How to Create Equivalent Equations

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Equivalent equations have the same solution but appear in different formats. There are several categories of equivalent equations. Some equations are true regardless of the choice numerical value; these are called identities. There is one solution, such as 5, for some equivalent equations. In such an event, all other choices are false. For some equivalent equations, however, there is no solution and the equations are known as contradictions. You need to take several steps to form equivalent equations and, if possible, obtain a solution.

Addition and Subtraction

  • Distribute any variables or terms within parentheses. Calculate 3(x-5), for example, to obtain 3x-15.

  • Combine like terms, or terms that contain the same variable on one side of the equal sign, regardless of the numerical coefficient.

  • Isolate the variable, such as x, to solve the equation. Subtract 3 from both sides of the equation x+3=15 to create the equivalent equation x=2.

  • Substitute your answer by replacing the variable with a numerical value in the equation. If x equals 24 in the equation 3x=72, replace the variable with 24. Both sides should equal 72.

Multiplication and Division

  • Multiply both sides of an equation by the same nonzero number to yield an equivalent equation. Multiply both sides by 5, for example, in the equation x/25=2. Solve for x, which is 50.

  • Isolate the variable by dividing both sides by its coefficient. Divide both sides by 3, since you multiply 3 by x in the equation 3x=72. Obtain the solution, which is 24.

  • Perform both multiplication and division operations when dealing with fractions. Divide both sides by 5, then multiply both sides by 2 in the equation (2/5)x=8. Verify that x equals 20.

  • Substitute your answer by replacing the variable with a numerical value, as in Section 1, Step 4.

Tips & Warnings

  • Some equations will have two or more solutions.
  • Check your solutions algebraically, as noted above, or graphically. The graphs of the equivalent equations should intersect at the point x.

References

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