solving systems by substitution (algebra 2)
7x + y = 12
y = 4x - 21​

(A) (m,n) = g(n,m) for all positive integers m,n,

(B) (m,n + 1) = g(m + 1,n) for all positive integers m,n,

(D) (2m,2n) = (g(m,n))2 for all positive integers m,n

[tex] \rm f(m,n,p) = \sum \limits_{i = 0}^{p} {}^{m} C_{i} \: \: {}^{n + i} C_{p} \: \: {}^{p + n} C_{p - i}[/tex]

[tex]\rm {}^{m} C_{i} \: \: {}^{n + i} C_{p} \: \: {}^{p + n} C_{p - i}[/tex]

[tex]\rm {}^{m} C_{i} \dfrac{(n + i)!}{p!(n - p + i)!} \times \dfrac{(n + p)!}{(p - i)!(n + i)!} [/tex]

[tex]\rm {}^{m} C_{i} \times \dfrac{(n + p)!}{p!} \times \dfrac{1!}{(n -p + i)!(p - i)!} [/tex]

[tex]\rm {}^{m} C_{i} \times \dfrac{(n + p)!}{p!} \times \dfrac{1!}{(n -p + i)!(p - i)!} [/tex]

[tex]\rm {}^{m} C_{i} \times \dfrac{(n + p)!}{p!n!} \times \dfrac{n!}{(n -p + i)!(p - i)!} [/tex]

[tex]\rm {}^{m} C_{i} \: \: {}^{n + p} C_{p} \: \: {}^{n} C_{p - i} \: \: \{{}^{m} C_{i} \: \: {}^{n } C_{p - i} = {}^{m + n} C_{p } \}[/tex]

[tex]\rm {}^{m} C_{i} \: \: {}^{n + p} C_{p} \: \: {}^{n} C_{p - i} \: \: \{{}^{m} C_{i} \: \: {}^{n } C_{p - i} = {}^{m + n} C_{p } \}[/tex]

[tex] \rm f(m,n,p) = {}^{n + p} C_{p}{}^{m + n} C_{p}[/tex]

[tex] \rm \dfrac{f(m,n,p)}{{}^{n + p} C_{p}} = {}^{m + n} C_{p}[/tex]

Now,

[tex] \rm g(m,n) = \sum \limits_{p = 0}^{m + n} \dfrac{f(m,n,p)}{{}^{n + p} C_{p}} [/tex]

[tex] \rm g(m,n) = \sum \limits_{p = 0}^{m + n} {}^{m + n} C_{p}[/tex]

[tex] \rm g (m,n) = {2}^{m + n} [/tex]

[tex] \rm(A) \: g(m,n) = q(n,m)[/tex]

[tex] \rm(B) \: g(m,n + 1) = {2}^{m + n + 1} [/tex]

[tex] \rm g(m + n,n ) = {2}^{m + 1 + n} [/tex]

[tex] \rm(D) \: g(2m,2n) = {2}^{2m + 2n} [/tex]

[tex] = \rm( {2}^{m + n} {)}^{2} [/tex]

[tex] = \rm(g(m,n)) {}^{2} [/tex]

Answer:

[tex]\textsf{1)} \quad t=\dfrac{48}{r}[/tex]

[tex]\textsf{2)} \quad \sf 2.4\:hours=2\:hours\:24\:minutes[/tex]

Step-by-step explanation:

[tex]\boxed{\sf Time=\dfrac{Distance}{Speed}}[/tex]

Given:

Substitute the given values into the formula:

[tex]\implies t=\dfrac{48}{r}[/tex]

Substitute r = 20 km/h into the derived equation from part 1 and solve for t:

[tex]\implies t=\dfrac{48}{20}=2.4\: \sf hours[/tex]

Note: 2.4 hours = 2 hours 24 minutes

Answer: 2.4 hours

Step-by-step explanation: t=time

         r=rate        so question one is equal to t=48/r

then question 2 is 48/20=2.4

Answer:

x ≥ 3

Step-by-step explanation:

2(x - 3) ≥ -3(-3 + x)

2x - 6 ≥ 9 - 3x

5x - 6 ≥ 9

5x ≥ 15

x ≥ 3

Answer:

x ≥ 3

Step-by-step explanation:

write, calculate, divide both sides

Answer:

A graphing calculator can help you.

Here is the graph, it is in the photo.

Answer:

Reflected function is y = - x³ + 3

Step-by-step explanation:

When a function f(x) is reflected about the y-axis, the reflected function becomes g(x) = f(-x)

f(x) = x³ + 3

f(-x) = (-x)³ + 3 =-x³ + 3

The rule f ( x ) - g ( x ) is given as x^3 - x^2 + 18

Given data

f(x) = x3 + 10, g(x) = x2 - 8, and h(x) = 5x + 4

f(x) - g(x) = ( x3 + 10 ) - ( x2 - 8 )

= ( x3 + 10 ) - ( x2 - 8 )

opening the parenthesis gives

= x3 + 10 - x2 + 8

collecting like terms gives

= x3 - x2 + 18

Hence we can say that f(x) - g(x) = x3 - x2 + 18

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complete question

The complete question is attached

Answer:

y = - 8x + 3

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 = - 8x - 2 ← is in slope- intercept form

with slope m = - 8

• Parallel lines have equal slopes , then

y = - 8x + c ← is the partial equation of the parallel line

to find c substitute (- 0.5, 7 ) into the partial equation

7 = 4 + c ⇒ c = 7 - 4 = 3

y = - 8x + 3 ← equation of parallel line

Answer:

(x/8)(19 - x)

Step-by-step explanation:

The quotient is the answer to a division problem.

The difference is the answer to a subtraction problem.

"Times" indicates multiplication, which you could look at it as a link between the two functions, or quantities (them being the constants and variables sectioned off into parenthesis).

Answer:

your quotient should be 57

so you would start with 38 and you gotta think hmm

C+(-19)=38

okay so you would change -19 to positive 19

and then add that to 38

and then you would get 57

so therefore C=57

so

57+(-19)=38

Answer:

57

Step-by-step explanation:

Yes; for each element of the domain, there is only one element of the range.

For a relation in coordinate form to be a function, there is must nor be repetition in the value of its coordinate.

Given the following relation as shown  { ( 2 , 5 ) , ( 4 , − 2 ) , ( 3 , 3 ) , ( 5 , 4 ) , ( − 2 , 5 ) }, since there is no repetition in its domain values, hence the relation is a function.

We can conclude therefore that Yes; for each element of the domain, there is only one element of the range.

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Answer:

- 1

Step-by-step explanation:

2x + 2y = 1

=> 2y = - 2x + 1

=> y = - x + 1

=> Slope = - 1

Answer:-

Paul has $64 pocket money each day.

Explanantion:-

Let amy ,flora and Paul 's pocket money be x,y and z respectively.

So,

By the question :

x = 3y

So, y = x/3 - eqn i

z = 7 + x - eqn ii

7(x + y + z) = 980

x + y + z = 140

Now,

x + y+ z = 140

x + x/3 + 7 + x = 140

or, 2x + x/3 = 133

or, 6x + x = 399

so, x = 399/7

so, x = $57

Now,

z = 7 + x

= 7 + $57

= $64

infinitely many solutions 

Example

Simplify left side using distributive property to 6x+16. Combine like terms (2x and 4x) on the right side to 6x+16. Both sides simplify to the same expression (left side = right side) so there are infinitely many solutions. You can plug in any real number for x and the left side will always equal the right side.

Answer:

The answer is yes. Since there is no x^2

The length of the field with area 119 square feet is 17 feet and the width of the feet is 7 feet.

The area is the total region or space in two dimension that is covered by a figure , surface or object. Area of a rectangle is calculated by the product of its length and width.

A rectangle has 4 sides where each pair of  opposite side is equal .The length of a rectangle is normally the longer side and the width defines the shorter side.

Given the area of the barn is 119 square feet.

Let us consider the width of the barn to be x feet. As the length is 10 feet longer than the width then we will find the length to be ( x +10 ) feet.

Area = length × width

or, 119 = x × (x+10)

or, 119 = x² +10x

or, x² +10x - 119=0

Now we will solve the quadratic equation thus formed by middle term-factorization method:

or, x² +17x - 7x - 119 = 0

or, x ( x + 17 ) - 7 ( x + 17) = 0

or, ( x + 17 ) ( x - 7 ) = 0

Now either x + 17 = 0 or x - 7 =0

Therefore either x = 7  

or, x = -17 (The width cannot be a negative value)

∴Width = 7 feet and length = 7 + 10 = 17 feet.

Therefore the length of the field is 17 feet and the width of the feet is 7 feet.

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The length and width is 32 inches and 10 inches.

Let us assume the width of rectangle be w. So, length of rectangle = 3w + 2. As per the known fact, the length and width of rectangle for perimeter is related as - Perimeter = 2 (length + width)

Keep the values in formula to find the value of width

84 = 2 (3w + 2 + w)

Performing addition in bracket

84 = 2 (4w + 2)

Performing multiplication

84 = 8w + 4

8w = 84 - 4

Performing subtraction

8w = 80

w = 80 ÷ 8

Performing division

w = 10 inches

Finding length now -

Length = 3×10 + 2

Length = 30 + 2

Length = 32 inches

Hence, the length and width is 32 inches and 10 inches.

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Answer:

A

Step-by-step explanation:

that should be an equal sign

Reason: The absolute value erases the negative sign.

Saying |-29| = 29 means that the number -29 is 29 units from 0 on the number line.

Answer: 29

Step-by-step explanation: The absolute value of -29 is 29. An absolute value is a number's distance from 0. -29 + 29 = 0, therefore the absolute value is 29.

Answer:

See below

Step-by-step explanation:

Rate = 20 sheets  / 14 seconds =   10/7  sheet / sec

every 200 sheets is 4 rest periods =  12 minutes

1000 sheets / (10/7 sheet/ sec)  +       12 min

  700 seconds + 12 minutes

   = 700 seconds + 720 seconds = 1420 seconds = 23 min 40 sec

The solution of the given equation 4m - m - t = 0  for m is t/3.

According to the given question.

We have a linear equation in two variables i.e. 4m - t = m.

As we know that, an equation is said to be linear equation in two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero.

Since, we have to solve the given equation 4m - t = m for m. So, the solution of the equation 4m - t = m for m is given by

4m - t = m

⇒ 4m - m - t = 0   (subtracting m both the sides)

⇒ 3m - t = 0

⇒  3m = t   (adding t both the sides)

⇒ m = t/3.

Hence, the solution of the given equation 4m - m - t = 0  for m is t/3.

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Answer: Yes

Step-by-step explanation: Kelly's results are likely to be correct mainly because she pulled votes from several different age groups. The fact that she asked several age groups whether or not they would be in favor of creating a local dog park shows that her results are likely to yield valid results.

The number of teachers that work in both schools is 97.

The addition is a way of combining things and counting them together as one large group.

We have,

Number of teachers in Alberts Elementary school = 76

Number of teachers in Bothel Elementary school = 24

Number of teachers that work in both schools = 3

The number of teachers that work in both schools:

= 76 + 24 - 3

We are subtracting 3 from the total because 3 teachers that work in both schools have been added twice.

= 100 - 3

= 97

Thus the number of teachers that work in both schools is 97.

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The amount that would be in the account kept with BPI bank in 3 years is P3,011.2640

It would 6.9422 years for the deposit to double if interest rate is 10,5%

What is future value of immediate amount?

The future value of deposit today in 3 years means its future equivalent when the amount has earned interest over the 3 years period

FV=PV*(1+r)^N

FV=future value after 3 years=unknown

PV=immediate deposit=3,000

r=annual interest rate=0.125%

N=number of years that the deposit lasted=3

FV=3000*(1+0.125%)^3

FV=P3,011.2640

Time taken to double:

FV=future value=3000*2=6000

PV=3000

r=annual interest rate=10.5%

N=number of years that it takes initial deposit to double=unknown

6000=3000*(1+10.5%)^N

6000/3000=(1.105)^N

2=(1.105)^N

take log of both sides

ln(2)=N*ln(1.105)

N=ln(2)/ln(1.105)

N=6.9422 years

Annuity:

FV=PMT*(1+r)^N-1/r

FV=future value of annuity=unknown

PMT=annual payment=20000

r=interest rate=3%

N=number of annual payments=7

FV=20000*(1+3%)^7-1/7%

FV= 153,249.2436

FV=PMT*(1+r)^N-1/r*(1+r)

FV=20000*(1+3%)^7-1/7%*(1+7%)

FV= 157,846.7209

20,000,000=1,250,000*(1+r)^20

20,000,000/1,250,000=(1+r)^20

16=(1+r)^20

(16)^1=(1+r)^20

divide indexes on both sides by 20

(16)^(1/20)=1+r

r=(16)^(1/20)-1

r=14.8698%

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Answer:

3k

Step-by-step explanation:

k+(k-1)+(k+1)=3k

The tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point

a)y=24x²+8x-2 at x=3.

b) y=(4x+2)/(6x-1) at x=3.

The tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point

Given

f at x=3 is y= 4x+2

g at x=3 is y=6x-1

f(3)= 4x+2  and g(3)=6x-1

y=f(x)g(x)

at x=3,

Solution for a:

y=f(3)g(3)

=(4x+2)(6x-1)

=24x²-4x+12x-2

=24x²+8x-2

So y=24x²+8x-2 at x=3.

Solution for b:

y=f(3)/g(3)

=(4x+2)/(6x-1)

So y=(4x+2)/(6x-1) at x=3.

Therefore y=24x²+8x-2 at x=3 for y = f(x)g(x) and  y=(4x+2)/(6x-1) at x=3 for y = f(x)/g(x).

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Answer:

X= -7

X = 3

Step-by-step explanation:

| x + 2| = 5

|-7 + 2| = 5

|-5| = 5

5 = 5

|x + 2| = 5

|3 + 2| = 5

|5| = 5

5 = 5

X= -7

X = 3