An equation is a mathematical statement that asserts the equivalence of two expressions. For example, the assertion that "two plus five equals seven" is represented by the equation $2 + 5 = 7$.

In many cases, an equation contains one or more variables. These are still written by placing each expression on either side of an equals sign ($=$). For example, the equation $x + 3 = 5$, read "$x$ plus three equals five", asserts that the expression $x+3$ is equal to the value 5.

It is possible for equations to have more than one variable. For example, $x + y + 7 = 13$ is an equation in two variables. However, this lesson focuses solely on equations in one variable.

Solving Equations

When an equation contains a variable such as $x$, this variable is considered an unknown value. In many cases, we can find the possible values for $x$ that would make the equation true. $$

For example, consider the equation we were talking about above: $x + 3 =5$. You have probably already guessed that the only possible value of $x$ is 2, because you know that $2 + 3 = 5$ is a true equation. We use an equals sign to show that we know the value of a given variable. In this case, we write $x=2$ (read as "$x$ equals two").

The values of the variables that make an equation true are called the solutions of the equation. In turn, solving an equation means determining what values for the variables make the equation a true statement. 

The equation above was fairly straightforward; it was easy for us to identify the solution as $x = 2$. However, it becomes useful to have a process for finding solutions for unknowns as problems become more complex.

Verifying Solutions 

If a number is found as a solution to an equation, then substituting that number back into the place of the variable should make the equation true. Thus, we can easily check whether a number is a genuine solution to a given equation.

For example, let's examine whether $x=3$ is a solution to the equation  $2x + 31 = 37$.

Substituting 3 for $x$, we have: 

$2(3) + 31 = 37 \\ 6 + 31 = 37$

This equality is a true statement. Therefore, we can conclude that $x = 3$ is, in fact, a solution to the equation $2x+31=37$.