Division Property of Equality
The Division Property of Equality states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. For example, if 6x = 24, dividing both sides by 6 gives x = 4. This property is one of the fundamental tools for solving linear equations in algebra. In Openstax Elementary Algebra 2E, students use the Division Property of Equality to isolate variables and solve equations with multiplication.
Key Concepts
Property For any numbers $a$, $b$, and $c$, and $c \neq 0$, if $a = b$, then $\frac{a}{c} = \frac{b}{c}$. When you divide both sides of an equation by any non zero number, you still have equality. The goal in solving an equation is to 'undo' the operation on the variable. If a variable is multiplied by a number, we divide both sides by that number to 'undo' the multiplication.
Examples To solve $9x = 72$, we divide both sides by 9. This gives $\frac{9x}{9} = \frac{72}{9}$, which simplifies to $x = 8$. To solve $ 6y = 48$, we divide both sides by $ 6$. This gives $\frac{ 6y}{ 6} = \frac{48}{ 6}$, which simplifies to $y = 8$. To solve $5p = 12$, we divide both sides by 5. This gives $\frac{5p}{5} = \frac{ 12}{5}$, which simplifies to $p = \frac{12}{5}$.
Explanation Think of this as fair sharing. If two sides of an equation are balanced, dividing both by the same non zero amount keeps them balanced. This is how we isolate a variable that's being multiplied by a number.
Common Questions
What is the Division Property of Equality?
The Division Property of Equality states that for any numbers a, b, and c where c is not zero, if a equals b, then a divided by c equals b divided by c. Dividing both sides of an equation by the same non-zero number keeps the equation true.
How do you use the Division Property of Equality?
When a variable is multiplied by a number, divide both sides by that number to isolate the variable. For example, to solve 5x = 20, divide both sides by 5 to get x = 4.
Why must you divide by a non-zero number?
Division by zero is undefined in mathematics. Dividing any number by zero has no meaningful result, so the Division Property of Equality only applies when dividing by a non-zero value.
What is the difference between the Division Property and Multiplication Property of Equality?
The Division Property divides both sides by the same number, while the Multiplication Property multiplies both sides by the same number. Both are used to isolate variables but apply to different equation forms.
When do students use the Division Property of Equality?
Students use this property whenever they solve equations where a variable is being multiplied by a coefficient. It is introduced in elementary algebra and remains a core technique throughout all levels of mathematics.