Question

How do you simplify the expression \[13+\left( -12 \right)-\left( -15 \right)\]?

Answer 1

Hint: Assume the given expressions as ‘E’. Now, use the algebraic conditions given as: - \[\left( -1 \right)\times 1=-1\] and \[\left( -1 \right)\times \left( -1 \right)=1\] to simplify the given expression. Perform simple arithmetic operations like: - addition and subtraction to get the answer.

Complete step by step solution:
Here, we have been provided with the expression: - \[13+\left( -12 \right)-\left( -15 \right)\] and we are asked to simplify it. That means we have to find the value of this expression.
Now, let us assume this expression as ‘E’, so we have,
\[\Rightarrow E=13+\left( -12 \right)-\left( -15 \right)\]
As we can see that there are three numbers and they have multiple arithmetic signs between them. There is a plus and then a minus sign between the first two numbers, 13 and 12. Now, there are two minus signs between the last two numbers 12 and 15. So, in order to simplify this expression. We have to convert these multiple operations between two numbers into a single arithmetic operation.
Let us consider two numbers ‘a’ and ‘b’ which can be positive or negative in nature. So, when we have expression like \[a+\left( -b \right)\] and \[a-\left( -b \right)\] then their simplified form is given by using the algebraic conditions: - \[\left( -1 \right)\times 1=-1\] and \[\left( -1 \right)\times \left( -1 \right)=1\], so the two expressions can be written as: -
(i) \[a+\left( -b \right)=a-b\]
(ii) \[a-\left( -b \right)=a+b\]
Using the above two results we can simplify the given expression ‘E’ as: -
\[\Rightarrow E=13-12+15\]
Now, we have the simplified form of the expression and we need to perform simple addition and subtraction to get the answer. Therefore, we have,
\[\begin{align}
  & \Rightarrow E=\left( 13+15 \right)-12 \\
 & \Rightarrow E=25-12 \\
 & \Rightarrow E=16 \\
\end{align}\]
Hence, our answer is 16.

Note:
One must remember the results we have obtained in (i) and (ii) otherwise you may get confused in multiple signs. As you may note that brackets are used to separate these signs because if we will write the expressions like: - \[a+-b\] or \[a-b\], it will look a bit odd. Remember the results: - \[\left( -1 \right)\times 1=-1\] and \[\left( -1 \right)\times \left( -1 \right)=1\] for the multiplication operation. Note that these two multiplication results are also valid for division operation.