Inverse Operations
Use inverse operations in Grade 9 algebra to undo two-step equations: work backward through operations in reverse order—undo the last operation first—to isolate the variable systematically.
Key Concepts
Property If an equation has two operations, use inverse operations and work backward to undo each operation one at a time. To reverse the order of operations, first add or subtract, and then multiply or divide.
Examples To solve $3x 2 = 10$, first add $2$ to both sides to get $3x = 12$. Then, divide by $3$ to find $x = 4$. To solve $4x + 5 = 17$, first subtract $5$ from both sides to get $4x = 12$. Then, divide by $4$ to find $x = 3$. To solve $30x + 90 = 1500$, first subtract $90$ to get $30x = 1410$. Then, divide by $30$ to find $x = 47$.
Explanation Think of it like getting undressed: you untie your shoes before you take them off! To solve an equation, you must reverse the order of operations. First, undo any addition or subtraction to clear out the constants. Then, undo multiplication or division to finally get the variable all by itself.
Common Questions
How do inverse operations work in two-step equations?
Reverse the order of operations. For 3x + 7 = 22: the variable was first multiplied by 3 then added 7. Undo in reverse: subtract 7 first (3x = 15), then divide by 3 (x = 5). Working backward maintains equation balance.
What pairs of operations are inverses of each other?
Addition and subtraction are inverses. Multiplication and division are inverses. Squaring and taking the square root are inverses. To solve, identify which operation was applied last and undo it first.
Why do you apply inverse operations to both sides of an equation?
An equation states two quantities are equal. Performing any operation on just one side destroys the equality. Applying the same inverse operation to both sides keeps the equation balanced while progressively isolating the variable.