Lesson 1: Solving Linear Equations I

Property

Addition and Subtraction Properties of Equality:
If the same quantity is added to or subtracted from both sides of an equation, the solution is unchanged.
In symbols, If $a = b$, then $a + c = b + c$ and $a - c = b - c$.
Multiplication and Division Properties of Equality:
If both sides of an equation are multiplied or divided by the same nonzero quantity, the solution is unchanged.
In symbols, If $a = b$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$, $c \neq 0$.

Examples

• If you have the equation $x - 5 = 10$, you can add 5 to both sides to get $x - 5 + 5 = 10 + 5$, which simplifies to $x = 15$.
• For the equation $4y = 28$, you can divide both sides by 4 to get $\frac{4y}{4} = \frac{28}{4}$, which simplifies to $y = 7$.
• If $z + 9 = 12$, you can subtract 9 from both sides to get $z + 9 - 9 = 12 - 9$, which simplifies to $z = 3$.

Explanation

To keep an equation balanced, you must perform the exact same operation on both sides. Whatever you add, subtract, multiply, or divide on one side, you must do to the other side too.

Property

To solve an equation, we isolate the variable. The process is:

1. Ask yourself: Which operation has been performed on the variable?
2. Perform the opposite operation on both sides of the equation to maintain balance.
3. Check your solution by substituting it into the original equation.

Examples

• To solve $x - 5 = 12$, we undo the subtraction by adding 5 to both sides: $(x - 5) + 5 = 12 + 5$, which simplifies to $x = 17$.

• To solve $y + 9 = 20$, we undo the addition by subtracting 9 from both sides: $(y + 9) - 9 = 20 - 9$, which simplifies to $y = 11$.