Question

Question If $a+b=16,a-c=4,andb-c=-4$, which statement is true?

$\begin{array}{c}{\rm I}.b+c=12\\ {\rm I}{\rm I}.a-b=8\\ {\rm I}{\rm I}{\rm I}.a+c=20\end{array}$

$\left(A\right){\rm I}$only

$\left(B\right){\rm I}{\rm I}$only

$\left(C\right){\rm I}\text{and}{\rm I}{\rm I}$only

$\left(D\right){\rm I},{\rm I}{\rm I},\text{and}{\rm I}{\rm I}{\rm I}$

Short answer

The option $\left(D\right)$ is true, the other options are false, because all the three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}$ are true.

Step-by-step solution

– Use the elimination method to get the system of equations in two variables.

Subtract $a-c=4$from $a+b=16$as follows:

$\begin{array}{l}\underset{\_}{\begin{array}{l}a+b=16\\ (-)a-c=4\end{array}}\\ 0+b+c=12\end{array}$

So, the resultant equation is $b+c=12$.

Subtract $b-c=-4$from $a+b=16$as follows:

$\begin{array}{l}\underset{\_}{\begin{array}{l}a+b=16\\ (-)+b-c=-4\end{array}}\\ a+0+c=20\end{array}$

So, the resultant equation is $a+c=20$.

Subtract $b-c=-4$from $a-c=4$as follows

$\begin{array}{l}\underset{\_}{\begin{array}{l}a-c=4\\ (-)+b-c=-4\end{array}}\\ a-b+0=8\end{array}$

So, the resultant equation is $a-b=8$.

– Check whether the option is true or false.

The given three equations are${\rm I}.b+c=12,{\rm I}{\rm I}.a-b=8,{\rm I}{\rm I}{\rm I}.a+c=20$.

From the above calculations in step 1, by using the equations $a+b=16,a-c=4,$and $b-c=-4$, one can determine the all the given three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}.$

Here, in the option $\left(A\right)$it is given that, ${\rm I}$only.

But, all the three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}$can be determined.

So, option $\left(A\right)$is false.

– Check whether the option B is true or false.

Here, in the option $\left(B\right)$it is given that, ${\rm I}{\rm I}$only.

But, all the three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}$can be determined.

So, option $\left(B\right)$is false.

– Check whether the option C is true or false.

Here, in the option $\left(C\right)$it is given that, ${\rm I}\text{and}{\rm I}{\rm I}$only.

But, all the three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}$can be determined.

So, option $\left(C\right)$is false.

– Check whether the option D is true or false.

Here, in the option$\left(D\right)$it is given that, ${\rm I}\text{,}{\rm I}{\rm I},\text{and}{\rm I}{\rm I}{\rm I}$.

But, all the three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}$can be determined.

So, option $\left(D\right)$is true.

Hence, the option $\left(D\right)$ is true, the other options are false, because all the three equations ${\rm I},{\rm I}{\rm I},{\rm I}{\rm I}{\rm I}$ are true.