Solving for Variables on Both Sides
Solve equations with variables on both sides in Grade 9 algebra: eliminate the variable term from one side using addition or subtraction, then isolate the remaining variable with inverse operations.
Key Concepts
Property If the variable being solved for is on both sides of the equation, the first step is to eliminate the variable on one side or the other.
Examples To solve $5y + 3b = 2y + 9b$ for y: $3y + 3b = 9b$, then $3y = 6b$, so $y = 2b$. To solve $10k a = 4k + 5a$ for k: $6k a = 5a$, then $6k = 6a$, so $k = a$. To solve $8p 2z = 3p + 8z$ for p: $5p 2z = 8z$, then $5p = 10z$, so $p = 2z$.
Explanation What happens when your target variable shows up to the party on both sides of the equation? You have to play matchmaker! Your first move is to use addition or subtraction to get all the terms with that variable to hang out together on one side. Once they're grouped, you can combine them and then proceed with isolating your chosen variable. It's cleanup time!
Common Questions
What is the strategy for solving equations with variables on both sides?
Choose one side to collect all variable terms. Add or subtract the variable term from one side to move it to the other. Then treat the resulting equation like a standard one-sided equation and isolate the variable.
How do you solve 5x + 3 = 2x + 12?
Subtract 2x from both sides: 3x + 3 = 12. Subtract 3: 3x = 9. Divide by 3: x = 3. You can collect variables on either side; choose the side that avoids negative coefficients.
What does it mean when variables on both sides cancel out completely?
If all variables cancel and you get a true statement (like 5 = 5), there are infinitely many solutions. If all variables cancel and you get a false statement (like 3 = 7), there is no solution.