Solving Linear Equations


1. Introduction

These equations are know as linear, because the monomial literal part doesn't have an exponent (for example, 3x can be part of a linear equation, but 3x2 can't because it's quadratic), so represented in a graph appear as a line. This fact assures us that if there is a solution, there can only be one solution (except in special cases were there are infinite solutions). We say if there is a solution because sometimes equations don't have solutions. For example, the equation x = x + 1 (which means a number equals the consecutive number) doesn't have a solution, because this is never true. In reality, this equation is reduced to 1 = 0, which is impossible.

2. Tips for solving the equations

  • If we obtain an impossible equality, there is no solution, like 1 = 0.
  • If we obtain an equality that is always true, whatever value gives a solution, then the solution is all real numbers. For example, if we obtain 0 = 0.
  • When there are denominators and we want to avoid them, we multiply the full equation by the same least common multiple of the denominators. This way when we simplify, they disappear.
  • To remove the parenthesis, we multiply the coefficient in front of it by all the elements it contains. This coefficient can be a negative sign (like –1, the content changes sign), a positive sign (like +1, the content doesn't change) or a positive or negative number or fractions (this number multiplies everything in the parentheses, changing the signs if its negative).
  • When we have a nested parenthesis, a parenthesis inside another parenthesis, we begging removing from outside to the inside. First, we remove the exterior parenthesis (multiplying it's content by the coefficients) and afterwards, we remove the rest the same way: from the most exterior to the interior. Realistically, there isn't any need to follow an order when removing the parenthesis, but it's recommended to follow one when we are learning.