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$Emily is solving the following equation for z. z-8=sqrt(3z-5) Her first few steps are given below. {:[(z-8)^(2)=(sqrt(3z-5))^(2)],[z^(2)-16 z+64=3z-5]:} Is it necessary for Emily to check her answers for extraneous solutions? Choose 1 answer: (A) Yes (B) No$

Emily is solving the following equation for $z$ . $\newline$ $z-8=\sqrt{3 z-5}$ $\newline$ Her first few steps are given below. $\newline$ $\begin{aligned} (z-8)^{2} & =(\sqrt{3 z-5})^{2} \\ z^{2}-16 z+64 & =3 z-5 \end{aligned}$ $\newline$ Is it necessary for Emily to check her answers for extraneous solutions? $\newline$ Choose $1$ answer: $\newline$ (A) Yes $\newline$ (B) No

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Math Problems

Algebra 1

Solve a system of equations using any method: word problems

Full solution

Q. Emily is solving the following equation for

z

\newline

z-8=\sqrt{3 z-5}

\newline

Her first few steps are given below.

\newline

\begin{aligned} (z-8)^{2} & =(\sqrt{3 z-5})^{2} \\ z^{2}-16 z+64 & =3 z-5 \end{aligned}

\newline

Is it necessary for Emily to check her answers for extraneous solutions?

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Equation setup: Emily starts with the equation $z - 8 = \sqrt{3z - 5}$ . To solve for $z$ , she squares both sides of the equation to eliminate the square root. $\newline$ $(z - 8)^2 = (\sqrt{3z - 5})^2$
Squaring both sides: By squaring both sides, Emily gets: $\newline$ $z^2 - 16z + 64 = 3z - 5$ $\newline$ This is a quadratic equation.
Quadratic equation: Emily needs to bring all terms to one side to set the equation to zero and solve the quadratic equation. $\newline$ $z^2 - 16z + 64 - 3z + 5 = 0$
Combining like terms: Combining like terms, Emily gets: $z^2 - 19z + 69 = 0$ This is the simplified form of the quadratic equation.
Solving the quadratic equation: Emily will now solve the quadratic equation for $z$ using methods such as factoring, completing the square, or the quadratic formula. $\newline$ However, before proceeding, we need to address the original question: Is it necessary for Emily to check her answers for extraneous solutions?
Checking for extraneous solutions: When an equation involves a square root and both sides are squared to eliminate the square root, there is a possibility of introducing extraneous solutions. These are solutions that do not satisfy the original equation. $\newline$ Therefore, it is necessary for Emily to check her answers against the original equation to ensure they are valid.