Solve 2x - 3y = -2; x y = 14

Question

Solve the system of equations

(x_{1}, y_{1}) = (- \frac{1 + 85}{2}, \frac{1 - 85}{3}) (x_{2}, y_{2}) = (\frac{- 1 + 85}{2}, \frac{1 + 85}{3})

Evaluate

{2 x - 3 y = - 2 x y = 14

Solve the equation for x

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{x = \frac{- 2 + 3 y}{2} x y = 14

Substitute the given value of x into the equation x y = 14

\frac{- 2 + 3 y}{2} \times y = 14

Simplify

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\frac{y ( - 2 + 3 y )}{2} = 14

Cross multiply

y (- 2 + 3 y) = 2 \times 14

Simplify the equation

y (- 2 + 3 y) = 28

Expand the expression

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- 2 y + 3 y^{2} = 28

Move the expression to the left side

- 2 y + 3 y^{2} - 28 = 0

Rewrite in standard form

3 y^{2} - 2 y - 28 = 0

Substitute a= 3,b= - 2 and c= - 28 into the quadratic formula y = \frac{- b \pm b ^{2} - 4 a c}{2 a}

y = \frac{2 \pm ( - 2 ) ^{2} - 4 \times 3 ( - 28 )}{2 \times 3}

Simplify the expression

y = \frac{2 \pm ( - 2 ) ^{2} - 4 \times 3 ( - 28 )}{6}

Simplify the expression

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y = \frac{2 \pm 340}{6}

Simplify the radical expression

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y = \frac{2 \pm 2 85}{6}

Separate the equation into 2 possible cases

y = \frac{2 + 2 85}{6} y = \frac{2 - 2 85}{6}

Simplify the expression

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y = \frac{1 + 85}{3} y = \frac{2 - 2 85}{6}

Simplify the expression

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y = \frac{1 + 85}{3} y = \frac{1 - 85}{3}

Evaluate the logic

y = \frac{1 + 85}{3} \cup y = \frac{1 - 85}{3}

Rearrange the terms

{x = \frac{- 2 + 3 y}{2} y = \frac{1 + 85}{3} \cup {x = \frac{- 2 + 3 y}{2} y = \frac{1 - 85}{3}

Calculate

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{x = \frac{- 1 + 85}{2} y = \frac{1 + 85}{3} \cup {x = \frac{- 2 + 3 y}{2} y = \frac{1 - 85}{3}

Calculate

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{x = \frac{- 1 + 85}{2} y = \frac{1 + 85}{3} \cup {x = - \frac{1 + 85}{2} y = \frac{1 - 85}{3}

Calculate

{x = - \frac{1 + 85}{2} y = \frac{1 - 85}{3} \cup {x = \frac{- 1 + 85}{2} y = \frac{1 + 85}{3}

Check the solution

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{x = - \frac{1 + 85}{2} y = \frac{1 - 85}{3} \cup {x = \frac{- 1 + 85}{2} y = \frac{1 + 85}{3}

Check the solution

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{x = - \frac{1 + 85}{2} y = \frac{1 - 85}{3} \cup {x = \frac{- 1 + 85}{2} y = \frac{1 + 85}{3}

Solution

(x_{1}, y_{1}) = (- \frac{1 + 85}{2}, \frac{1 - 85}{3}) (x_{2}, y_{2}) = (\frac{- 1 + 85}{2}, \frac{1 + 85}{3})

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