - 1):
Rearranging the equation, we get:
y(x² - 1) = 2xy + 2x
Expanding the equation, we have:
x²y - y = 2xy + 2x
Combining like terms, we get:
x²y - 2xy - y - 2x = 0
This equation doesn't have a general solution in terms of elementary functions. It can be solved numerically or using approximation methods.
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Related Questions
Multiply. 1. (5.2x)(1.9x)
Answers
Answer:
9.88x
Step-by-step explanation:
multiply both numbers together
pls help thank you!!!!
Answers
Item Cost $
2 18
4 36
6 54
8 72
Find the constant rate of change for the table above.
Constant rate of change = $ per 1 item
Answers
Answer:
Hi... The constant rate of change is 9
(a) Attendance at the Accra Sports Stadium was alysed by the General Secretary, Prosper Harrison Addo. The analysis demonstrated that spectators consisted of 70% males. If seven people are randomly selected from the spectators during a football match, What is the probability that 4 of them are males? (3 marks) i 11. Find the probability that at most 5 of them are females (4 marks)
Answers
a) The probability of randomly selecting 4 males out of 7 spectators, given that 70% of the spectators are males, can be calculated using the binomial probability formula.
b) To find the probability that at most 5 of the randomly selected spectators are females, we need to calculate the cumulative probability of selecting 0, 1, 2, 3, 4, and 5 females from the total number of selected spectators.
a) To calculate the probability of selecting 4 males out of 7 spectators, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- n is the total number of trials (number of people selected)
- k is the number of successful trials (number of males selected)
- p is the probability of success in a single trial (probability of selecting a male)
- C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k! * (n - k)!)
In this case, n = 7, k = 4, and p = 0.70 (probability of selecting a male). Therefore, the probability of selecting 4 males out of 7 spectators is:
P(X = 4) = C(7, 4) * (0.70)^4 * (1 - 0.70)^(7 - 4)
b) To find the probability that at most 5 of the selected spectators are females, we need to calculate the cumulative probability of selecting 0, 1, 2, 3, 4, and 5 females. This can be done by summing the individual probabilities for each case.
P(X ≤ 5 females) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
To calculate each individual probability, we use the same binomial probability formula as in part a), with p = 0.30 (probability of selecting a female).
Finally, we sum up the probabilities for each case to find the probability that at most 5 of the selected spectators are females.
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2 3 4 5 6 7 8 9 10
TIME REMAINING
01:44:40
A piece of fabric costs $5.29 a yard. Which is an estimated cost of 16.3 yards of fabric?
$21
$80
$165
$800
Mark this and return
Answers
5.29*16.3=86.227 which is closest to 80
giving brainliest!!! please help if you know the answer.
Answers
Answer:
(x + 5) (x - 5)
Step-by-step explanation:
x² - 25
This is known as the difference of squares
(x + 5) (x - 5)
Expand it to see the x term cancels out
x² - 5x + 5x - 25
x² - 25
HeLP me ASAP PLEASE ASAP i will give 50 points for the right answer
Answers
which series of transformations shows that paralelogram abcd is congruent to parallelogram abcd by superimposing one onto the other
Answers
The series of transformations that shows that parallelogram ABCD is congruent to parallelogram A'B'C'D' by superimposing one onto the other is a combination of translation and rotation that involves 150 degrees. A transformation refers to the process of changing an object's position, size, or shape.
Translation, reflection, rotation, dilation, and skew are the five basic transformation types that modify an object.In geometry, congruent shapes have the same shape and size. To superimpose one shape onto another means to make them coincide with each other.
The following series of transformations would show that parallelogram ABCD is congruent to parallelogram A'B'C'D' by superimposing one onto the other: Step 1: Translation. Translate the parallelogram ABCD by 5 units to the right and 6 units down to obtain parallelogram A1B1C1D1.Step 2: Rotation. Rotate parallelogram A1B1C1D1 by 150 degrees counterclockwise around point A1 to obtain parallelogram A2B2C2D2.Step 3: Translation. Translate parallelogram A2B2C2D2 by 8 units to the right and 3 units up to obtain parallelogram A'B'C'D'.Thus, the series of transformations that shows that parallelogram ABCD is congruent to parallelogram A'B'C'D' by superimposing one onto the other involves translation and rotation, with a rotation of 150 degrees.
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Must complete all parts below
(a) Draw the graph of the function f(x) = 2^x+3
(b) Translate the graph down 4 units and draw the new graph.
(c) What is the equation of the new graph?
Answers
To answer this question, we need to follow the steps given to us. First, we need to draw the graph of the function f(x) = 2^x+3.
To do this, we can start by making a table of values and plotting points on the coordinate plane.
x | f(x)
--|-----
-2 | 1/8
-1 | 1/4
0 | 1
1 | 8
2 | 11
3 | 16
Once we have a few points plotted, we can connect them with a smooth curve to get the graph of f(x) = 2^x+3.
Next, we need to translate the graph down 4 units and draw the new graph. To do this, we simply subtract 4 from the function. So the new function is g(x) = 2^x-1.
We can repeat the process of making a table of values and plotting points to get the new graph.
x | g(x)
--|-----
-2 | 1/8 - 1
-1 | 1/4 - 1
0 | 1 - 1
1 | 8 - 1
2 | 11 - 1
3 | 16 - 1
Again, we can connect the points with a smooth curve to get the new graph.
Finally, we need to find the equation of the new graph. We already know that the new function is g(x) = 2^x-1, so that is the equation of the new graph.
In summary, we first drew the graph of f(x) = 2^x+3, then translated it down 4 units to get the new graph of g(x) = 2^x-1. The equation of the new graph is g(x) = 2^x-1.
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Chords AB and CD intersect at E and AE = EB. A semicircle is drawn with diameter CD. EF, perpendicular to CD, meets this semicircle at E. Prove that AE = EF.
help me pls.
Answers
The proof that AE = EF is done from the congruency theorem of triangles and the Pythagorean identity.
How can it be proved that AE = EF?In order to prove that AE = EF, note that angle CEF is a right angle since EF is perpendicular to the diameter CD and intersects it at E.
Since AE = EB, triangles AEB and CED are congruent (SAS theorem)
Therefore, angle EAF is congruent to angle ECF.
Also, angle EAF is a right angle, since it is inscribed in a semicircle with a diameter CD.
Therefore, triangles EAF and ECF are similar by the Angle-Angle (AA) theorem.
Since they are similar, the ratio of corresponding side lengths is the same.
Since AE = EB, we have that:
AE / EF = EC / CF
But we also know that EC = ED + DC = ED + 2EF, since CD is the diameter of the semicircle and EF meets it at E perpendicularly.
Similarly, we have CF = CD + DF
CF = 2EF + DF, where DF is the other leg of the right triangle CDF.
Substituting these expressions for EC and CF in the above equation, we get:
AE / EF = (ED + 2EF) / (2EF + DF)
Simplifying this equation, we get:
AE / EF = (ED/EF) + 2 / (2 + DF/EF)
But ED/EF = sin(DEF) and DF/EF = sin(EFD) by the definition of sine in right triangle DEF. Since the angles DEF and EFD add up to 90 degrees, their sines are complementary.
Therefore, we have that:
ED/EF = cos(EFD) and DF/EF = sin(DEF)
Substituting these expressions in the above equation, we get:
AE / EF = cos(EFD) + 2 / (2 + sin(DEF))
But cos(EFD) + 2 = sin(DEF) (Pythagorean identity).
Therefore, we have:
AE / EF = sin(DEF) / (2 + sin(DEF))
Multiplying both sides by (2 + sin(DEF)), we get:
AE = EF
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The Winchester Opera House sold 4,600 tickets this year, but it has been experiencing a 10%drop in sales from one year to the next. If this trend continues, then how many tickets willthe opera house sell 8 years from now?
Answers
Exponential decay formula:
\(y=a(1-r)^t\)y is the number of tickets sold at t years
a is the initial number of tickets sold
r is the rate of drop (in decimal form)
t is the time in years
\(y=4600(1-0.1)^t\)8 years from now:
\(\begin{gathered} y=4600(1-0.1)^8 \\ \\ y=4600(0.9)^8 \\ \\ y=1980.149 \end{gathered}\)Then, if the trend continues, 8 years from now the number of tickets sell will be approximately 1980Use the triangle angle-sum theorem and other angle relationships to find the missing angle measures
Answers
(90+40+x=180)
Second is 40 by the vertical angles theorem
Finally the third one is 45
I NEED SOME HELP PLEASE!!!:)
Pamela received some gift cards for music and movie downloads for her birthday. Using one of them, she downloaded 4 songs and 1 movie, which cost a total of $16. Using another, she purchased 6 songs and 1 movie, which cost a total of $18. How much does each download cost?
Downloads cost $______for a song and $________for a movie.
Answers
Answer:
Downloaded music 1 EACH: $1
Movie 1 EACH: $12
Step-by-step explanation:
4 songs and 1 movie = $16
6 songs and 1 movie = $18
if we minus the two formulas we get:
2 songs = $2
Try it out on the formula, it will work.
$4 + x = $16
x = $12
$6 + $12 = $18
Chandy is decorating for the
e school dance. If she wants to wrap
the clock in floral garland and its
r diameter is 13 in, how.
much garland does
she need? Use 3.14
for pi and round
to the nearest
tenths place.
Answers
Chandy wants to wrap 40.82 inches the clock in floral garland.
What is Circle?The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.
Given that;
Chandy is decorating for the school dance.
And, She wants to wrap the clock in floral garland and its diameter is 13 in.
Hence, The radius is,
⇒ 13 / 2
⇒ 6.5 inches
Now, Circumference of circle = 2πr
Where, 'r' is radius of circle.
Hence, We get;
⇒ Circumference of circle = 2πr
⇒ Circumference of circle = 2 × 3.14 × 6.5
⇒ Circumference of circle = 40.82 inches
Thus, Chandy wants to wrap 40.82 inches the clock in floral garland.
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consider a large block of iced in the shape of a cube. at the time the block is 1 ft on each side, the lengths of each side are increasing at a rate of 2 ft per hour. at what rate is the volume of the block increasing at this time
Answers
The volume of the block is increasing at a rate of \(6 ft^3/hour\) at this time.
Space in three dimensions is quantified by volume. It is frequently expressed as a numerical value using SI-derived units, other imperial units, or US customary units. Volume definition and length definition are connected.
The area occupied inside an object's three-dimensional bounds is referred to as its volume. The item's capacity is another name for it. A three-dimensional object's volume, which is expressed in cubic metres, is the quantity of space it takes up.
Let's start by finding the formula for the volume of a cube with side length s:
V = \(s^3\)
Now, let's differentiate both sides with respect to time (t):
dV/dt = \(3s^2(ds/dt)\)
We know that ds/dt = 2 ft/hour, and when s = 1 ft, we have:
dV/dt = \(3(1^2)(2) = 6 ft^3/hour\)
Therefore, the volume of the block is increasing at a rate of \(6 ft^3/hour\) at this time.
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What is the Next number in this series 7 11 19 35?
Answers
Answer:
67
Step-by-step explanation:
we have a gap between 1st and 2nd terms of 4. a gap between 2nd and 3rd of 8. 16 for next gap. the gap doubles every time.
so we expect gap between 4th and 5th terms to be 2 X 16 = 32.
35 + 32 = 67. that is the next number in the sequence.
How do you feel about the distance formula?
Answers
The question said how do i feel about the distance formular.
First let's know what the distance formular is all about.
The distance formular is an algebraic expression that gives the distnces between pairs of points in terms of their coordinates.
In two and three dimensional Euclidean space, the distance formular for points in rectangular coordinates are based on the Pythagorean theorem. The distance between the points (a, b) and (c, d) is given by;
√(a - c)² + (b - d)².
In three dimensional space, the distance between the points (a, b, c) and (d, e, f) is
√(a - d)² + (b - e)² + (c - f)².
From the explanation and definition about the distance formular above, what i feel is that it enables getting points and distances within a plane possible and easy.
A survey was conducted to investigate whether alcohol consumption and smoking are related. In a random
sample of 300 smokers, 196 said they had consumed alcohol at least once in the past week. In an
independent random sample of 300 non-smokers, 159 said they had consumed alcohol in the past week. If
P, is the proportion of smokers in the population who have had a drink in the past week and P is the
corresponding proportion of non-smokers, then a test of the hypotheses H, P, -P-0 against the two-sided alternative produces a test statistic of z=3.07 and a P-value of 0.002. If we had instead analyzed these results with a chi-square test of homogeneity, which of the following would be the test statistic and P-value?
a-942, P-value = 0.002
b. -942, P-value-0.004
-3.07, P-value - 0.004
d. -1.75, P-value = 0.002
e-1.75, P-value=0.004
Answers
e) The test statistic and P-value for the chi-square test of homogeneity would be -1.75 and 0.004, respectively.
A chi-square test of homogeneity is a useful tool for comparing two or more categorical variables. In this case, the two variables are smoking (smokers and non-smokers) and alcohol consumption (those who had consumed alcohol in the past week and those who hadn't).
The chi-square statistic is calculated by finding the difference between the observed and expected frequencies of the two groups and squaring it. The expected frequencies are found by multiplying the sample size by the overall probability of success (in this case, drinking alcohol).
The P-value is then calculated based on the chi-square statistic and the degrees of freedom (in this case, one). In this case, the chi-square statistic is -1.75 and the P-value is 0.004.
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What is the X intercept of the equation y-1= -(x+5)?
Answers
Answer:
X-intercept is -4
Step-by-step explanation:
What is the volume of the two boxes?
Answers
Answer:
volume = 2520 mm³
Step-by-step explanation:
v = 2x12x7x15 = 2520 mm³
The ordered pair corresponding to f(-1)=-2.8 is
Answers
Answer:
(-1, -2.8)
Step-by-step explanation:
When a function is defined as:
\(f(x) = y\)
\(x\) is the function's input, and \(y\) is the function's output, usually in terms of x.
From the given information:
\(f(-1) = -2.8\)
we can create an ordered pair in the form (input, output):
input = -1
output = -2.8
↓
(-1, -2.8)
there are 18 major sea islands in the queen elizabeth islands of canada. there are 15 major lakes in saskatchewan, canada. (a) if you are planning a trip to visit one of these islands, followed by one of these lakes, how many different trips could you make? (b) if you plan to visit either one of these lakes or one of these islands, how many different visits could you make?
Answers
(a) Total number of trips = 270
(b) There are 33 different visits that could make.
We have the information:
There are 18 major sea islands in the queen Elizabeth islands of Canada. There are 15 major lakes in Saskatchewan, Canada.
Let us say that A indicates the major sea Islands and B indicates the major lakes.
Taking both the cases as events, we will use multiplication principle.
Based on the information, the event A occurs in m ways and the event B will occur in n ways. So, the two events will occur in m×n ways.
(a) A × B = 18 × 15
Total number of trips = 270
(b) |A| = 18
|B| = 15
A ∩B = θ (as we cannot visit both an island and lake)
Addition Principle: There are |A| + |B| ways to choose an element form
A ∪ B , when A and B are the finite sets with A ∩B = θ
|A ∪ B| = |A| + |B| = 18 + 15 = 33
Thus, There are 33 different visits that could make.
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Gold has a density of 0.70 pounds per cubic inch and copper has a density of 8.96 grams per cubic centimeter. How much will 1
cubic foot of each metal weigh? 1 ft(3) = 1728 in(3), 1 ft(3) = 28,316.8 cm cubed, and 1 lb = 453.5 g.
"use the equation p=m/v
Answers
Answer:
1 cubic foot of gold will weigh 1209.6 pounds
1 cubic foot of copper will weigh 559 pounds.
Step-by-step explanation:
Density = Mass/Volume
Mass = Density × Volume
Volume of each metal = 1 ft³
a) Gold
Density of Gold = 0.70 pounds per cubic inch(lb/in³)
We will be converting the Volume of gold from cubic foot to cubic inches
The Volume of gold given in the question = 1 cubic foot = 1 ft³
1 cubic foot = 1728 cubic inches
Hence, the volume of Gold = 1728 in³
Mass of Gold = Density of Gold × Volume of Gold
Mass of Gold = 0.70Ib/in³ × 1728 in³
= 1209.6 lb(pounds)
2) Copper
Density of copper = 8.96 grams per cubic centimeter
Ib = pounds
We convert this density in g/cm³ to ib/in³
453.5g = 1 Ib
8.96g = y Ib
Cross Multiply
= 453.5g × y Ib =8.96g × 1 Ib
= y Ib = 8.96g × 1 Ib/453.5g
y Ib = 0.0197574421 Ib
Density of Copper = 0.0197574421 Ib/cm³
28,316.8 cm³ = 1 ft³
1 cm³ = x ft³
x ft³ = 1 cm³ × 1 ft³/28,316.8cm³
x ft³ = 0.0000353147 ft³
1 cm³ = 0.0000353147 ft³
We have to convert the density of Copper from Ib/cm³ to Ib/ft³
0.0197574421 Ib/cm³ = 0.0197574421 Ib/0.0000353147 ft³
559.46793026 Ib/ft³
Volume of copper = 1ft³
Mass = Density × Volume
Mass of copper = 559.46793026 Ib/ft³ × 1 ft³
= 559.46793026 Ib or pounds
≈ 559 pounds
Therefore , the weight of gold is 1209.6 pounds and the weight of copper will be 559 pounds.
How did you convert units to solve each word problem
Answers
To convert units in order to solve word problems, We use conversion factors and dimensional analysis.
Conversion factors are ratios that represent the relationship between two units. For example, 1 meter is equal to 3.28 feet, so the conversion factor between meters and feet is 3.28 feet per meter or 1 meter per 3.28 feet.
Dimensional analysis is a method that uses conversion factors to convert between different units. To use dimensional analysis, we set up the conversion factors so that the units cancel out and leave us with the desired units. For example, if we want to convert 10 meters to feet, we can use the conversion factor of 3.28 feet per meter:
10 meters × 3.28 feet/meter = 32.8 feetThe meters cancel out, leaving us with feet.
By using conversion factors and dimensional analysis, we can convert units and solve word problems that involve different units of measurement.
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A green gold bracelet weighs 56g. The bracelet is made from 75% gold, 20% silver and 5% copper.
(a) What is the mass of the gold in the bracelet?
(b) What is the mass of the copper in the bracelet?
Answers
Answer:
a. 42 g
b. 2.8 g
Step-by-step explanation:
a. Since the entire thing weighs 56g and gold is 75% of the mass you simply find 75% of the mass. This is done by converting the 75% into decimal form by dividing it by 100 to get 0.75 and then multiply it by the 56 g to get (0.75 * 56 g) = 42 g
b. same thing here as the previous answer. 5%/100 = 0.05. (0.05 * 56 g) = 2.8 g
I need to show the work. Please help
Answers
Answer:
18
Step-by-step explanation:
125 = 7x - 1 ...... because if the two lines are parallel alternate interior angles are congruent
125 + 1 = 7x
126 = 7x
\( \frac{126}{7} = \frac{7x}{7} \)
x = 18
hope it helps
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Explain the FOIL method when multiplying two binomials. Use the following example to
help you explain in words.
(2x+3)(3x-1)
Answers
Answer:
6x²+7x-3
Step-by-step explanation:
(2x+3)(3x-1)
(2x)(3x)+(2x)(-1)+(3)(3x)+(3)(-1)
6x²-2x+9x-3
6x²+7x-3
Basically, what we are doing is taking 2x and distributing it to the 3x and -1. Then, we distribute the 3 to the 3x and -1. Add up these two totals and you have your expanded binomial.
FOIL means First, Outer, Inner, Last, so this makes sense.
Solve the equation. Linear Equations:Variable on Both Sides
-7x-3x+2=-8x-8
Solve it step by step
Answers
Let f(x)=2sqrt(x) If g(x) is the graph of f(x) shifted up 2 units and right 5 units, write a formula for g(x) g(x)=
Answers
The function after the given translations is g(x)=2√(x+5)+2.
The given function is f(x)=2√x.
When the shape is moved towards the right by k units, then replace x with x + k.
When the shape is moved up by k units, then replace y with y + k.
Here, g(x)=f(x+5)+2
Now, g(x)=2√(x+5)+2
Therefore, the function after the given translations is g(x)=2√(x+5)+2.
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Find an equation of the line having the given slope and containing the given point. m=2 (-5,-6)
Answers
The equation of the line with slope 2 and containing the point (-5, -6) is y = 2x + 16.
We can use the point-slope form of the equation of a line to find the equation of the line:
y - y1 = m(x - x1)
where m is the slope of the line, and (x1, y1) is a point on the line.
Substituting m = 2 and (-5, -6) for (x1, y1), we get:
y - (-6) = 2(x - (-5))
Simplifying and solving for y, we get:
y + 6 = 2(x + 5)
y = 2x + 16
Therefore, the equation of the line with slope 2 and containing the point (-5, -6) is y = 2x + 16.
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