Question

If \(\mathrm{xy}=28, \mathrm{yz}=18, \mathrm{zx}=14\) and \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are \(>0\), the value of \(\mathrm{x}+\mathrm{y}+\mathrm{z}\) is equal to (a) \(\frac{41}{9}\) (b) \(\frac{41}{3}\) (c) \(\frac{21}{4}\) (d) \(\frac{21}{8}\)

Short answer

Based on the given step by step solution, create a short answer question: Question: Given that \(xy = 28\), \(yz = 18\), and \(xz = 14\), where x, y, and z are positive values, find the value of \(x + y + z\). Answer: \(\frac{31}{6}\)

Step-by-step solution

Write the equations from the given information

We have the following equations given: 1. \(xy = 28\) 2. \(yz = 18\) 3. \(xz = 14\)

Solve for one variable in terms of another

We can start by solving equation (1) for x: \(x = \frac{28}{y}\) Now, let's substitute this expression for x in equation (3) and solve for z: \(z = \frac{14}{x} = \frac{14}{\frac{28}{y}} = \frac{14y}{28} = \frac{1}{2}y\)

Substitute in equation (2) and solve for y

Now we have expressions for x and z in terms of y. Let's substitute these expressions into equation (2) and solve for y: \(yz = 18\) \(\left(\frac{28}{y}\right)\left(\frac{1}{2}y\right) = 18\) Simplify and solve for y: \(\frac{14}{1} = 18\) \(y = \frac{7}{9}\)

Solve for x and z

With the value of y, we can now solve for x and z: \(x = \frac{28}{y} = \frac{28}{\frac{7}{9}} = 4\) \(z = \frac{1}{2}y = \frac{1}{2} \cdot \frac{7}{9} = \frac{7}{18}\)

Find the sum of x, y, and z

Finally, we can find the value of x + y + z: \(x + y + z = 4 + \frac{7}{9} + \frac{7}{18} = \frac{36}{9} + \frac{7}{9} + \frac{7}{18}=\frac{43}{9}+\frac{7}{18}\) Let's find a common denominator by finding the least common multiple (LCM) of 9 and 18, which is 18. Then add the fractions: \(x+y+z= \frac{86}{18}+\frac{7}{18}=\frac{93}{18}= \boxed{\frac{31}{6}}\) The correct answer is not among the given options.

Key concepts

Algebraic Equations

Tackling algebraic equations is like playing detective with numbers and symbols. An algebraic equation is a statement where two expressions are set equal to each other, often involving variables like \( x \), \( y \), and \( z \). In our detective work, we're not just looking for what \( x \), \( y \), and \( z \) are, we're finding values that make the equation true.

In the context of IIT-JEE mathematics, these equations aren't just puzzles; they're gateways to understanding complex problems. The quintessential step begins with isolating variables to uncover their values. As we saw in the exercise, we manipulated the first equation to express \( x \) in terms of \( y \), setting us on a path to find the other variables. It's like a mathematical chain reaction; finding one variable helps us to discover the next until all pieces of the puzzle fit together to reveal the big picture.

When delving into algebraic equations for your IIT-JEE preparation, remember to:

Systems of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. Our goal is to find a common solution to all equations, meaning a set of values for the variables that satisfy every equation at once. Think of it as finding the precise point where different paths cross in a labyrinth.

The exercise we've explored is a perfect example of this. We have three equations, each forming a different 'path'. By substituting variables and using elimination methods, we find where these paths meet - the values of \( x \), \( y \), and \( z \). In competitive exams like IIT-JEE, solving systems of equations quickly and accurately can set you apart. Here are some key points to remember:

• Substitution is your friend: Replace variables where possible to simplify.
• Stay organized: Label each step clearly as the complexity grows.
• Practice different methods: Substitution, elimination, and graphing are all valuable tools.

Keep these strategies in mind, and you'll untangle any system of linear equations that comes your way.

Rational Expressions

Now let's look at the journey of handling rational expressions. In layman's terms, think of these as fractions where both the numerator (top number) and the denominator (bottom number) are algebraic expressions. They follow the same rules as fractions but involve more variables and often require simplification to find solutions.

In the provided solution, the process of simplifying the equation \( \left(\frac{28}{y}\right)\left(\frac{1}{2}y\right) = 18 \) by eliminating \( y \) from the numerator and denominator is a clear use of handling rational expressions. Simplification is a powerful step in harnessing these expressions, allowing us to pare down complex forms to something much more manageable.

Enhance your rational expression skills for the IIT-JEE by keeping in mind to:

• Identify and factor denominators and numerators wherever possible.
• Look for common factors to cancel out and simplify the expression.
• Understand the domain of the expression (the set of values that \( y \) can take without making the denominator zero).

Armed with these skills, those daunting rational expressions will start to look a lot less intimidating.