Equivalent Equations
Equivalent equations are mathematical expressions or equations that have different forms or representations but have the same solution or solutions
Equivalent equations are mathematical expressions or equations that have different forms or representations but have the same solution or solutions. In other words, if two equations are equivalent, their solutions are identical. Changing the format or the arrangement of an equation does not alter the solutions of the equation.
There are several ways in which equations can be made equivalent. Here are a few common methods:
1. Addition or Subtraction: Adding or subtracting the same value on both sides of an equation maintains the equivalence of the equation. For example, 2x = 5 and 2x + 3 = 5 + 3 are equivalent equations.
2. Multiplication or Division: Multiplying or dividing both sides of an equation by the same nonzero value does not change the solutions of the equation. For instance, 3x = 9 and (1/3)(3x) = (1/3)(9) are equivalent equations.
3. Distributive Property: Using the distributive property of multiplication over addition or subtraction allows us to simplify equations while keeping their solutions the same. For example, 2(x + 3) = 10 and 2x + 6 = 10 are equivalent equations.
4. Combining Like Terms: Combining or simplifying like terms on both sides of an equation maintains its equivalence. For instance, 4x + 2 = 3x + 5 and x = 3 are equivalent equations.
It is important to note that not all equations can be made equivalent using these methods. Some equations may require additional algebraic manipulations or transformations to become equivalent. Additionally, while equivalent equations have the same solutions, they may not necessarily have the same number of solutions.
Solving equations using equivalent equations can be beneficial when simplifying or rearranging equations to make them easier to solve. By applying these methods, you can transform an equation into a different but equivalent form without altering the solutions.
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