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(1) 8x -y=20
(2) -5x +y =-8
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(1)+(2) 3x +12
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x=4,,,,,,,subst in (1)
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8(4) -y =20
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-y = 20-32= -12
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y =12
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checking,,,(4, 12),,,,(1) 8(4) -(12) =20,,,,,,ok
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(2) -5(4) +(12) =-8,,,,-20 +12 =-8,,,,ok
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Let m = amt milk,,,,and c = amt of cream,,,,both in gallons
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.02m +.09c = .04 (35)
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but m+c =35,,,,,or m=35-c,,,,subst
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.02(35-c) + .09 c = .04(35)
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.7 -.02c + .09c = 1.4
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.07c = .7
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c= 10 gal,,,,,,,,,m=35-10=25 gal
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checking,,,,,.02(25) + .09(10) = .04(35)
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.5 + .9 = 1.4,,,,,ok
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starting with the standard slope intercept form,,,y=mx+b
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if m= (6/7) ,,,and (8, -7),,,,,,,,,substituting
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(-7) = (6/7)(8) +b
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-7 -48/7= b
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-7 -6 6/7 =b
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b= -13 6/7
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reviewing the std eqn,,,y = 6/7 *x - 13 6/7,,,answer in slope intercept form
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multiplying by 7
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7y =6x - 97
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or,,,,-6x +7y = -97,,,,mult by -1
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6x -7y =97,,,,,answer in std form..
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checking,,( 8,-7),,,,,6(8) -7(-7) = 97,,,,,ok
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If the axis of a graph can be set up (labeled) as fractions, just use axis,,
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for example, if x axis is labeled every 1/16,,,and you want to locate x=1/2, just use 8/16 as location.
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If the graph is labeled as whole numbers, just convert fraction to decimal, by dividing numerator by denominator, and then estimate location between whole numbers.
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5/3 = 1.667,,,,,on number line it is between +1 and +2,,,,or more specifically,,it is 2/3 of way between +1 and +2 on number line
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odd(5)-even(2) yields odd(3)
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odd(3) divided by odd (7) yields odd (3/7)
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70 m * (100cm)/(1m) * (1 in)/(2.54 cm) = 2755.9 in
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Rough check, 1 m = 39.36 in approx , 70*39.36 = 2755.2 in ok
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Note: dimensions only,,,(m) * (cm)/(m) * (in)/ (cm) shows cancellation of dim's
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hence,,,the name,,,,,,dimensionless analysis,,,,
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if we review,,,(a+b)^2 = (a+b) * (a+b) = a^2 +2ab +b^2
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looking at coefficients, 2given / 2(half) =1,,,1^2 = 1(3rd coefficient)
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therefore if we have 2nd coefficient, take half and square it to get 3rd coefficient.
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x^2 -3x + 2.25,,,,,,,-3/2=-1.5,,,,,-1.5^2 = 2.25
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check,,,,(x-1.5)^2 = (x-1.5)(x-1.5) = x^2 -1.5x -1.5x +2.25 = x^2 -3x +2.25
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note,,,1.5 = 3/2
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as 2qt = 1 qt is UNTRUE,,,there is NO SOLUTION to this problem
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50 cm^2 * (1 m/100cm)^2= 50/100^2 m^2 = .005 m^2
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4i(3i-2),,,,distribute
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4*3*i*i - 4*i*2
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12i^2 - 8i,,,,but i^2 =-1,,,,subst
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-12 -8i,,,,answer in std form(a+bi)
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Length = K / Width
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example is ,,,12 = K/18,,,,,K=216,,,
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problem,,,,16=216/w,,,,,,w=13.5 ft
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check: K=16*13.5 = 216,,,,ok
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Let x= amt added in liters
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.1(20) + .05x = .08(20+x),,,drug amt,,,,,note total volume is x+20
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.(2) +.05x = 1.6 + .08x
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.(.4) = .03 x
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13.33333 = x in liters
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check,,,.1*20 +.05*13.33333= .08*33.33333
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2+.667 = 2.6667,,,ok
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Let x = greater number,,,y= other number
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x+y= 55,,,,,or y=55-x
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x=(17*y) +1,,,,,subst
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x=17*(55-x) +1
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x=935 -17x +1
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18x = 936
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x=52,,,,,,,y=55-52 = 3
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check,,,,52+3=55,,,,ok
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52 = 17*3 +1 = 51+1 =52,,,,ok
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Let x = number
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(2/3)*x = -6,,,,,,multiply by 3/2
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x=-6*3/2 =-9
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check ,,,,(2/3) * -9 = -6,,,,,,ok
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Velocity = Distance / time = 400 miles / 6 hrs = 66.67 miles per hr
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ln 36.9 = 3.608211 = 3.6082
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technique,, 36.9, enter, ln, enter.
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the standard form for a circle is,,,(x-h)^2 +(y-k)^2 = r^2
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with (h,k) as center of circle,,and r as radius
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with center at (2, -4),,,,,h= 2,,,k= -4,,,and radius is the x distance from center to x=-2,,,|(-2-2)| = |(-4)| = 4
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substituting, (x-2)^2 + (y-(-4) )^2 =4^2 =16
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(x-2)^2 +(y+4)^2 = 16,,,,answer
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a rough sketch shows how this is possible on the x-y coordinate system.
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Let p = cost of one pillow
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Total cost = 4*p + (4p*.05) = $ 46.20
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4p + .2p = 46.20
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4.2 p =46.20
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p= $11,,,,,answer
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alternate solution,,,1.05 *4p = 46.20
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p= $ 11
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note increases in cost, like tip or taxes, use 1+ increase
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for decreases in cost, like discounts, use 1- decrease
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check...46.20/44 = 1.05,,,or 5% increase,,,ok
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the distance formula is ,,,,d= sqrt{y2-y1)^2 + (x2-x1)^2 }
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for (6,-30,,,,(-1,4)
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d= sqrt( -30 -4)^2 + (6-(-1) )^2 }
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d= sqrt { (-34)^2 + (7)^2 }
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d=sqrt { 1156 +49 }
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d=sqrt { 1205)
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d=34.713
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a rough sketch always helps clarify.
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the change in y = 34,,,and the change in x= 7,,,c^2 = a^2+b^2,, d^2=34^2+7^2,,,d=34.713,,,ok
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Area = pi*r^2 =pi*8^2 =64*pi
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Area of Sector = Area * radians of sector/ 2*pi radians
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a= ( 64 * pi) * (5/6 pi) / 2 (pi) = 26.67*pi = 83.7758= 83.8 cm ^2
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check ,,,5/6 pi * 180/pi = 150 degrees, a= 64 pi * 150/360 = 83.7758 ,,,,ok
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% increase is the amt of increase divided by original number
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(1425 - 281)/281 = 1144/281 =4.071 = 407.1 %
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185/30=4.625/x,,,,,,where x = # inches for 30 miles
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cross multiply,,,,185 x = 30*4.625
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x=30*4.625/185 = .75 = 3/4 inch
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check,,,,,185/4.625 =30/.75 = 40,,,,,ok
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Let x= distance for day 1,,,subsequent days are,,x+3,,x+6,,x+9,,,x+12
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as total for 5 days is 30,,
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x +x+3 +x+6 +x+9 =x+12 =30
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5x +30 =30
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5x =0
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x= 0,,,,,,,or he doesn't have to walk at all 1st day,,
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check,,0+3+6+9+12 = 30,,,,ok
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y=K*(x*z)/w,,,,,,,subst in values known
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3/2 = K*{ 2*3/4} = K* 6/4 =K*3/2
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K=1
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expression,,,y= x*z/w
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75(x/60) =60( (x+10)/60 ) = x+10
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mult thru by 60 to clear fractions
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75x = 60x +600
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15x = 600
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x=40
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check,,,75{ (40)/60} = 60{ [(40)+10]/60 }
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50 = 60 {50/60} = 50,,,,,ok
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Let a = amt of almonds,,,,,and c= amt of chocolate,,,,,in #'s
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a+c=30,,,a=30-c
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.75*c + .50*a =.65 * 30,,,,,subst
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.75c +.5(30-c) = .65*30
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.75c +15 -.5c = 19.5
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.25c =4.5
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c=18 #,,,,,a=30-18=12 #
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check,,,,.75(18) +.5(12) =.65*30
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13.5 + 6 = 19.5,,,,ok

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Let a,,b,,c,,,be the pieces,,,a,, largest,,,c,,,smallest,,,,in inches
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a+b+c=4ft = 48 inches
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a=2c+12,,,,,b= a-6,,,,,,,substituting
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(2c+12) +(a-6) +c =48,,,,,,but a=(2c+12)
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(2c+12+ +{(2c+12)-6} +c = 48
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2c+12 +2c+12 -6 +c = 48
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5c = 30
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c=6,,,,a=2(6)+12= 24,,,,b=(26-6)= 18
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check,,,,,a+b+c=48,,,,,24+18+6=48,,,,ok
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In fractions the numerator is the number of pieces,,,the denominator is the size of the pieces.
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To compare, usually the denominator is the same
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3/8 vs 7/16,,,,,lets use LCD = 16
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3/8 = 6/16....16/8=2,,,2*3=6
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now problem is 3/8 =6/16 vs 7/16,,,,,with 7/16 Larger than 6/16 or 3/8
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7/16 > 3/8

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Let x = amount added in gallons
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.2*1 + .68*x = .24(1+x)
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.2 +.68x =.24 +.24x
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.44x = .04
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x=.091 gallons
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check,,,.2 +.68*.091 =1.091*.24
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.2618 = .2618,,,,ok
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Let x = angle,,,,,Let y = supplement = 180-x
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x/3 = 4+y = 4+(180-x) = 184 -x
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1.33333x=184
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x=138,,,,,,,y=42
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check,,,138/3 = 46=4+42=46,,,,,ok
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