Question

If the values of A, B and C are given as \[A=5{{p}^{2}}-3{{q}^{2}}+{{r}^{2}}\], $B=-2{{q}^{2}}+3{{p}^{2}}-4{{r}^{2}}$ and $C=-7{{r}^{2}}+3{{p}^{2}}+2{{q}^{2}}$, then find the value of $A+B-C$?

Answer 1

Hint: We start solving the problem by writing the order of the variables in the order ${{p}^{2}}$ coming first followed by ${{q}^{2}}$ and ${{r}^{2}}$ in each of the function A, B and C. We then substitute the values in $A+B-C$ and make addition and subtractions operations for the coefficients of variables ${{p}^{2}}$, ${{q}^{2}}$ and ${{r}^{2}}$ to get the required result.

Complete step by step answer:
According to the problems, we have defined A, B and C as \[A=5{{p}^{2}}-3{{q}^{2}}+{{r}^{2}}\], $B=-2{{q}^{2}}+3{{p}^{2}}-4{{r}^{2}}$ and $C=-7{{r}^{2}}+3{{p}^{2}}+2{{q}^{2}}$. We need to find the value of $A+B-C$.
Let us write every equation by having ${{p}^{2}}$ terms first, followed by ${{q}^{2}}$ terms and ${{r}^{2}}$ terms.
So, we have $B=-2{{q}^{2}}+3{{p}^{2}}-4{{r}^{2}}$ and we rewrite it as $B=3{{p}^{2}}-2{{q}^{2}}-4{{r}^{2}}$. We also have $C=-7{{r}^{2}}+3{{p}^{2}}+2{{q}^{2}}$ and we rewrite it as $C=3{{p}^{2}}+2{{q}^{2}}-7{{r}^{2}}$.
We need to find the value of $A+B-C$.
We have $A+B-C=\left( 5{{p}^{2}}-3{{q}^{2}}+{{r}^{2}} \right)+\left( 3{{p}^{2}}-2{{q}^{2}}-4{{r}^{2}} \right)-\left( 3{{p}^{2}}+2{{q}^{2}}-7{{r}^{2}} \right)$.
$\Rightarrow A+B-C=5{{p}^{2}}-3{{q}^{2}}+{{r}^{2}}+3{{p}^{2}}-2{{q}^{2}}-4{{r}^{2}}-3{{p}^{2}}-2{{q}^{2}}+7{{r}^{2}}$.
We know that whenever we get involved in addition or subtraction with different variables, we add or subtract the coefficients of that variable.
$\Rightarrow A+B-C=\left( 5+3-3 \right){{p}^{2}}+\left( -3-2-2 \right){{q}^{2}}+\left( 1-4+7 \right){{r}^{2}}$.
$\Rightarrow A+B-C=5{{p}^{2}}+\left( -7 \right){{q}^{2}}+\left( 4 \right){{r}^{2}}$.
$\Rightarrow A+B-C=5{{p}^{2}}-7{{q}^{2}}+4{{r}^{2}}$.
We have found the value of $A+B-C$ as $5{{p}^{2}}-7{{q}^{2}}+4{{r}^{2}}$.

∴ The value of $A+B-C$ is $5{{p}^{2}}-7{{q}^{2}}+4{{r}^{2}}$.

Note: We should not add coefficients directly without checking the variables that are present after the coefficients that we are adding. We should make sure that the multiplication of signs is done properly. We should not make mistakes while making addition and subtraction operations. Similarly, we expect problems to find the multiplication of the given equations.