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${:[" FIND "x","y","z],[-(3)/(3+4y)=(1)/(x-z)],[(3)/(x+y-1)=-z^(-1)],[(6z)/(5y+8)=-1]:}$

$\begin{array}{l}\text { FIND } x, y, z \\ -\frac{3}{3+4 y}=\frac{1}{x-z} \\ \frac{3}{x+y-1}=-z^{-1} \\ \frac{6 z}{5 y+8}=-1\end{array}$

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Algebra 2

Simplify variable expressions using properties

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\begin{array}{l}\text { FIND } x, y, z \\ -\frac{3}{3+4 y}=\frac{1}{x-z} \\ \frac{3}{x+y-1}=-z^{-1} \\ \frac{6 z}{5 y+8}=-1\end{array}

Solve for z: Solve the third equation for z. $\newline$ The third equation is $\frac{6z}{5y+8} = -1$ . To find z, we multiply both sides by $5y+8$ to get $6z = -(5y+8)$ . Then we divide both sides by $6$ to isolate z. $\newline$ Calculation: $6z = -(5y+8)$ implies $z = -\frac{5y+8}{6}$ .
Substitute z into second equation: Substitute the expression for z from Step $1$ into the second equation. $\newline$ The second equation is $\frac{3}{x+y-1} = -z^{-1}$ . We substitute $z = -\frac{5y+8}{6}$ into the equation to get $\frac{3}{x+y-1} = -\left(-\frac{6}{5y+8}\right)$ . $\newline$ Calculation: $\frac{3}{x+y-1} = \frac{6}{5y+8}$ .
Cross-multiply for x: Cross-multiply to solve for x in terms of y. $\newline$ Cross-multiplying the equation from Step $2$ gives us $3(5y+8) = 6(x+y-1)$ . $\newline$ Calculation: $15y + 24 = 6x + 6y - 6$ .
Solve for x: Simplify the equation from Step $3$ to solve for x. $\newline$ We simplify the equation $15y + 24 = 6x + 6y - 6$ by subtracting $6y$ from both sides and adding $6$ to both sides to isolate terms with x on one side. $\newline$ Calculation: $15y - 6y + 24 + 6 = 6x$ , which simplifies to $9y + 30 = 6x$ , and then $x = \frac{9y + 30}{6}$ .
Simplify x expression: Simplify the expression for x. $\newline$ We simplify the expression $x = \frac{9y + 30}{6}$ by dividing both the numerator and the denominator by $3$ . $\newline$ Calculation: $x = \frac{3(3y + 10)}{6}$ , which simplifies to $x = \frac{3y + 10}{2}$ .
Substitute z and x into first equation: Substitute the expressions for z and x from Steps $1$ and $5$ into the first equation. $\newline$ The first equation is $-\frac{3}{3+4y} = \frac{1}{x-z}$ . We substitute $x = \frac{3y + 10}{2}$ and $z = -\frac{5y+8}{6}$ into the equation. $\newline$ Calculation: $-\frac{3}{3+4y} = \frac{1}{\frac{3y + 10}{2} - \left(-\frac{5y+8}{6}\right)}$ .
Find common denominator: Find a common denominator to combine the terms in the denominator of the right side of the equation from Step $6$ . $\newline$ We need to find a common denominator for $\frac{3y + 10}{2}$ and $\frac{5y+8}{6}$ to combine them. $\newline$ Calculation: The common denominator is $6$ , so we rewrite the equation as $-\frac{3}{3+4y} = \frac{1}{\frac{9y + 30}{6} + \frac{5y+8}{6}}$ .
Combine denominator terms: Combine the terms in the denominator on the right side of the equation from Step $7$ . $\newline$ We combine the terms to get a single fraction in the denominator. $\newline$ Calculation: $-\frac{3}{3+4y} = \frac{1}{\frac{14y + 38}{6}}$ .
Invert and multiply: Invert the denominator on the right side of the equation from Step $8$ and multiply by $1$ . $\newline$ We invert the fraction in the denominator and multiply by $1$ to get the right side of the equation in the same form as the left side. $\newline$ Calculation: $-\frac{3}{3+4y} = \frac{6}{14y + 38}$ .
Cross-multiply for y: Cross-multiply to solve for y. $\newline$ We cross-multiply the equation from Step $9$ to solve for y. $\newline$ Calculation: $-3(14y + 38) = 6(3+4y)$ .
Solve for y: Simplify the equation from Step $10$ to solve for y. $\newline$ We distribute the multiplication on both sides and then combine like terms. $\newline$ Calculation: $-42y - 114 = 18 + 24y$ .
Simplify y fraction: Move all terms involving y to one side and constant terms to the other side. $\newline$ We add $42y$ to both sides and subtract $18$ from both sides to isolate y. $\newline$ Calculation: $-42y + 42y - 114 + 18 = 18 + 24y + 42y - 18$ , which simplifies to $-96 = 66y$ .
Substitute y into x expression: Solve for y. $\newline$ We divide both sides by $66$ to find y. $\newline$ Calculation: $y = \frac{-96}{66}$ .
Simplify x expression: Simplify the fraction for y. $\newline$ We simplify the fraction $\frac{-96}{66}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $6$ . $\newline$ Calculation: $y = \frac{-16}{11}$ .
Simplify x fraction: Substitute the value of y back into the expression for x from Step $5$ . $\newline$ We substitute $y = \frac{-16}{11}$ into $x = \frac{3y + 10}{2}$ to find x. $\newline$ Calculation: $x = \frac{3(\frac{-16}{11}) + 10}{2}$ .
Simplify x further: Simplify the expression for x. $\newline$ We simplify the expression by performing the multiplication and addition. $\newline$ Calculation: $x = \frac{-48 + 110}{22}$ .
Substitute y into z expression: Simplify the fraction for x. $\newline$ We simplify the fraction $\frac{-48 + 110}{22}$ by combining the terms in the numerator and then dividing by the denominator. $\newline$ Calculation: $x = \frac{62}{22}$ .
Simplify z expression: Simplify the fraction for x further. $\newline$ We simplify the fraction $\frac{62}{22}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . $\newline$ Calculation: $x = \frac{31}{11}$ .
Simplify z fraction: Substitute the value of y back into the expression for z from Step $1$ . $\newline$ We substitute $y = \frac{-16}{11}$ into $z = -\frac{5y+8}{6}$ to find z. $\newline$ Calculation: $z = -\frac{5(\frac{-16}{11})+8}{6}$ .
Simplify z further: Simplify the expression for z. $\newline$ We simplify the expression by performing the multiplication and addition. $\newline$ Calculation: $z = -\frac{80 - 88}{66}$ .
Simplify z further: Simplify the expression for z. $\newline$ We simplify the expression by performing the multiplication and addition. $\newline$ Calculation: $z = -\frac{80 - 88}{66}$ .Simplify the fraction for z. $\newline$ We simplify the fraction $\frac{80 - 88}{66}$ by combining the terms in the numerator and then dividing by the denominator. $\newline$ Calculation: $z = \frac{-8}{66}$ .
Simplify z further: Simplify the expression for z. $\newline$ We simplify the expression by performing the multiplication and addition. $\newline$ Calculation: $z = -\frac{80 - 88}{66}$ .Simplify the fraction for z. $\newline$ We simplify the fraction $\frac{80 - 88}{66}$ by combining the terms in the numerator and then dividing by the denominator. $\newline$ Calculation: $z = \frac{-8}{66}$ .Simplify the fraction for z further. $\newline$ We simplify the fraction $\frac{-8}{66}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . $\newline$ Calculation: $z = \frac{-4}{33}$ .

FIND x,y,z−33+4y=1x−z3x+y−1=−z−16z5y+8=−1 \begin{array}{l}\text { FIND } x, y, z \\ -\frac{3}{3+4 y}=\frac{1}{x-z} \\ \frac{3}{x+y-1}=-z^{-1} \\ \frac{6 z}{5 y+8}=-1\end{array} FIND x,y,z−3+4y3​=x−z1​x+y−13​=−z−15y+86z​=−1​

$\begin{array}{l}\text { FIND } x, y, z \\ -\frac{3}{3+4 y}=\frac{1}{x-z} \\ \frac{3}{x+y-1}=-z^{-1} \\ \frac{6 z}{5 y+8}=-1\end{array}$