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How to Solve Linear Relationships

Linear relationships or linear equations are simple functions where y is dependent on some function f (x), or y = f (x), that obey the following rules: f(x) has one or two variables; no variables in f (x) is raised to a power greater than 1 and graphically plotting a linear function will result in a straight line. To solve linear functions, you mathematically manipulate the function to get y on the left side of the equal sign and the portion of the equation which is a function of x on the right side of the equal sign.

Instructions

- 1
  Write out the beginning linear equation clearly. As an example, choose the equation 5x + 7y = 20
- 2
  Get the equation in the form y = f (x) where y is a function of x. Using the example:
  
  5x + 7y = 20
  
  Subtract 5x from both sides to get 7y on the left side by itself
  
  5x + 7y - 5x = 20 - 5x
  
  5x cancels on the left side leaving
  
  7y = 20-5x
  
  Divide both sides by 7 to get y on the left side by itself
  
  7y/7 = (20-5x)/7
  
  7 divided by 7 is 1 leaving just y on the left side
  
  y = (20-5x)/7
  
  This shows y as a function of x where, when x changes, y changes.
- 3
  Check your work by substituting the value of y into the original equation to see if it checks out.
  
  5x + 7y = 20
  
  Substituting y = (20-5x)/7 in this equation
  
  5x + 7 [(20-5x)/7] = 20
  
  Rewriting for clarity
  
  5x + 7/7 [(20-5x)] = 20
  
  7/7 is 1, leaving
  
  5x + 20 - 5x = 20
  
  5x cancels on the left side leaving
  
  20 = 20
  
  The left side equals the right side. This checks out.

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