Simultaneous Equations Definition

• Simultaneous equations are two or more multivariable equations that can be solved at the same time because their definitions are related to each other. A set of equations might include two equations, with two variables each. The answer of each equation is dependent on the other, and there is more than one variable per equation, so it's not as easy as solving one equation and then moving on to solve the other. An example of simultaneous equations would be 5x + 3y = 15 and 8x + 2y = 24.
  
  

Substitution Method

• The substitution method of solving simultaneous equations is the easiest way to solve sets with two equations of two variables ("x" and "y") each. The method involves setting one of the equations equal to "y," plugging that value in for "y" in the other equation and solving for the answer of "x." Plug the answer for "x" back into the simplified first equation and solve for "y." Double-check the answers by placing the "x" and "y" values into the original equations to see if the equations remain true.
  
  

Simple Example

• Solve the simultaneous equations 6x + 2y = 12 and y = x + 2. The second equation is already set equal to "y," so plug the value into the first equation for the variable: 6x + 2(x + 2) = 12. Distribute the 2 through the parentheses: 6x + 2x + 4 = 12. Combine like terms: 8x + 4 = 12. Subtract 4 from both sides: 8x = 8. Divide 8 from both sides: x = 1. Substitute 1 for "x" in the other equation to solve for "y": y = 1 + 2 = 3.
  
  

Complex Example

• Solve the simultaneous equations 3x + 6y = 24 and 10x + 5y = 15. Work on isolating the "y" in the second equation. Subtract both sides by 10x: 5y = 15 - 10x. Divide 5 from each side: y = 3 - 2x.
  
  Plug this value in for the "y" in the other equation; 3x + 6(3 - 2x) = 24. Distribute the 6 through the parentheses: 3x + 18 - 12x = 24. Combine like terms; 18 - 9x = 24. Subtract 18 from both sides: -9x = 6. Divide -9 from both sides: x = -6/9 or -2/3.