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${:[x^(2)+y^(2)=10],[x-2y=5]:} If (x,y) is a solution to the system of equations shown, which of the following are y-coordinates of the solutions? Choose 1 answer: (A) 0 and sqrt10 (B) 0 and -sqrt10 (c) -3 and -1 (D) 3 and 1$

$\begin{aligned} x^{2}+y^{2} & =10 \\ x-2 y & =5 \end{aligned}$ $\newline$ If $(x, y)$ is a solution to the system of equations shown, which of the following are $y$ -coordinates of the solutions? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ and $\sqrt{10}$ $\newline$ (B) $0$ and $-\sqrt{10}$ $\newline$ (C) $-3$ and $-1$ $\newline$ (D) $3$ and $1$

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Full solution

\begin{aligned} x^{2}+y^{2} & =10 \\ x-2 y & =5 \end{aligned}

\newline

(x, y)

is a solution to the system of equations shown, which of the following are

y

-coordinates of the solutions?

\newline

Choose

1

answer:

\newline

(A)

0

and

\sqrt{10}

\newline

(B)

0

and

-\sqrt{10}

\newline

(C)

-3

and

-1

\newline

(D)

3

and

1

Equations: We have the system of equations: $\newline$ $1$ . $x^2 + y^2 = 10$ $\newline$ $2$ . $x - 2y = 5$ $\newline$ To find the y-coordinates of the solutions, we can solve the second equation for x: $\newline$ $x = 2y + 5$
Substituting x: Now we substitute $x = 2y + 5$ into the first equation: $\newline$ $(2y + 5)^2 + y^2 = 10$ $\newline$ Expand the squared term: $\newline$ $4y^2 + 20y + 25 + y^2 = 10$ $\newline$ Combine like terms: $\newline$ $5y^2 + 20y + 25 = 10$
Expanding the equation: Subtract $10$ from both sides to set the equation to zero: $\newline$ $5y^2 + 20y + 15 = 0$ $\newline$ Now we can divide the entire equation by $5$ to simplify: $\newline$ $y^2 + 4y + 3 = 0$
Setting the equation to zero: Factor the quadratic equation: $\newline$ $(y + 1)(y + 3) = 0$ $\newline$ Set each factor equal to zero and solve for y: $\newline$ $y + 1 = 0$ or $y + 3 = 0$ $\newline$ So, $y = -1$ or $y = -3$