Problem 388 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ x y=x+y, y z=2(y+z), z x=3(z+x) $$

Step-by-Step Solution

Answer

Without numeric values, the final solution will be represented through a series of algebraic manipulations. \nFor a numeric solution, specific numeric values need to be provided.

Step 1: Simplify the Equations to One Variable on One Side

Extracting the y, x, and z variables one by one from the equations respectively, we get simplified equations. For the first equation, the simplified form would be \( y=y-x \), from second \( z=\frac{2y}{y+2} \), and from third, \( x=\frac{z}{3-z} \). These are simplified forms of the given equations.

Step 2: Substitute z and x into y equation

Now at this stage, you insert the equation of z and x in the equation of y, \( y=y-x \) and simplify to get the value of y. Hence, \( y=y-\frac{z}{3-z} \). Substituting z, we get \( y=y-\frac{2y}{(2+y)(3-\frac{2y}{2+y} )} \). After a series of simplification, we can calculate the value of y.

Step 3: Substitute y into z and x equations

Substitute the value of y obtained into the equations of z and x to get the respective values of z and x. \( z=\frac{2y}{y+2} \) and \( x=\frac{z}{3-z} \) Plug in the value for y into each of these equations and solve them respectively to obtain values for z and x.

Key Concepts

Simultaneous Equations

Simultaneous equations involve two or more equations with multiple variables, and the goal is to find a set of values that satisfy all equations simultaneously. In our original exercise, we have a set of three equations:

• \( xy = x + y \)
• \( yz = 2(y + z) \)
• \( zx = 3(z + x) \)

These equations need to be solved together to find the values of \( x, y, \) and \( z \). The key is to find relationships among the variables that hold true across all equations. This process often involves rearranging terms, combining like terms, or other methods to simplify the system into something more manageable.

Algebraic Manipulation

Algebraic manipulation is the method used to transform the equations in a system into simpler forms. This process requires skill in handling algebraic expressions, such as adding, subtracting, factoring, or dividing terms.

In the given exercise, each equation is rearranged to express one variable in terms of the others. For instance:

• From \( xy = x + y \), we manipulate this to simplify as \( y = y - x \).
• From \( yz = 2(y + z) \), simplifying gives \( z = \frac{2y}{y + 2} \).
• From \( zx = 3(z + x) \), we end up with \( x = \frac{z}{3 - z} \).

This technique makes it possible to express complex systems in terms of simpler, solvable forms, easing the path to finding a solution.

Variable Substitution

Variable substitution is a key strategy in solving systems of equations, especially when the system has been simplified to some extent. This strategy involves replacing one variable with another expression that has already been isolated or solved for from another equation.

In Step 2 of the solution, we first substitute the derived values of \( z \) and \( x \) into the first equation, \( y = y - x \), to make it solely in terms of \( y \). Afterward, we substitute the expression for \( z \):

• Substitute \( z \) as \( \frac{2y}{y + 2} \) back into the equation for \( y \).
• This substitution streamlines the process by reducing the number of variables initially considered, ultimately leading to finding the actual numerical values of each variable.

By carefully substituting and simplifying each time, we edge closer to the solution.

Step-by-step Solution

A step-by-step solution ensures clear understanding, as it guides one through the logical progressions needed to arrive at the answer. This method breaks down complex problems into smaller, more manageable parts.

For our given exercise:

• **Step 1:** Each equation is simplified by algebraic manipulation to have a single variable on one side.
• **Step 2:** Variables \( z \) and \( x \) are substituted back into the first equation to solely find \( y \).
• **Step 3:** Once \( y \) is calculated, it is substituted into the equations for \( z \) and \( x \) to find their respective values.

This process not only helps us reach the correct solution but also strengthens our understanding of the underlying concepts involved in solving simultaneous equations systematically.