b (a+b) - a (a-b) simplify

Answer:

The concentric circles are

[tex]3x^{2}+3y^{2}+12x-6y-21=0[/tex]  and [tex]4x^{2}+4y^{2}+16x-8y-308=0[/tex]

[tex]5x^{2}+5y^{2}-30x+20y-10=0[/tex]  and [tex]3x^{2}+3y^{2}-18x+12y-81=0[/tex]

[tex]4x^{2}+4y^{2}-16x+24y-28=0[/tex]  and [tex]2x^{2}+2y^{2}-8x+12y-40=0[/tex]

[tex]x^{2}+y^{2}-2x+8y-13=0[/tex]  and  [tex]5x^{2}+5y^{2}-10x+40y-75=0[/tex]

Step-by-step explanation:

we know that

The equation of the circle in standard form is equal to

[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex]

where

(h,k) is the center and r is the radius

Remember that

Concentric circles, are circles that have the same center

so

Convert each equation in standard form and then compare the centers

The complete answer in the attached document

Part 1) we have

[tex]3x^{2}+3y^{2}+12x-6y-21=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](3x^{2}+12x)+(3y^{2}-6y)=21[/tex]

Factor the leading coefficient of each expression

[tex]3(x^{2}+4x)+3(y^{2}-2y)=21[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]3(x^{2}+4x+4)+3(y^{2}-2y+1)=21+12+3[/tex]

[tex]3(x^{2}+4x+4)+3(y^{2}-2y+1)=36[/tex]

Rewrite as perfect squares

[tex]3(x+2)^{2}+3(y-1)^{2}=36[/tex]

[tex](x+2)^{2}+(y-1)^{2}=12[/tex]

therefore

The center is the point (-2,1)                                  

Part 2) we have

[tex]5x^{2}+5y^{2}-30x+20y-10=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](5x^{2}-30x)+(5y^{2}+20y)=10[/tex]

Factor the leading coefficient of each expression

[tex]5(x^{2}-6x)+5(y^{2}+4y)=10[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]5(x^{2}-6x+9)+5(y^{2}+4y+4)=10+45+20[/tex]

[tex]5(x^{2}-6x+9)+5(y^{2}+4y+4)=75[/tex]

Rewrite as perfect squares

[tex]5(x-3)^{2}+5(y+2)^{2}=75[/tex]

[tex](x-3)^{2}+(y+2)^{2}=15[/tex]

therefore

The center is the point (3,-2)      

Part 3) we have

[tex]x^{2}+y^{2}-12x-8y-100=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](x^{2}-12x)+(y^{2}-8y)=100[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex](x^{2}-12x+36)+(y^{2}-8y+16)=100+36+16[/tex]

[tex](x^{2}-12x+36)+(y^{2}-8y+16)=152[/tex]

Rewrite as perfect squares

[tex](x-6)^{2}+(y-4)^{2}=152[/tex]

therefore

The center is the point (6,4)      

Part 4) we have

[tex]4x^{2}+4y^{2}-16x+24y-28=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](4x^{2}-16x)+(4y^{2}+24y)=28[/tex]

Factor the leading coefficient of each expression

[tex]4(x^{2}-4x)+4(y^{2}+6y)=28[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]4(x^{2}-4x+4)+4(y^{2}+6y+9)=28+16+36[/tex]

[tex]4(x^{2}-4x+4)+4(y^{2}+6y+9)=80[/tex]

Rewrite as perfect squares

[tex]4(x-2)^{2}+4(y+3)^{2}=80[/tex]

[tex](x-2)^{2}+(y+3)^{2}=20[/tex]

therefore

The center is the point (2,-3)  

Part 5) we have

[tex]x^{2}+y^{2}-2x+8y-13=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](x^{2}-2x)+(y^{2}+8y)=13[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex](x^{2}-2x+1)+(y^{2}+8y+16)=13+1+16[/tex]

[tex](x^{2}-2x+1)+(y^{2}+8y+16)=30[/tex]

Rewrite as perfect squares

[tex](x-1)^{2}+(y+4)^{2}=30[/tex]

therefore

The center is the point (1,-4)  

Part 6) we have

[tex]5x^{2}+5y^{2}-10x+40y-75=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](5x^{2}-10x)+(5y^{2}+40y)=75[/tex]

Factor the leading coefficient of each expression

[tex]5(x^{2}-2x)+5(y^{2}+8y)=75[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]5(x^{2}-2x+1)+5(y^{2}+8y+16)=75+5+80[/tex]

[tex]5(x^{2}-2x+1)+5(y^{2}+8y+16)=160[/tex]

Rewrite as perfect squares

[tex]5(x-1)^{2}+5(y+4)^{2}=160[/tex]

[tex](x-1)^{2}+(y+4)^{2}=32[/tex]

therefore

The center is the point (1,-4)  

Part 7) we have

[tex]4x^{2}+4y^{2}+16x-8y-308=0[/tex]

Group terms that contain the same variable, and move the constant to the opposite side of the equation

[tex](4x^{2}+16x)+(4y^{2}-8y)=308[/tex]

Factor the leading coefficient of each expression

[tex]4(x^{2}+4x)+4(y^{2}-2y)=308[/tex]

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

[tex]4(x^{2}+4x+4)+4(y^{2}-2y+1)=308+16+4[/tex]

[tex]4(x^{2}+4x+4)+4(y^{2}-2y+1)=328[/tex]

Rewrite as perfect squares

[tex]4(x+2)^{2}+4(y-1)^{2}=328[/tex]

[tex](x+2)^{2}+(y-1)^{2}=82[/tex]

therefore

The center is the point (-2,1)  

Part 8) Part 9) and Part 10)  in the attached document

Soccer ball reaches its highest position when its equation turns from positive slope to zero and then negative. Mathematically it is where the derivative of the its function equals to zero. so:

d(-3x^2 + 6x + 3) / dx = -6x + 6

-6x + 6 = 0 -> x = 1 ->

[tex]y = 6[/tex]

it seems Paige's ball reaches higher than Viaola's with only 6 meters height.

Answer: 90 degrees.

Step-by-step explanation: since 360 degrees is a whole circle, subtract 360 from 270 to get 90 degrees.

Answer:

98.1% chance of being accepted

Step-by-step explanation:

Given:

sample size,n=56

acceptance condition= at most 2 batteries do not meet specifications

shipment size=7000

battery percentage in shipment that do not meet specification= 1%

Applying binomial distribution

In this formula, a is the acceptable number of defectives;

 n is the sample size; 

p is the fraction of defectives in the population.  

Now putting the value

a= 2

n=56

p=0.01

[tex]\frac{56!}{0!\left(56-0\right)!}\left(0.01\right)^0\:\left(1-0.01\right)^{\left(56-0\right)} + \frac{56!}{1!\left(56-1\right)!}\left(0.01\right)^1\:\left(1-0.01\right)^{\left(56-1\right)} +[/tex][tex]\:\frac{56!}{2!\left(56-2\right)!}\left(0.01\right)^2\:\left(1-0.01\right)^{\left(56-2\right)}[/tex]

=0.56960+0.32219+0.08949

After summation, we get 0.981 i.e. a 98.1% chance of being accepted.  As this is such a high chance, we can expect many of the shipments like this to be accepted!

Answer:

(3v-7)(2v^2+5)

Step-by-step explanation:

To factor 6v^3-14v^2+15v-35 by grouping we are going to try pair to up the pair two terms and also the last two terms. Like this:

(6v^3-14v^2)+(15v-35)

Now from each we factor what we can:

2v^2(3v-7)+5(3v-7)

Now there are two terms: 2v^2(3v-7) and 5(3v-7).

These terms contain a common factor and it is (3v-7).

We are going to factor (3v-7) out like so:

2v^2(3v-7)+5(3v-7)

(3v-7)(2v^2+5)

Answer:

Step-by-step explanation:

Actually, you have not "given" the domain.

The domain of F(x)=x^2+2 is "the set of all real numbers," because F(x)=x^2+2 is a polynomial.

F(x^2) = (x^2) + 2 = x^4 + 2.  Again, this is a polynomial and the domain is "the set of all real numbers."

Double check to ensure that you have copied down this problem correctly.

Answer:

yes

Step-by-step explanation:

let me explain more in depth. 1: yes, it's the point where the function crosses the x axis. 2: the absence of acceleration. 3: I think it can be negative

Answer:

(3,2)

Step-by-step explanation:

We are given the system:

3x-y=7

2x-2y=2.

We are asked to solve this by substitution.  We need to pick an equation and pick a variable from that equation to solve for that variable.

I really like either for this.  Some people might go with the first one though. Let's do that.  I will solve the first one for y.

3x-y=7

Subtract 3x on both sides:

   -y=-3x+7

Divide both sides by -1:

     y=3x-7

Now we are ready for substitution.  We are going to plug this equation into the second equation giving us:

2x-2y=2 with y=3x-7 gives us:

2x-2(3x-7)=2

Distribute:

2x-6x+14=2

Combine like terms:

-4x+14=2

Subtract 14 on both sides:

-4x      =2-14

Simplify:

-4x       =-12

Divide both sides by -4:

  x       =-12/-4

Simplify:

 x         =3

So using y=3x-7 and x=3, I will find y now.

y=3x-7 if x=3

y=3(3)-7   (I inserted 3 for x since we had x=3)

y=9-7       (Simplified)

y=2           (Simplified)

The answer is (x,y)=(3,2).

Step-by-step explanation:

Draw a picture (like the image below).

Notice that triangles ABC and ACD both contain right angles, and both contain angle A (20°).  Since angles of a triangle add up to 180°, that means their third angle must also be the same (70°).

Also notice that triangles ABC and CBD both contain right angles, and both contain angle B (70°).  So their third angle must also be the same (20°).

Therefore:

m∠CDB = 90°

m∠CBD = 70°

m∠BCD = 20°

m∠CDA = 90°

m∠CAD = 20°

m∠ACD = 70°

[tex]\bf (\sqrt{5x+6})^2\implies \sqrt{(5x+6)^2}\implies 5x+6[/tex]

Answer:

[tex]c=7[/tex]

Step-by-step explanation:

They are just asking you to compare

[tex]ax^2+bx+c=0[/tex] to

[tex]3x^2+5x+7=0[/tex].

What constant values are in the place of [tex]a,b, \text{ and } c[/tex].

[tex]a=3[/tex]

[tex]b=5[/tex]

[tex]c=7[/tex]

Step-by-step explanation:

exponential decay functions are written in the form :

[tex]y=ab^{x}[/tex]

where b is less than 1

if we look at the 3rd choice and consider the term on the right.

[tex](8/7)^{-x}[/tex]

= [tex](7/8)^{x}[/tex]

If we compare this to the general form above,

b = 7/8 (which is less than 1)

hence the 3rd choice is correct.

The function which is an exponential decay function is:

                       [tex]f(x)=\dfrac{3}{2}(\dfrac{8}{7})^{-x}[/tex]

We know that an exponential  function is in the form of:

          [tex]f(x)=ab^x[/tex]

where a>0 and  if 0<b<1 then the function is a exponential decay function.

and if b>1 then the function is a exponential growth function.

a)

[tex]f(x)=\dfrac{3}{4}(\dfrac{7}{4})^x[/tex]

Here

[tex]b=\dfrac{7}{4}>1[/tex]

Hence, the function is a exponential growth function.

b)

[tex]f(x)=\dfrac{2}{3}(\dfrac{4}{5})^{-x}[/tex]

We know that:

[tex]a^{-x}=(\dfrac{1}{a})^x[/tex]

Hence, we have the function f(x) as:

[tex]f(x)=\dfrac{2}{3}(\dfrac{5}{4})^x[/tex]

Here

[tex]b=\dfrac{5}{4}>1[/tex]

Hence, the function is a exponential growth function.

c)

[tex]f(x)=\dfrac{3}{2}(\dfrac{8}{7})^{-x}[/tex]

We know that:

[tex]a^{-x}=(\dfrac{1}{a})^x[/tex]

Hence, we have the function f(x) as:

[tex]f(x)=\dfrac{3}{2}(\dfrac{7}{8})^x[/tex]

Here

[tex]b=\dfrac{7}{8}<1[/tex]

Hence, the function is a exponential decay function.

d)

[tex]f(x)=\dfrac{1}{3}(\dfrac{9}{2})^x[/tex]

Here

[tex]b=\dfrac{9}{2}>1[/tex]

Hence, the function is a exponential growth function.

Answer:

y=(3/2)x+6

If your equation is in a different form, let me know.

Step-by-step explanation:

So the slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.

Parallel lines have the same slope, m (different y-intercept (b) though).

So we need to find the slope going through (1,6) and (-7,-6).

To do this you could use [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex].

Or, what I like to do is line the points up vertically and subtract vertically then put 2nd difference over first difference. Like so:

(  1  ,  6)

-( -7,  -6)

---------------

  8     12

So the slope of our line is 12/8.

Let's reduce it! Both numerator and denominator are divisible by 4 so divide top and bottom by 4 giving 3/2.

Again parallel lines have the same slope.  

So we know the line we are looking for is in the form y=(3/2)x+b where we don't know the y-intercept (b) yet.

But we do know a point (x,y)=(2,9) that should be on our line.

So let's plug it in to find b.

y=(3/2)x+b  with (x,y)=(2,9)

9=(3/2)2+b

9=3       +b

Subtract 3 on both sides:

9-3=b

6=b

So the equation in slope intercept form is y=(3/2)x+6

Answer:

Step-by-step explanation:

Two consecutive odd intergers: x, x + 2.

The sum: 36

The equation:

x + (x + 2) = 36

x + x + 2 = 36

2x + 2 = 36             subtract 2 from both sides

2x = 34      divide both sides by 2

x = 17

x + 2 = 17 + 2 = 19

Answer:

C, E, F

Step-by-step explanation:

The range of the function [tex]g(x)=x^2[/tex] is [tex]y\in [0,\infty)[/tex], the range of the function [tex]h(x)=-x^2[/tex] is [tex](-\infty,0][/tex]

This means that for any value of x, the value of [tex]g(x)[/tex] is always greater or equal to the value of [tex]h(x)[/tex] (the values are equal at x=0).

So, options A and B are false, because at x=0 the values are equal and h(x) cannot be greater than g(x)

Options C, E and F are true, because for all non-zero x, g(x)>h(x).

Option D is false (the reason is the same as for option B)

Answer:

[tex]y+6=m(x+2)[/tex]

where I would have to look at the table to know [tex]m[/tex].

Step-by-step explanation:

Point-slope form of a line is

[tex]y-y_1=m(x-x_1)[/tex]

where [tex]m \text{ is the slope and } (x_1,y_1) \text{ is a point on that line}[/tex]

You are given [tex](x_1,y_1)=(-2,-6) \text{, but no value for }m[/tex].

So we know we are looking for an equation that looks like this:

[tex]y-(-6)=m(x-(-2))[/tex]

If you simplify this looks like:

[tex]y+6=m(x+2)[/tex]

Answer:

d

Step-by-step explanation:

The answer to the given expression when [3(1/6) - 1(5/8)] divided by [8{3/4} - 1.35) will be equal to 0.00896.

Convert mixed numbers to fractions:

[tex]3(\frac{1}{6}) = \frac{(3 \times 6 + 1) }{6} = \frac{19 }{ 6}\\\\8(\frac{3}{4}) = \frac{(8 \times 4 + 3) }{ 4} = \frac{35 }{ 4}[/tex]

Substitute the fractions into the expression:

[tex]\frac{[ (\frac{19}{6}) - 1(\frac{5}{8}) ] }{ [ (\frac{35}{4}) - 1.35 ]}[/tex]

Simplify the expression:

Numerator:

Common denominator for (19/6) and (5/8) is 24

19/6 - 1(5/8) = (19/6) - (15/8) = (19*4 - 6*15)/24 = 1/24

Denominator:

Convert 1.35 to fraction: 1.35 = 135/100 = 27/20

Common denominator for (35/4) and (27/20) is 20

(35/4) - 1.35 = (35/4) - (27/20) = (35 * 5 - 27 * 1) / 20 = 123/20

Divide the numerator and denominator by their greatest common divisor (GCD):

GCD(1, 24) = 1

GCD(123, 20) = 1

Simplify the expression:

[tex]\frac{(\frac{1 }{ 24}) }{ (\frac{123 }{ 20})}\\\\ = \frac{1 }{ \frac{24 \times 123 }{ 20}}\\\\ = \frac{1 }{ 111.6}[/tex]

1 / 111.6 ≈ 0.00896

The methods that we can use to calculate the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,15) are: Option B: Count by hand, Option D: Dividie 15 by 2

Midpoint of a line segment that lies in the mid of that line segment, as the name 'midpoint' suggests.

If the endpoints of the considered line segments are (a,b), and (c,d), then the coordinates of the midpoint would be:

[tex](x,y) = \left(\dfrac{c-a}{2}, \dfrac{d-b}{2}\right)[/tex]

We're specified here that:

Since the line is vertical, we can easily find its midpoint by going up by half of the length of the line segment.

The y-coordinate starts from 0 and goes to 15 and x-coordinate is still all along the line as the line is vertical, so the length's half is (15-0)/2 =15/2 = 7.5 units. This gives the y-coordinate of the midpoint as visible in the formula specified above.

If we go this units up, we will reach the midpoint. Since x-coordinates of the points in the line segment are fixed to 0, so the midpoint's coordinates are (0, 7.5)

We can also count by hand as there is motion only in y-coordinates, so move half of the total motion upwards from (0,0) or half of the total length downwards from (0,15).

So we see, that the second method and the fourth method listed in the option can be used.

Thus, the methods that we can use to calculate the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,15) are: Option B: Count by hand, Option D: Dividie 15 by 2

Learn more about midpoint of a line segment here:

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Final answer:

To find the y-coordinate of a vertical line segment midpoint between (0,0) and (0,15), you can count by hand to get an approximate location, add the y-coordinates of the endpoints and then divide by 2, or simply divide 15 by 2 to get the correct value of 7.5.

Explanation:

To calculate the y-coordinate of the midpoint of a vertical line segment with endpoints at (0,0) and (0,15), you can use the following methods:

Option d is the most direct mathematical approach to finding the midpoint's y-coordinate. Option b is a valid but less precise method that relies on visual estimation. Option c is essentially part of the formula used in option d.

Answer:

see explanation

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

Here (h, k) = (3, 1), hence

y = a(x - 3)² + 1

To find a substitute (- 2, - 4) into the equation

- 4 = a(- 2 - 3)² + 1

- 4 = 25a + 1 ( subtract 1 from both sides )

25a = - 5 ( divide both sides by 25 )

a = - [tex]\frac{5}{25}[/tex] = - [tex]\frac{1}{5}[/tex]

y = - [tex]\frac{1}{5}[/tex] (x - 3)² + 1 ← in vertex form

The equation that reveals the minimum value for the equation is 2(x + 3)² = 32

Which reveals the minimum value for the equation

From the question, we have the following parameters that can be used in our computation:

2x² + 12x − 14 = 0

Rewrite as

2x² + 12x = 14

So, we have

2(x² + 6x) = 14

Take the coefficient of x

k = 6

Divide by 2

k/2 = 3

Square both sides

(k/2)² = 9

So, we have

2(x² + 6x + 9) = 14 + 2 * 9

2(x² + 6x + 9) = 32

Express as squares

2(x + 3)² = 32

Hence, the equation that reveals the minimum value for the equation is 2(x + 3)² = 32

Question

Which of the following reveals the minimum value for the equation 2x^2 + 12x - 14 = 0?

2(x + 6)^2 = 26

2(x + 6)^2 = 20

2(x + 3)^2 = 32

Answer:

x=8 is a true solution of the radical equation

Step-by-step explanation:

we have

[tex]x-3=\sqrt{4x-7}[/tex]

Solve for x

squared both sides

[tex](x-3)^{2}=4x-7\\\\x^{2}-6x+9=4x-7\\\\ x^{2}-10x+16=0[/tex]

Convert to factored form

[tex]x^{2}-10x+16=(x-2)(x-8)[/tex]

The solutions are x=2 and x=8

Verify the solutions

For x=2

Substitute in the original equation

[tex]2-3=\sqrt{4(2)-7}[/tex]

[tex]-1=1[/tex] ----> is not true

therefore

x=2 is not a true solution of the radical equation

For x=8

Substitute in the original equation

[tex]8-3=\sqrt{4(8)-7}[/tex]

[tex]5=5[/tex] ----> is true

therefore

x=8 is a true solution of the radical equation

The true solution to the radical equation is x = 8, after checking the potential solutions by substituting them back into the original equation.

The student is asked to find the true solution(s) to the radical equation x – 3 = square root of 4x-7. The student has solved the equation, but it is essential to check the solutions by substituting them back into the original equation to ensure they are not extraneous. The student got x = 2 and x = 8 as potential solutions. We must substitute these values back into the original equation to determine their validity:

Hence, the only true solution to the radical equation x – 3 = square root of 4x-7 is x = 8.

Answer:

The measure of angle ABC is 34°

Step-by-step explanation:

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

so

∠ABC=(1/2)[major arc AC-minor arc AC]

∠ABC=(1/2)[major arc AC-146°]

Find the measure of major arc AC

major arc AC=360°-146°=214°

substitute

∠ABC=(1/2)[214°-146°]=34°

Answer:

4

Step-by-step explanation:

The GCF of 52 and 84 is 4.

52 = 2 × 2 × 13

84 = 2 × 2 × 3 × 7

Therefore, GCF = 2 × 2

GCF = 4

The GCF of 52 and 84 is found by finding the prime factors of each number and identifying the common factors. The GCF is 4.

The GCF (Greatest Common Factor) of two numbers is the largest number that can evenly divide both numbers. To find the GCF of 52 and 84, we need to find the prime factors of each number and then find the highest common factor of these sets of prime factors.

The common factors are 2 and 2, so the GCF of 52 and 84 is 2 x 2, which equals 4.

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so the price difference is 19500 - 16770 = 2730.

if we take 19500 to be the 100%, what is 2730 off of it in percentage?

[tex]\bf \begin{array}{ccll} amount&\%\\ \cline{1-2} 19500&100\\ 2730&x \end{array}\implies \cfrac{19500}{2730}=\cfrac{100}{x}\implies \cfrac{50}{7}=\cfrac{100}{x}\implies 50x=700 \\\\\\ x=\cfrac{700}{50}\implies x=14[/tex]

Answer:

The answer is 14%

Step-by-step explanation:

1) Divide 19500 by 16770

16770/19500= 0.86

2) Multiply by 100 (this is the percentage between the original and find prices of the car)

0.86(100)= 86%

3) Subtract 86% from 100% to find the change in percentage

100-86= 14%

Therefore, the percentage decrease of the price of the car is 14%.

Hope this helps!

Answer:

C

Step-by-step explanation:

[tex]y=x^2-6x+6\\y=x^2-6x+9-3\\y=(x-3)^2-3[/tex]

The vertex form of the quadratic equation is y = (x-3)^2 + 3.

The vertex form of a quadratic function is given by y = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex. To convert the quadratic equation y = x^2-6x+6 into vertex form, we need to complete the square.

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Answer:

(1, 1 )

Step-by-step explanation:

Given the 2 equations

2x + 3y = 5 → (1)

3x + 2y = 5 → (2)

We can eliminate the term in x by multiplying (1) by 3 and (2) by - 2

6x + 9y = 15 → (3)

- 6x - 4y = - 10 → (4)

Add (3) and (4) term by term

(6x - 6x) + (9y - 4y) = (15 - 10), that is

5y = 5 ( divide both sides by 5 )

y = 1

Substitute y = 1 in either (1) or (2) and solve for x

Substituting in (1), then

2x + (3 × 1) = 5

2x + 3 = 5 ( subtract 3 from both sides )

2x = 2 ( divide both sides by 2 )

x = 1

Solution is (1, 1 )

Let c be children and a adults.

3.5c + 7a = 1771 (the total revenue is equal to the amounts made off of people)

c + a = 331 (total number of people)

The second formula becomes a = 331 - c. This can be substituted into the first formula.

3.5c + 7(331 - c) = 1771 = 7*331 - 3.5c = 1771. 7*331 = 2317, so 3.5c = 2317 - 1771 = 546.

546/3.5 = 156 = c (number of children).

c + a = 156 + a = 331 => a = 331 - 156 = 175 (number of adults).

There are 156 children and 175 adults

Answer:

156 children

175 adults

Step-by-step explanation:

Let's call x the number of children admitted and call z the number of adults admitted.

Then we know that:

[tex]x + z = 331[/tex]

We also know that:

[tex]3.50x + 7z = 1,771.00[/tex]

We want to find the value of x and z. Then we solve the system of equations:

-Multiplay the first equation by -7 and add it to the second equation:

[tex]-7x - 7z = -2,317[/tex]

[tex]3.50x + 7z = 1,771[/tex]

----------------------------------

[tex]-3.5x = -546[/tex]

[tex]x =\frac{-546}{-3.5}\\\\x=156[/tex]

Now we substitute the value of x in the first equation and solve for the variable z

[tex]156 + z = 331[/tex]

[tex]z = 331-156[/tex]

[tex]z = 175[/tex]

Step-by-step explanation:

:

  2-csc²A

▬▬▬▬▬▬▬

csc²A + 2cotgA

  2 - 1/sin²A

= ▬▬▬▬▬▬▬▬▬▬

 1/sin²A + 2cosA/sinA

  2sin²A - 1

= ▬▬▬▬▬▬▬

  1 - 2cosAsinA

    sin²A + sin²A - 1

= ▬▬▬▬▬▬▬▬▬▬▬▬

 sin²A - 2cosAsinA + cos²A

  sin²A - cos²A

= ▬▬▬▬▬▬▬

  (sinA - cosA)²

 (sinA - cosA)(cosA + sinA)

= ▬▬▬▬▬▬▬▬▬▬▬▬

   (sinA - cosA)²

 sinA + cosA

= ▬▬▬▬▬▬ <-- Let check "+" and "-"

 sinA - cosA

By taking a common Equation denominator and simplifying, we will see that LHS = RHS. To prove that the given equation is true, we can simplify both sides step by step and show that they are equal.

To show that the left-hand side (LHS) is equal to the right-hand side (RHS) of the given equation, let's simplify both sides step by step.

LHS = 2 - cosec²A / cosec²A + 2cotA

= 2 - (1/sin²A) / (1/sin²A) + 2cosA/sinA

= 2 - 1/sin²A / 1/sin²A + 2cosA/sinA

RHS = sinA - cosA / sinA + cosA

By taking a common denominator and simplifying, we will see that LHS = RHS.

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The probable questionShow that the LHS = RHS.

2-cosec^2A/cosec^2A+2cotA=sinA-cosA/sinA+cosA may be: