Question

How do inverse operations help solve equations?

Answer 1

Hint: Addition and subtraction are inverse operations of each other as are multiplication and division are inverse operations of each other. That means addition is the inverse of subtraction and vice versa, also multiplication is the inverse of division and vice versa.

Complete step by step answer:
To solve an equation means to isolate the unknown \[x\] on the left of the $=$ sign.
\[x~=\]The solution.
So, numbers therefore must be shifted from one side of an equation to the other. We can do that by writing them on the other side with the help of inverse operation.
So, if an equation was like:
\[x~+~a~=~b\]
The equation will look like this after inverse operation:
\[x~+~a~-a=~b-a\]
then the solution will be:
\[x~=~b~-~a\]
Since \[a\] was added on the left, we have to subtract it on both sides.
This is the relationship between addition and subtraction and similarly for each of the operations and their inverses.

For example,$1$ (for addition and subtraction)
If we have to solve: $x-7=8$
Here as we can see that $7$ has been subtracted to the variable $x$.
As we know the inverse of subtraction is addition, and so by applying inverse operation we will add $7$ on both sides, we get:
$x-7=8$
$x-7+7=8+7$
$x=15$
As we can see from above equation we get value of $x$ as $15$ and that is because $x$ on the left hand side is just plain $x$ and so by cancelling $7$on left hand side we get:
$x=15$ as the solution

Example $2$, (for multiplication and division)
Solve: $7x=56$
Here, as can see that the variable $x$ is multiplied by 7.
Therefore, we will apply the inverse of multiplication that is division. So we'll divide both sides by $7$.
$7x=56$
$\dfrac{7x}{7}=\dfrac{56}{7}$
So, we get:
$x=8$ as the solution.

Note: Inverse operation allows us to “undo” what has been done to the variable.
We can see in example $1$ , $7$ has been subtracted to the variable $x$ and as we know the inverse of subtraction is addition, so by adding $7$ we can undo the subtraction.
After $7$ was subtracted on the left hand side, the result was equal to $8$. We undo the subtraction, by adding $7$ and hence we can see that, the starting value that is $x$ was $15$.
When using inverse operation to solve an algebraic equation, it is important to keep the order of operations that is $PEMDAS$(Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) in mind.