An equation is a statement that shows the equality of two mathematical expressions. An equation always contains an equal sign and has a **left side** and a **right side**.
Examples of equations:
\( 3x + 3 = 2x + 4 \) The left side is \( 3x + 3 \) and the right side is \( 2x + 4 \).
\( 2x + 3y = 2 - 2x \) This is an equation in two variables \( x \) and \( y \).
If we substitute \( x = -3 \) in the equation \( 2x + 8 = -2x - 4 \), we obtain:
Left side: \[ 2x + 8 = 2(-3) + 8 = -6 + 8 = 2 \]
Right side: \[ -2x - 4 = -2(-3) - 4 = 6 - 4 = 2 \]
Since both sides equal 2, the substitution gives a true statement. Therefore, \( -3 \) is the solution or root of the equation \( 2x + 8 = -2x - 4 \). The set of all solutions is called the solution set.
To solve an equation means to find all its solutions.
Two equations are equivalent if they have exactly the same solutions.
Example: The following equations all have solution \( x = 0 \), so they are equivalent:
\( -3x + 2 = x + 2 \)
\( -3x = x \)
\( x = 0 \)
If we add the same number or expression to both sides of an equation, the solution set does not change.
\[ \text{If } A = B,\ \text{then } A + C = B + C. \]
Example:
The equations
\( 2x + 3 = 5 \)
and
\( 2x + 3 + (-3) = 5 + (-3) \)
both have the same solution \( x = 1 \).
If we multiply both sides of an equation by the same nonzero number or expression, the solution set does not change.
\[ \text{If } A = B,\ \text{then } C \cdot A = C \cdot B, \quad C \ne 0. \]
Example:
The equations
\( \dfrac{x}{2} = 4 \)
and
\( 2 \cdot \dfrac{x}{2} = 2 \cdot 4 \)
both have the same solution \( x = 8 \).