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Which value for the constant
c makes
z=4 an extraneous solution in the following equation?

{:[sqrt((3)/(2)z+10)=cz+8],[c=]:}

Which value for the constant $c$ makes $z=4$ an extraneous solution in the following equation? $\newline$ $\begin{array}{l} \sqrt{\frac{3}{2} z+10}=c z+8 \\ c=\square \end{array}$

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Identify equivalent linear expressions

Full solution

Q. Which value for the constant

c

makes

z=4

an extraneous solution in the following equation?

\newline

\begin{array}{l} \sqrt{\frac{3}{2} z+10}=c z+8 \\ c=\square \end{array}

Substituting $z=4$ : Let's first substitute $z=4$ into the left side of the equation to see what value we get. $\newline$ $\sqrt{\left(\frac{3}{2}\right)z + 10} = \sqrt{\left(\frac{3}{2}\right)\cdot 4 + 10}$
Calculating the value inside the square root: Now, let's calculate the value inside the square root. $\newline$ $(\frac{3}{2})\cdot 4 + 10 = 6 + 10 = 16$
Taking the square root of $16$ : Next, we take the square root of $16$ . $\newline$ $\sqrt{16} = 4$
Substituting $z=4$ into the right side: Now, let's substitute $z=4$ into the right side of the equation. $cz + 8 = c\cdot4 + 8$
Setting up the inequality: Since $z=4$ is an extraneous solution, the right side of the equation should not equal $4$ when $z=4$ . Therefore, we set up the equation $c\cdot 4 + 8 \neq 4$ .
Isolating $c$ in the inequality: We solve for $c$ by isolating it on one side of the inequality. $\newline$ $c \times 4 \neq 4 - 8$ $\newline$ $c \times 4 \neq -4$
Solving for c: Divide both sides by $4$ to solve for c. $\newline$ $c \neq \frac{-4}{4}$ $\newline$ $c \neq -1$

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