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If
(x,y) is a solution to the system of equations shown, what is the product of the
y-coordinates of the solutions?

x^(2)+4y^(2)=40

x+2y=8

If $(x, y)$ is a solution to the system of equations shown, what is the product of the $y$ -coordinates of the solutions? $\newline$ $x^{2}+4 y^{2}=40$ $\newline$ $x+2 y=8$

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Evaluate recursive formulas for sequences

Full solution

Q. If

(x, y)

is a solution to the system of equations shown, what is the product of the

y

-coordinates of the solutions?

\newline

x^{2}+4 y^{2}=40

\newline

x+2 y=8

Solve linear equation for x: Solve the linear equation for $x$ . The linear equation is $x + 2y = 8$ . We can solve for $x$ by isolating it on one side of the equation. $x = 8 - 2y$
Substitute expression for $x$ into quadratic equation: Substitute the expression for $x$ into the quadratic equation. $\newline$ The quadratic equation is $x^2 + 4y^2 = 40$ . We substitute $x = 8 - 2y$ into this equation to get: $\newline$ $(8 - 2y)^2 + 4y^2 = 40$
Expand and simplify the equation: Expand the squared term and simplify the equation. $\newline$ Expanding $(8 - 2y)^2$ gives us $64 - 32y + 4y^2$ . Now we have: $\newline$ $64 - 32y + 4y^2 + 4y^2 = 40$ $\newline$ Combine like terms to get: $\newline$ $64 - 32y + 8y^2 = 40$
Move all terms to one side: Move all terms to one side to set the equation to zero. $\newline$ Subtract $40$ from both sides to get: $\newline$ $8y^2 - 32y + 24 = 0$
Simplify the equation by dividing all terms: Simplify the equation by dividing all terms by $8$ . $\newline$ Dividing each term by $8$ gives us: $\newline$ $y^2 - 4y + 3 = 0$
Factor the quadratic equation: Factor the quadratic equation. $\newline$ The equation $y^2 - 4y + 3$ factors into: $\newline$ $(y - 3)(y - 1) = 0$
Solve for $y$ : Solve for $y$ by setting each factor equal to zero. $\newline$ Setting each factor equal to zero gives us two possible $y$ -values: $\newline$ $y - 3 = 0$ or $y - 1 = 0$ $\newline$ So, $y = 3$ or $y = 1$
Find the product of the y-coordinates: Find the product of the y-coordinates. $\newline$ The product of the y-coordinates of the solutions is: $\newline$ $3 \times 1 = 3$

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