Solve the simultaneous equation 4x^2 - 9y^2 = 19 2x + 3y= 1​

The line 2x +3y = 1 is tangent to 4x^2 -9y^2 = 19

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Solve the linear equation for y

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1) y = -2x/3 +1/3

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substitute for y in the equation of the hyperbole

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4x^2 -9(-2x/3 +1/3)^2 = 19

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4x^2 -9(4x^2/9 -4x/9 +1/9) = 19

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4x^2 -4x^2 +4x -1 = 19

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4x -20 = 0

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4x = 20

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x = 5

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use equation 1 to get the value of y

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y = -2(5)/3 +1/3 = -3

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solution is x = 5, y = -3 which is the point (5, -3)

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check the solution

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hyperbole

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4(5^2) -9(-3)^2 = 19

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100 -81 = 19

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19 = 19

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Line

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2(5) +3(-3) = 1

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10 -9 = 1

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1 = 1

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