Intro to equations

An equation is a statement that two expressions are equal. For example, the expression $5 + 3$5, plus, 3 is equal to the expression $6 + 2$6, plus, 2 (because they both equal $8$8), so we can write the following equation:

All equations have an equal sign ($=$equals). The $=$equals sign is not an operator like addition ($+$plus) or subtraction ($-$minus) symbols. The equal sign doesn't tell us what to do. It only tells us that two expressions are equal. For example, in:

The $-$minus sign tells us what to do with $6$6 and $2$2: subtract $2$2 from $6$6. However, the $=$equals sign does not tell us what to do with $6-2$6, minus, 2 and $3+1$3, plus, 1. It only tells us that they are equal.

Let's make sure we know the difference between an expression and an equation.

All of the equations we just looked at were true equations because the expression on the left-hand side was equal to the expression on the right-hand side. A false equation has an $=$equals, but the two expressions are not equal to each other. For example, the following is a false equation.

When we see an equation that's not true, we can use the not equal sign ($\neq$does not equal) to show that the two expressions are not equal:

Let's make sure we understand what a true equation is.

All of the equations that we've looked at so far have included only numbers, but most equations include a variable. For example, the equation $x + 2 = 6$x, plus, 2, equals, 6 has a variable in it. Whenever we have an equation like this with a variable, we call it an algebraic equation.

For an algebraic equation, our goal is usually to figure out what value of the variable will make a true equation.

For the equation $x + 2 = 6$x, plus, 2, equals, 6, notice how $\purpleD{x = 4}$start color #7854ab, x, equals, 4, end color #7854ab creates a true equation and $\maroonD{x = 3}$start color #ca337c, x, equals, 3, end color #ca337c creates a false equation.

The value of the variable that makes a true equation is called a solution to the equation. Going back to our example, $x=\purpleD{4}$x, equals, start color #7854ab, 4, end color #7854ab is a solution of $x + 2 = 6$x, plus, 2, equals, 6 because it makes the equation true.