Question

Solve:
 \[
  \dfrac{x}{{40 - 25}} = \dfrac{y}{{10 - 4}} = \dfrac{z}{{5 - 20}} \\
  \dfrac{x}{{15}} = \dfrac{y}{6} = \dfrac{{ - z}}{{15}} \Rightarrow \dfrac{x}{5} = \dfrac{y}{2} = \dfrac{z}{{ - 5}} = k \\
 \]

Answer 1

Hint:
This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. Also, we need to know how to find common factors between two or more terms. We need to know how to multiply a variable with a constant term, a variable with a variable, and a constant term with a constant term to make the easy calculation.

Complete step by step solution:
The given question is shown below,
\[
  \dfrac{x}{{40 - 25}} = \dfrac{y}{{10 - 4}} = \dfrac{z}{{5 - 20}} \\
  \dfrac{x}{{15}} = \dfrac{y}{6} = \dfrac{{ - z}}{{15}} \Rightarrow \dfrac{x}{5} = \dfrac{y}{2} = \dfrac{z}{{ - 5}} = k \\
 \]
Let’s take the final equation from the question,
\[\dfrac{x}{5} = \dfrac{y}{2} = \dfrac{z}{{ - 5}} = k \to \left( 1 \right)\]
We know that,
If, \[\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = z\]
Then it can also be written as,
\[
  \dfrac{a}{b} = z \\
  \dfrac{c}{d} = z \\
  \dfrac{e}{f} = z \\
 \]
By using this formula we can modify the equation\[\left( 1 \right)\], as given below,
\[\left( 1 \right) \to \dfrac{x}{5} = \dfrac{y}{2} = \dfrac{z}{{ - 5}} = k\]
\[\dfrac{x}{5} = k \to \left( 2 \right)\]
\[\dfrac{y}{2} = k \to \left( 3 \right)\]
\[\dfrac{z}{{ - 5}} = k \to \left( 4 \right)\]
To find\[x\]value, let’s solve the equation\[\left( 2 \right)\]
\[\left( 2 \right) \to \dfrac{x}{5} = k\]
Let’s move the term\[5\]from LHS to RHS of the above equation, we get
\[x = 5k\]
To find\[y\]value, let’s solve the equation\[\left( 3 \right)\],
\[\left( 3 \right) \to \dfrac{y}{2} = k\]
Let’s move the term\[2\] from LHS to RHS of the above equation, we get
\[y = 2k\]
To find the value of \[z\], let’s solve the equation \[\left( 4 \right)\], we get
\[\left( 4 \right) \to \dfrac{z}{{ - 5}} = k\]
Let’s move the term \[ - 5\] from LHS to RHS of the above equation we get,
\[z = - 5k\]
So, the final answer is,

\[x = 5k\], \[y = 2k\] and \[z = - 5k\]

Note:
This question describes the arithmetic operations like addition/ subtraction/ multiplication/ division. Note that when we move one term from LHS to RHS or RHS to LHS of the equation the arithmetic operation can be modified as follows,
1) The addition operation is converted into the subtraction process.
2) The subtraction process is converted into the addition process.
3) The multiplication process is converted into a division process.
4) The division process is converted into a multiplication process.