Maintaining Balance: The Properties of Equality
Maintaining Balance: The Properties of Equality explains that an equation remains true when the same quantity is added, subtracted, multiplied, or divided on both sides. Taught in Yoshiwara Elementary Algebra Chapter 1: Variables, these addition, subtraction, multiplication, and division properties of equality are the fundamental rules that justify every algebraic equation-solving step for Grade 6 students. The equation is treated like a balanced scale that must remain in balance.
Key Concepts
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.
Common Questions
What are the properties of equality?
The properties of equality say you can add, subtract, multiply, or divide the same nonzero quantity to both sides of an equation and it remains true.
Why must you do the same thing to both sides?
Because an equation is like a balance scale. Changing one side without matching it on the other side makes the equation unequal.
What is the addition property of equality?
If a = b, then a + c = b + c for any number c. Adding the same value to both sides preserves equality.
Where are the properties of equality in Yoshiwara Elementary Algebra?
They are introduced in Chapter 1: Variables of Yoshiwara Elementary Algebra.
How do the properties of equality help solve equations?
Each step in solving an equation is justified by one of these properties. For example, subtracting 3 from both sides of x + 3 = 7 uses the subtraction property to get x = 4.