Question

How do you solve $4x + 8 = 7x - 7$ ?

Answer 1

Hint: One must remember in algebra, only the terms having the same degree or power of the variable can be added or subtracted. Here, we shift all the terms with the same power on the same side of the equation and then performing the necessary operations, the value of $x$ can be found.

Complete step-by-step answer:
As we know in algebra, only the terms having the same power on the variable can be added or subtracted.
Here, the terms $4\;x$ and $7\;x$ have the same power of $x$ , and hence they can be added or subtracted from each other.
Similarly, the terms $\;8$ and $ - 7$ have the same power of $x$ , which is $\;0$ because any number with a power of $\;0$ is equal to $\;1$ .
Hence, these terms can also be written as $8{x^0}$ and $ - 7{x^0}$ . As they have the same power, these terms can be added or subtracted from each other.
To simplify the equation, let’s start by shifting the same power terms on the same side of the equation
The equation we are given here is
$\Rightarrow$ $4x + 8 = 7x - 7$
Now, adding $\;7$ on both the sides of the equation
$ \Rightarrow 4x + 8 + 7 = 7x - 7 + 7$
$ \Rightarrow 4x + 15 = 7x + 0$
Now, subtracting $4\;x$ from both sides of the equation
$ \Rightarrow 4x + 15 - 4x = 7x - 4x$
Rearranging the terms,
$ \Rightarrow 4x - 4x + 15 = 7x - 4x$
$ \Rightarrow 0 + 15 = 7x - 4x$
Now, to explain the subtraction, let’s expand the terms $7\;x$ and $4\;x$ .
We know that $7\;x$ means $\;7$ times $x$ . Hence, expanding $7\;x$ as shown
$ \Rightarrow 7x = x + x + x + x + x + x + x$
Similarly, for $4\;x$ we can write
$ \Rightarrow 4x = x + x + x + x$
Substituting these expansions in the equation,
$ \Rightarrow 15 = (x + x + x + x + x + x + x) - (x + x + x + x)$
Opening the brackets,
$ \Rightarrow 15 = x + x + x + x + x + x + x - x - x - x - x$
$ \Rightarrow 15 = x + x + x$
Here we have $\;3$ times $x$ which can be written as $3\;x$ ,
$ \Rightarrow 15 = 3x$
Dividing both the sides by $\;3$ ,
$ \Rightarrow \dfrac{{15}}{3} = x$
Factoring the numerator,
$ \Rightarrow \dfrac{{5 \times 3}}{3} = x$

$ \Rightarrow x = 5$
This is the solution for the given equation.

Note:
Another method for subtracting the terms with $x$ is by taking the common factor i.e. variable $x$ common and subtracting the numbers in the bracket. If one wants to check the accuracy of the solution, one can substitute the obtained value in the given equation and check if it satisfies the equation.