Write the equation y - 3 = 4(x + 1) in standard form.
Write the equation y + 7 = -2(x - 3) in slope-intercept form.
The equation y - 5 = 3(x - 2)
is written in point-slope form. Write this equation in the
following ways:
a. slope-intercept form
b. standard form
So remember that slope-intercept form is y = mx+b (m = slope, b = y-intercept) and standard form is ax+by = c. (a and b are coefficients and c is the constant)
Starting with the first question: 9x + 2y = 6
Firstly, subtract 9x on both sides: 2y = -9x + 6
Next, divide both sides by 2, and your equation will be y = -4.5x + 3
Now with the next equation: y - 3 = 4(x + 1)
Firstly, foil 4(x + 1): y - 3 = 4x + 4
Next, subtract y on both sides: -3 = 4x - y + 4
Next, subtract 4 on both sides, and your answer will be: -7 = 4x - y
Next equation: y + 7 = -2(x - 3)
Firstly, foil -2(x - 3): y + 7 = -2x + 6
Next, subtract 7 on both sides of the equation, and your answer will be: y = -2x - 1
Next: y - 5 = 3(x - 2) (part a)
Firstly, foil 3(x - 2): y - 5 = 3x - 6
Next, add 5 on both sides, and your equation is: y = 3x - 1
Next, y - 5 = 3(x - 2) (part b)
Firstly, foil 3(x - 2): y - 5 = 3x - 6
Next, subtract y on both sides of the equations: -5 = 3x - y - 6
Next, add 6 on both sides of the equation, and your answer will be: 1 = 3x - y
Answer:
In both parts of the question, you're asked to find the limit as
n approaches infinity for certain probabilities involving the estimation of the unknown probability of success
π. Given that
=
0.5
π=0.5, we can simplify the expressions and apply limit properties.
Step-by-step explanation:
Let's start with part (a):
a) Find
lim
�
→
∞
�
(
�
(
�
^
−
�
)
≤
�
)
lim
n→∞
P(n(
π
^
−π)≤x), if
�
=
0.5
π=0.5.
In a Bernoulli distribution, the variance of the estimator
�
^
π
^
is given by
Var
(
�
^
)
=
�
(
1
−
�
)
�
Var(
π
^
)=
n
π(1−π)
. Since
�
=
0.5
π=0.5, this variance simplifies to
Var
(
�
^
)
=
1
4
�
Var(
π
^
)=
4n
1
.
We can use the Central Limit Theorem (CLT) here. The CLT states that as
�
n approaches infinity, the distribution of the sample mean approaches a normal distribution with mean
�
μ (population mean) and variance
�
2
�
n
σ
2
, where
�
2
σ
2
is the population variance. Since we have
�
=
0.5
π=0.5 and
Var
(
�
^
)
=
1
4
�
Var(
π
^
)=
4n
1
, we can treat
�
^
π
^
as a sample mean of Bernoulli trials with
�
=
0.5
π=0.5.
Now, let's rewrite the expression
lim
�
→
∞
�
(
�
(
�
^
−
�
)
≤
�
)
lim
n→∞
P(n(
π
^
−π)≤x) as a z-score (standard score) and find the limit:
lim
�
→
∞
�
(
�
(
�
^
−
�
)
Var
(
�
^
)
≤
�
Var
(
�
^
)
)
lim
n→∞
P(
Var(
π
^
)
n(
π
^
−π)
≤
Var(
π
^
)
x
)
Substitute the values:
�
=
0.5
π=0.5 and
Var
(
�
^
)
=
1
4
�
Var(
π
^
)=
4n
1
:
lim
�
→
∞
�
(
2
�
(
�
^
−
0.5
)
1
4
�
≤
�
1
4
�
)
lim
n→∞
P(
4n
1
2n(
π
^
−0.5)
≤
4n
1
x
)
Simplify:
lim
�
→
∞
�
(
4
�
(
�
^
−
0.5
)
≤
2
�
)
lim
n→∞
P(4n(
π
^
−0.5)≤2x)
Notice that the left-hand side now resembles a z-score. As
�
n goes to infinity, the expression will converge to the standard normal distribution's cumulative distribution function (CDF). Therefore, the limit is:
lim
�
→
∞
�
(
4
�
(
�
^
−
0.5
)
≤
2
�
)
=
Φ
(
2
�
)
lim
n→∞
P(4n(
π
^
−0.5)≤2x)=Φ(2x)
where
Φ
Φ represents the standard normal cumulative distribution function.
This limit is not dependent on
�
π and will approach the value of
Φ
(
2
�
)
Φ(2x) as
�
n goes to infinity.
For part (b), the approach is similar, but it involves the logit transformation. The logit transformation of
�
^
π
^
is
logit
(
�
^
)
=
log
(
�
^
1
−
�
^
)
logit(
π
^
)=log(
1−
π
^
π
^
). You would follow a similar process of simplifying and finding the limit as
�
n approaches infinity.
-3y+4=-11
20x + 25y = 345
How much more can Paul shovel in 1 minute than Melinda?
3 square feet per minute
6 square feet per minute
9 square feet per minute
15 square feet per minute
Answer: Hello mate!
we have the system of equations
30x + 30y = 450
20x + 25y = 345
where x is the number of square feet of snow that Melinda shovels in one minute, and y is the number of square feet of snow that Paul shovels in one minute.
Let's solve the system of equations, first, isolate one of the therms in the first equation, for example, x.
30x+30y = 450
x + y = 450/30 = 15
x = 15 - y
now we can replace it in the second equation and solve it for y.
20x + 25y = 345
20(15 - y) + 25y = 345
300 - 20y + 25y = 345
5y = 45
y = 45/5 = 9
Paul can shovel 9 square feets of snow in one minute, now replace it in the first equation and solve it for x.
x + y = 450/30 = 15
x + 9 = 15
x = 15 - 9 = 6
so melinda sholves 6 square feet of snow in one minute.
The question is:
How much more can Paul shovel in 1 minute than Melinda?
We need to see the difference y - x = 9 - 6 = 3
this means that paul shovels 3 more square feet per minute than Melinda.
The coordinates will be (0, 3) and (1.5, 0), and mark these points on the paper and make a straight line.
A Linear system is a system in which the degree of the variable in the equation is one. It may contain one, two, or more than two variables.
The linear function is given below.
y = -2x + 3
Then the graph of the equation will be
Put x = 0, then the value of y will be
y = -2(0) + 3
y = 3
Put y = 0, then the value of x will be
2x = 3
x = 1.5
Then the coordinates will be (0, 3) and (1.5, 0).
Mark these points on the paper and make a straight line.
The graph is given below.
More about the linear system link is given below.
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