Solving Equations

Key Principles in Solving Equations

Types of Equations

  • Linear equations are of the form ax + b = c whereas quadratic equations are of the form ax^2 + bx + c = 0.

  • Simultaneous equations involve a set of equations with multiple variables that are solved together.

  • Inequalities are statements about the relative sizes of two expressions that use signs like ‘<’, ‘>’, ‘≤’, and ‘≥’.

Strategies to Solve Linear Equations

  • To solve linear equations, aim to isolate the variable on one side of the equation.

  • Use the BIDMAS order of operation rules to progressively isolate the variable.

  • For example, in the equation 3x + 2 = 8, subtract 2 from both sides first (follows BIDMAS) to get 3x = 6, then divide by 3 to get x = 2.

Approaches for Quadratic Equations

  • Quadratic equations are typically solved by factoring, completing the square, or using the quadratic formula.

  • Factoring involves writing the quadratic as a product of two brackets, i.e ax^2 + bx + c = (dx + e)(fx + g) = 0.

  • Completing the square transforms the quadratic into a perfect square trinomial.

  • The quadratic formula is x = [-b ± sqrt(b^2 - 4ac)] / 2a, which can be used when factoring or completing the square is not feasible.

Solving Simultaneous Equations

  • Simultaneous equations can be solved by substitution, elimination, or graphical methods.

  • Substitution involves substituting the expression of one variable from one equation into the other.

  • Elimination balances the coefficients to subtract or add equations and eliminate one of the variables.

  • Graphical methods involves plotting the two equations on a graph; where they intersect is the solution.

Working with Inequalities

  • Solve inequalities just like equations, but remember, if you multiply or divide by a negative number, switch the inequality sign.

  • The solution of an inequality is often a range of values rather than a single value.

Checking and Practising

  • Once you’ve solved an equation, check your answer by substituting it back into the original equation to see if both sides balance.

  • Regular practice of different types of equations, through solving exercises and past paper questions, improves understanding and efficiency.

Remember, equations provide a framework for understanding how variables are related. An ability to manipulate and solve these equations opens up multidimensional ways of approaching mathematical problems.