Equations with Variables on Both Sides
Learn how to solve equations where the variable appears on both sides of the equals sign.
Definition
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Worked Examples
Solve:
Identify variable terms on both sides
Left: , Right: → Variables on both sides
Subtract from both sides
→
Add to both sides
→
Divide both sides by
→
Check the solution
gives \checkmark → Solution verified!
Answer:
Common Mistakes
Subtracting variables incorrectly: instead of
Why it's wrong: When combining like terms, only the coefficients are subtracted: .
Correct: , not . The variable stays!
Forgetting to perform operations on BOTH sides
Why it's wrong: An equation is like a balance scale. If you only change one side, it becomes unbalanced (and wrong).
Correct: Always write out both sides:
Sign errors when moving terms: becoming
Why it's wrong: When a term crosses the equals sign, its sign changes. But stays positive if it's not moving.
Correct: only if you're subtracting from both sides.
Not distributing before collecting like terms
Why it's wrong: You cannot combine terms inside and outside parentheses until you distribute.
Correct: Always distribute first: , then collect like terms.
Interactive Visual
Balance Scale
Solution: x = 4
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Practice Problems
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Why It Matters
- Comparing costs: When does renting become cheaper than buying?
- Break-even analysis: At what point do two options cost the same?
- Distance problems: When do two objects meet if traveling toward each other?
- Science: Balancing chemical equations, physics formulas
Real World Applications
Comparing Phone Plans
Phone companies offer different pricing structures. Equations help find the break-even point.
Example:
Plan A costs 20 euros plus 0.10 euros per minute. Plan B costs 10 euros plus 0.15 euros per minute. When do they cost the same?
Plan A: 20 + 0.10m euros. Plan B: 10 + 0.15m euros, where m = minutes used.
At how many minutes do both plans cost the same?
Step 1: Write the mathematical expression
Set the costs equal:
Catching Up in a Race
When two runners start at different positions or speeds, we can calculate when they'll meet.
Example:
Runner A starts 50 meters ahead but runs at 6 m/s. Runner B runs at 8 m/s. When does B catch A?
Position A: 50 + 6t meters. Position B: 8t meters (t = seconds).
After how many seconds does Runner B catch Runner A?
Step 1: Write the mathematical expression
Set positions equal:
Temperature Conversion
Celsius and Fahrenheit scales meet at a specific temperature.
Example:
At what temperature does Celsius equal Fahrenheit? Use .
If , then:
At what temperature are Celsius and Fahrenheit equal?
Step 1: Write the mathematical expression
Solve:
Key Takeaways
- 1Equations with variables on both sides require collecting variable terms on one side first
- 2Use inverse operations: subtract/add variable terms, then subtract/add constants
- 3If there are parentheses, distribute before collecting like terms
- 4Always check your solution by substituting back into the original equation
- 5Whatever you do to one side, you must do to the other side
Frequently Asked Questions
Glossary
- Like terms
- Terms with the same variable raised to the same power (e.g., and are like terms)
- Coefficient
- The number multiplied by a variable (e.g., in , the coefficient is )
- Inverse operation
- An operation that undoes another (addition/subtraction, multiplication/division)
- Identity equation
- An equation that is true for all values of the variable (e.g., )
- Contradiction
- An equation with no solution because it's never true (e.g., )
Formula Card
Solving Strategy
Steps to solve equations with variables on both sides