Understanding equations with variables on both sides involves interpreting each equation as a statement of equality and preserving that equality while using the distributive property, combining like terms, and applying inverse operations to isolate the variable. The solution set may contain one value, no value, or every value when simplification produces a true or false statement; solutions can be checked by substitution and related to intersections of linear graphs. The scope is one-variable linear equations with rational coefficients, excluding nonlinear equations, variable denominators, and general systems.
An equation with variables on both sides can be solved by simplifying both sides and then moving all variable terms to one side.
Remember: whatever operation you do to one side, do to the other side to keep the equation balanced.
Example:
1. Distribute.
Multiply by each term inside the parentheses:
2. Combine like terms.
On the left, :
3. Move the variable terms to one side.
Subtract from both sides:
4. Isolate the variable.
Add to both sides:
5. Check the solution.
Substitute for in the original equation:
Since the statement is true, the solution is:
When solving, simplify first, collect all variable terms on one side, and then use inverse operations to isolate the variable.
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