How do you solve for x in 5x + 7x = 72?

Nov 7, 2014

I think it's important not only to be able to solve, but to understand why a particular method can be used to solve.

The first step to solve the above equation is to simplify it using one of the laws of arithmetic - the distributive law that states that
$a \cdot c + b \cdot c = \left(a + b\right) \cdot c$

Using this law, we can say that
$5 \cdot x + 7 \cdot x = \left(5 + 7\right) \cdot x$
Since $5 + 7 = 12$, we can rewrite our equation in a form
$12 \cdot x = 72$

The next step involves a transformation of this equation using another law that states: if two numeric values are equal, you can divide both of them by the same number not equal to zero, and the results will be equal.

Let's divide both side of the equation above by $12$, the result is
$\frac{12 \cdot x}{12} = \frac{72}{12}$

Performing obvious operations, we get
$x = 6$
This is the solution since we have determined the value of an unknown variable $x$.

I would strongly recommend to check the answer obtained, no matter how simple the transformation steps were. In this case we can substitute the value $x = 6$ into original equation and check if it's true.
Indeed, $5 \cdot 6 + 7 \cdot 6 = 30 + 42 = 72$, which proves that the solution we obtained $x = 6$ is correct.