How to Find the Equivalent System of Equations?

You can find an equivalent system of equations by applying operations that don't change the solution set of the original system. The most common methods involve manipulating the equations to create a new system with the same solutions.

Here's a breakdown of how to do it:

Methods to Create Equivalent Systems:

• Replacing an equation by a multiple of itself: This involves multiplying both sides of an equation by a non-zero constant.

  • Example: If you have the equation x + y = 5, multiplying both sides by 2 gives you 2x + 2y = 10. The equation 2x + 2y = 10 is equivalent to x + y = 5.

• Replacing one equation by the sum (or difference) of itself and another equation: This involves adding (or subtracting) a multiple of one equation to another equation.

  • Example: Consider the system:

    x + y = 5
    x - y = 1

    You can replace the first equation with the sum of the two equations:
    (x + y) + (x - y) = 5 + 1, which simplifies to 2x = 6.
    The new system is:

    2x = 6
    x - y = 1

    This system is equivalent to the original system because it has the same solution (x=3, y=2).

• Interchanging the order of the equations: Simply swapping the positions of the equations in the system doesn't change the solution set.

  • Example: The system:

    x + y = 5
    x - y = 1

    is equivalent to:

    x - y = 1
    x + y = 5

• Substitution: While technically not creating a new system, solving one equation for a variable and substituting that expression into another equation helps in solving the system. The process maintains equivalence because each step preserves the solutions.

Why these methods work:

These operations preserve the solution set because they don't introduce any new solutions or eliminate any existing ones. Multiplying an equation by a constant simply scales the relationship defined by the equation. Adding or subtracting equations combines the existing relationships in a way that still satisfies the original conditions.

Example:

Original System:

3x + 2y = 7
x - y = 0

1. Multiply the second equation by -2: -2x + 2y = 0
2. Add this modified second equation to the first equation: (3x + 2y) + (-2x + 2y) = 7 + 0 which simplifies to x + 4y = 7.

Equivalent System:

x + 4y = 7
x - y = 0

Solving either system will give you the same solution: x = 7/5 and y = 7/5.