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Expressions in mathematics are combinations of variables, numbers, and symbols that represent values. They don't include an equal sign. For example,

• \(-6x + 5\)
• \(3x - 13\)

These expressions consist of variables (like \(x\)) and constants (numbers like \(5\) and \(-13\)).
Expressions can vary in complexity from simple ones containing a single number or variable to complex ones involving several operations like addition, subtraction, multiplication, or division.

The key component is that an expression does not compare two quantities, it simply describes a value or a range of values. Expressions do not make a claim of equality or inequality, they are more like mathematical phrases.

The equal sign \(=\) is a crucial component in mathematics, serving as a bridge between two expressions to form an equation.
When you see \(=\), it signifies that the values on either side of it are identical in value. It acts as a statement of balance, saying "these two expressions equal each other."

Without an equal sign, you only have two separate expressions. For instance, in the statement \(-6x + 5 = 3x - 13\), the equal sign connects the left expression \(-6x + 5\) with the right expression \(3x - 13\).
This connection through the equal sign transforms expressions into something more significant, an equation, allowing for the formulization of problems and the establishment of relationships between variables and constants.

A mathematical statement can be either an expression or an equation and is intended to express an idea or relationship in the language of mathematics.
If the statement lacks an equal sign, it is simply an expression. But when an equal sign is included, like in the case of \(-6x + 5 = 3x - 13\), it becomes an equation.

A mathematical statement involving an equation asserts a relationship between two expressions.

• An equation is an assertion that two expressions are equal, turning it into a verifiable claim.
• Equations are often solved to find the values of variables that make the statement true.

Understanding the difference between expressions and equations within mathematical statements is crucial, as it helps to effectively interpret and solve mathematical problems.