How do you simplify ${{\left( 7y-x \right)}^{2}}$ ?

Hint: To solve these questions just apply the basic algebraic identities which satisfy the given expression in the question. After applying the correct identity, expand the expression according to the identity and simplify it further to get the final answer.

Complete step by step answer:
Algebraic identities are nothing but algebraic equations which always hold true for any value of variables in them. They contain expressions with variables and constants on both sides of the equation and the left-hand side of the equation is equal to the right-hand side of the equation.
Given expression ${{\left( 7y-x \right)}^{2}}$ can be compared to the algebraic identity,${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ where $a=7y$ and $b=x$ .
Now expanding the expression in accordance with the algebraic identity given above we get,
$\Rightarrow {{\left( 7y-x \right)}^{2}}={{\left( 7y \right)}^{2}}+{{x}^{2}}-2\left( 7y \right)\left( x \right)$
Simplifying the above expression further by multiplying the terms together and taking the square of some terms to get,
$\Rightarrow {{\left( 7y \right)}^{2}}+{{x}^{2}}-2\left( 7y \right)\left( x \right)=49{{y}^{2}}+{{x}^{2}}-14xy$
Now, by rearranging the terms of the above expression we get,
$\Rightarrow 49{{y}^{2}}+{{x}^{2}}-14xy={{x}^{2}}+49{{y}^{2}}-14xy$

Hence, on simplifying and expanding the expression${{\left( 7y-x \right)}^{2}}$, we get the final answer as${{x}^{2}}+49{{y}^{2}}-14xy$.

Note: The errors to look out for while solving these types of questions include not using parentheses when required which then leads to miscalculations and hence a wrong answer, therefore it is always advised to use parenthesis whenever and wherever required. It also becomes crucial to correctly identify the identity that is applicable to the given expression and can be expressed in that specific format.