Question

Find the value of the given expression \[\left( -157 \right)\times \left( -19 \right)+157=\_\_\_\_\_.\]

Answer 1

Hint: To solve this question, we will assume the variables for – 157, – 19, and 157 as a, b and - a respectively. Then we will try to obtain the above equation in terms of the variable. Then, we will simplify by taking the term which is common to the two terms hence obtained. We will be able to solve the given expression by performing the addition, subtraction, and multiplication operations.

Complete step-by-step solution:
We are given our expression as
\[\left( -157 \right)\times \left( -19 \right)+157......\left( i \right)\]
Let us assume a variable for each of these terms as
\[a=-157\]
\[b=-19\]
\[\Rightarrow -a=-\left( -157 \right)=157\]
So we get,
\[\Rightarrow -a=157\]
Now, we will write the equation (i) in terms of the variables a and b. We have,
\[\left( i \right)ab-a\]
So, finally, we have to calculate the value of ab – a. We can see that the term a is common to both the terms ab and a. So, taking ‘a’ common in the above expression, we get,
\[ab-a=a\left( b-1 \right)\]
We have already assumed the values of a and b. Now substituting a = – 157 and b = – 19 in the above expression, we get,
\[\Rightarrow ab-a=a\left( b-1 \right)\]
\[\Rightarrow ab-a=-157\left( -19-1 \right)\]
First, we will compute the value inside the bracket and then multiply. So, we have -19 and 1 and we have to subtract them. It is the same as adding -19 and -1. So, adding the terms inside the bracket, we will get
\[\Rightarrow ab-a=-157\left( -20 \right)\]
Now, we will multiply the terms. As we know that \[-\left( -x \right)=x\] for any x number, we will get
\[\Rightarrow ab-a=3140\]
Therefore, we get the value of \[\left( -157 \right)\times \left( -19 \right)+157=3140.\]

Note: Another simple method to solve this question can be without assuming any variables and without taking the common terms. The method is as below.
\[\left( -157 \right)\times \left( -19 \right)+157=157\times 19+157\]
\[\Rightarrow \left( -157 \right)\times \left( -19 \right)+157=2983+157\]
\[\Rightarrow \left( -157 \right)\times \left( -19 \right)+157=3140\]
Therefore, we get the same answer as obtained above.