Understanding equations with variables on both sides involves interpreting equality as a balance and transforming equivalent linear expressions through the distributive property, combining like terms, and inverse operations while preserving the solution set. Learners identify whether a linear equation has one solution, no solution, or infinitely many solutions, and may connect these cases to intersections or coincident lines; the scope is limited to linear equations with numerical coefficients and does not include nonlinear equations or more advanced parameter-based analysis.
An equation with variables on both sides can be solved by treating both sides like a balanced scale. Whatever operation you perform on one side, perform on the other side too.
Solve:
1. Use the distributive property.
Multiply by each term inside the parentheses:
2. Combine like terms.
Combine and :
3. Move the variable terms to one side.
Subtract from both sides:
4. Move the constant to the other side.
Add to both sides:
5. Check the solution.
Substitute for in the original equation:
Since the statement is true, the solution is:
If all variable terms cancel and a true statement remains, there are infinitely many solutions. If all variable terms cancel and a false statement remains, there is no solution.
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