solve the system of equation
2x+y=5
y^2-2y=-3x+5
2 x + y = 5 Subtract 2 x to both sides
2 x + y - 2 x = 5 - 2 x
y = 5 - 2 x
y = - 2 x + 5
y ^ 2 - 2 y = - 2 x + 5
( - 2 x + 5 ) ^ 2 - 2 * ( - 2 x + 5 ) = - 3 x + 5
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Remark :
( a - b ) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2
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( - 2 x ) ^ 2 - 2 * 2 x * 5 + 5 ^ 2 - 2 * ( - 2 x ) - 2 * 5 = - 3 x + 5
4 x ^ 2 - 20 x + 25 + 4 x - 10 = - 3 x + 5
4 x ^ 2 - 16 x + 15 = - 3 x + 5 Add 3 x to both sides
4 x ^ 2 - 16 x + 15 + 3 x = - 3 x + 5 + 3 x
4 x ^ 2 - 13 x + 15 = 5 Subtract 5 to both sides
4 x ^ 2 - 13 x + 15 - 5 = 5 - 5
4 x ^ 2 - 13 x + 10 = 0
The solutions of are :
x = 5 / 4
and
x = 2
So :
x = 5 / 4
y = - 2 x + 5
y = - 2 * 5 / 4 + 5 = - 10 / 47 + 5 =
- 2 * 5 / ( 2 * 2 ) + 5 = - 5 / 2 + 5 =
- 5 / 2 + 10 / 2 = 5 / 2
and
x = 2
y = - 2 x + 5
y = - 2 * 2 + 5 = - 4 + 5 = 1
Two pairs of the solutions :
x = 5 / 4 , y = 5 / 2
and
x = 2 , y = 1
P.S.
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4 x ^ 2 - 13 x + 10 = 0
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4 x ^ 2 - 13 x + 10 = 0
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