Solve the system x^2+y^2=25 and x+2y=5, and another question...?

Question

Please provide solutions for the above.

Another question:

A flagpole stands on top of a building. The angle of elevation of the foot of the pole 30m from the foot of the building is 60 degrees and 80m farther, the top of the pole has an elevation of 30 degrees. How high is the pole?

Please provide solutions. Someone answered (on the 2nd question), "Let H be the height of the building and h the height of the pole. The equations are

H= 30 tan 60 degrees
(H+h)=110 tan 30 degrees

I don't know anything about those. Please help and thank you.

Answer 1

Question 1:

Intuition.

Note that x^2 + y^2 = 25 is a circle of radius 5 centered at (0,0)
Note also that x + 2y = 5 is a straight line with slope -1/2 and y-intercept at (0, 5/2)
Guess: the line will intersect the circle twice (not always the case though, but a rough sketch of the above tells us this might be true) so we will (hopefully) get two solutions

Solution.

You can use a simple substitution to solve this problem.
So the second equation gives x = 5 - 2y.
Substituting for x into the first equation we have:
(5-2y)^2 + y^2 = 25
25 - 10y - 10y + 4y^2 + y^2 = 25
5y^2 - 20y = 0
5y(y-4)=0
y=0 or 4

Substitute these two values of y into x = 5 - 2y to get your x-values:
x = 5-2(0) = 5 or x = 5-2(4) = -3

So the solutions to this system are: (5, 0) and (-3, 4)


Question 2:

Note that you can form two 30:60:90 triangles, whose sides have a x:x*Sqrt(3):x*2 relationship.

Triangle whose angle of elevation 30m from foot of the building to the foot of the pole gives us that x = 30 m (since this is the side opposite the 30 degree angle - draw it to believe it). Thus, the three sides have lengths 30, 30*Sqrt(3), and 60 with 60 being the length of the hypotenuse.

Triangle whose angle of elevation 110m from foot of the building (80m + 30m) to the top of the pole gives us that x*Sqrt(3) = 110 (since this is the side opposite the 60 degree angle - draw it to believe it). This means that x = 110/Sqrt(3). Thus, the three sides have lengths 110/Sqrt(3), 110, and 220/Sqrt(3) with 220/Sqrt(3) being the length of the hypotenuse.

Okay, we only really care about the height of the pole.
Notice that height of pole = height of (pole+building) - height of building.
From the first triangle, height of building = 30*Sqrt(3)
From the second triangle, height of (pole+building) = 110/Sqrt(3)
So height of pole = 110/Sqrt(3) - 30*Sqrt(3) = 20/Sqrt(3)

Answer 2

For the system x^2 + y^2 = 25, x + 2y = 5:

All you have to do is use substition; solve the linear equation
x + 2y = 5 in terms of x, and plug into the other equation.

x = 5 - 2y; therefore

(5 - 2y)^2 + y^2 = 25
(25 - 20y + 4y^2) + y^2 = 25
5y^2 - 20y + 25 = 25
5y^2 - 20y = 0
5y(y - 4) = 0, therefore, y = {0, 4}

When y = 0, x = 5 - 2y = 5 - 2(0) = 5
When y = 4, x = 5 - 2y = 5 - 2(4) = -3

Therefore, (x = -3, y = 4) and (x = 5, y = 0) are solutions.

Answer 3

x² + y² = 25- - - - - -Equation 1
x + 2y = 5- - - - - - -Equation 2
- - - - - - -

Substitute Method equation 2

x + 2y= 5

x + 2y - 2y = 5 - 2y

x = 5 - 2y

The answer is x = 5 - 2y

insert the x value into equation 1

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

x² + y² = 25

(5 - 2y)² + y² = 25

25 - 20y + 4y² + y² = 25

25 - 20y + 5y² - 25 = 25 - 25

- 20y + 5y² = 0

(-4 + y)5y = 0

- - - - - - - - - - -

Roots

5y = 0

5y / 5 = 0 / 5

y = 0

- - - - - - -

-4 + y = 0

- 4 + y + 4 = 0 + 4

y = 4

The answer is y = 4

Insert the y value into equation 1

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

x² + y² = 25

x ²+ (4)² = 25

x² + 16 = 25

x² + 16 - 16= 25 - 16

x² = 9

x² = ± √9

x = ± 3

The answer is x = ± 3

Insert the x value into equation 1

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

Check for equation 1

x² + y² = 25

(3)² + (4)² = 25

9 + 16 = 25

25 = 25

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

Check for equation 2

x + 2y = 5

- 3 + 2(4) = 5

- 3 + 8 = 5

5 = 5

- - - - - - -

When x = 0, y = 5

When x = 5, y = 0

When x = - 3, y = 4

- - - - - - - - - -s-

Answer 4

the biggest element this equation is attempting to augment, i think of, is how order of operations works so we will take it in steps. a million. exchange 8 for X on your equation: y = (8*2 / 4) - 2. Now resolve for y. 2. remember which you do the stuff interior parenthesis first. With the stuff interior the parenthesis carry out multiplication and branch first, going from left to genuine. So do 8 * 2 first this is sixteen. Then divide sixteen through four this is 4. This leaves you with y = 4 - 2 and ... 3. y = 2. you could verify your answer through substituting 2 for y on your unique equation: 2 = (8*2/4)-2. Do the math and you will locate that 2 = 2 this is genuine and subsequently proves your answer is genuine. good success, Amanda

Answer 5

i dont know how to solve first equation!!!!

H=30*root 3.................1
H+h=110*1/root3........2
from 2
H=110/root3 - h................3
equating 1 & 3
30*root3=110/root3 - h
h=110/root3-30*root 3
L.C.M
h=(110-30*3)/root 3
h=20/root3
h=11.547
Do tell me that what is the answer?