Solve each system. (Hint: Let (1)/(x)=t and (1)/(y)=u.) (2)/(x)+(3)/(y)-(2)/(z)=-1 (8)/(

Step 1:u000ALetu0027s substitute $u005Cfrac{1}{x}u003Dt$, $u005Cfrac{1}{y}u003Du$ and $u005Cfrac{1}{z}u003Dv$ into the system of

Solve each system. (Hint: Let (1)/(x)=t and (1)/(y)=u.) ...

Solve each system. (Hint: Let $\frac{1}{x}=t \text { and } \frac{1}{y}=u$.)
$$\begin{aligned}&\frac{2}{x}+\frac{3}{y}-\frac{2}{z}=-1\\&\frac{8}{x}-\frac{12}{y}+\frac{5}{z}=5\\&\frac{6}{x}+\frac{3}{y}-\frac{1}{z}=1\end{aligned}$$

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Key Concepts

System of Equations

A system of equations involves multiple equations with several unknowns, and the goal is to determine the set of values that satisfy all of the equations simultaneously. Solving such systems is foundational in algebra and requires methods that reduce complexity and allow for exact solutions.

Substitution Method

The substitution method is a technique used to solve systems by expressing one variable in terms of another and then substituting that expression into the other equations. This step-by-step reduction of variables simplifies the problem into one or more single-variable equations.

Change of Variables

Changing variables is a strategy that involves replacing complicated expressions with simpler variables, which can transform a non-linear system into a linear one. This makes it easier to apply standard solution techniques, especially when dealing with reciprocal or fractional relationships.

Reciprocal Transformation

Reciprocal transformation is a specific instance of variable substitution where variables expressed as inverses (such as 1/x or 1/y) are replaced with new variables. This is particularly useful for handling equations where the original variables are in the denominator, converting the equations into a more tractable form.

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