Question

How do you simplify $(7 + x) + 7x$ ?

Answer 1

In algebra, the rule for addition and subtraction is that only the terms having the same power or degree of the variable considered can be added or subtracted. Hence, here we have to combine the terms with the same power of $x$ to simplify the equation.

Complete step-by-step answer:
Here in the given equation algebraic operations of addition and multiplication of a number and a variable are shown.
In the first term $(7 + x)$ , the addition of a number and a variable is shown.
This type of addition is not possible as the number is a constant value, but the variable can take up any value. Hence it is impossible to add a value to an unknown variable value.
Also, as we know the addition rule for algebra, where only terms with the same power of variable can be added.
Here, $\;7$ can also be written as $7 \times {x^0}$ , because we know that any number with a power of $\;0$ is equal to $\;1$ . Hence a term with power $\;0$ cannot be added to a term with power $\;1$ .
We can express the addition as $(7 + x)$ , but it makes no sense till the value of the variable $x$ is known.
In the second term $7\;x$ , the multiplication of a number and a variable is shown which can also be expressed as $7 \times x$ .
As $7\;x$ means $\;7$ times $x$ , it can also be expressed as
$\Rightarrow 7x = x + x + x + x + x + x + x$
As per the addition rule i.e. only terms with the same power can be added, if $x$ is added $\;7$ times it results in $7\;x$ .
Substituting the value of $7\;x$ from the above equation in the given equation
$\Rightarrow 7 + x + x + x + x + x + x + x + x$
$\Rightarrow 7 + \left( {{\text{8 times }}x} \right)$
As we know $\;8$ times $x$ can be expressed as $8\;x$

$\Rightarrow 7 + 8x$. This is the simplified value of the given equation.

Note:
Another method for simplifying the equation is by taking the common factor i.e. variable $x$ common and adding the numbers in the bracket. One must always remember the fact that $7 + x \ne 7x$ , except for a unique value of $x$ where $7 + x = 7x$.