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Zorluk: ZorSimultaneous Linear and Quadratic Equations

Given the simultaneous equations x+2y=7x + 2y = 7 and x2+3xy+y2=19x^2 + 3xy + y^2 = 19, find the product of all possible values of yy that satisfy the system.

Cevap: -30

Cevap

The product of all possible values of y that satisfy the system is -30.
Rearranging the linear equation gives x = 7 - 2y. Substituting this into the quadratic equation x^2 + 3xy + y^2 = 19 produces (7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19. Expanding and combining like terms yields y^2 + 7y - 30 = 0. Solving for y gives y = 3 and y = -10. Multiplying these values together gives a product of -30.

Adım Adım Çözüm

1
Isolate x in the linear equation
x = 7 - 2y
Expressing one variable in terms of the other enables substitution into the quadratic equation.
2
Substitute x into the quadratic equation and expand
(7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19
This reduces the system to a single quadratic equation in terms of y.
3
Simplify the resulting expression into standard quadratic form
y^2 + 7y - 30 = 0
Expanding yields (49 - 28y + 4y^2) + (21y - 6y^2) + y^2 = 19, which reduces to y^2 + 7y - 30 = 0.
4
Calculate the product of the roots of y
y_1 * y_2 = -30
By Vieta's formulas, the product of roots for y^2 + ay + b = 0 is b/1 = -30 (or factoring gives y = 3 and y = -10, with product 3 * (-10) = -30).

Anahtar Kavram

Solving simultaneous linear and quadratic equations via substitution and applying quadratic root properties
Tahmini Süre:2m 0s
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Konular

Simultaneous Linear and Quadratic Equations

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