Solving equations

Solving Equations

Understanding Equations

  • An equation is a mathematical statement where two expressions are equal, often representing a real-world problem.
  • Equations contain variables, constants and arithmetic operations.
  • The goal when solving an equation is to find the values of the variables that make the equation true.

Principle of Solving Equations

  • The fundamental principle of solving equations is the concept of balance. Whatever you do to one side of the equation, you must do to the other, maintaining this balance.
  • Simple linear equations involve balancing the equation to isolate the variable on one side and constants on the other. For example, in the equation 2x = 10, to solve for x, divide both sides by 2 giving x = 5.
  • Equations with exponents, like quadratic equations, require special techniques like factoring, completing the square, or the quadratic formula.

Techniques of Solving Equations

  • Solving linear equations: To solve an equation like 3x + 4 = 13, first isolate the terms with the variable, by subtracting 4 from both sides to get 3x = 9. Then, divide both sides by 3 to solve for x, getting x = 3.
  • Quadratic equations: These are solved by factoring, completing the square or using the quadratic formula. For an equation like x² - 5x + 6 = 0, you can factorise to (x - 2)(x - 3) = 0 and solve for x giving the roots x = 2, 3.
  • Simultaneous equations: These are a set of equations with multiple variables. Solve them by substitution or elimination. For example, with the equations 2x + y = 3 and x - y = -1, you can add the two equations together to eliminate y, and allow you to solve for x.

Checking Solutions

  • After solving an equation, always check the solution by substituting it back into the original equation to check it holds true.
  • Checking solutions ensures the correctness of your answer and verifies if the problem has one solution, multiple solutions, or no solution.

Always remember, understanding the type of equation and being familiar with the correct method to solve it is the key to tackling algebraic problems effectively.