Algebraic Invariance Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
What changes and what stays the same when multiplying both sides of 2x=62x = 6 by 3?

Solution

  1. 1
    New equation: 6x=186x = 18. Solution: x=3x = 3 (unchanged).
  2. 2
    The coefficients changed, but the solution set is invariant.

Answer

Coefficients change; solution set is invariant.
Multiplying both sides of an equation by a nonzero constant changes the equation's appearance but preserves its solution set. This invariance is what makes the operation 'legal' in equation-solving.

About Algebraic Invariance

Algebraic properties or quantities that remain unchanged when specific algebraic transformations are applied to an expression or system.

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More Algebraic Invariance Examples

Example 1 medium

The polynomial [formula] can be rewritten as [formula]. What is invariant?

Example 2 hard

Show that the discriminant [formula] is invariant under the substitution [formula] in [formula].

Example 3 easy

When you factor [formula], what is invariant?

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