Solve the system y=2x-6; y=-1/2x+4

x = 108/5 , y = 186/5

Step-by-step explanation:

Solve the following system:

{y = 2 x - 6 | (equation 1)

y = 48 - x/2 | (equation 2)

Express the system in standard form:

{-(2 x) + y = -6 | (equation 1)

x/2 + y = 48 | (equation 2)

Add 1/4 × (equation 1) to equation 2:

{-(2 x) + y = -6 | (equation 1)

0 x+(5 y)/4 = 93/2 | (equation 2)

Multiply equation 2 by 4:

{-(2 x) + y = -6 | (equation 1)

0 x+5 y = 186 | (equation 2)

Divide equation 2 by 5:

{-(2 x) + y = -6 | (equation 1)

0 x+y = 186/5 | (equation 2)

Subtract equation 2 from equation 1:

{-(2 x)+0 y = (-216)/5 | (equation 1)

0 x+y = 186/5 | (equation 2)

Divide equation 1 by -2:

{x+0 y = 108/5 | (equation 1)

0 x+y = 186/5 | (equation 2)

Collect results:

 {x = 108/5 , y = 186/5

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)