Equations

Simultaneous Equations

  • Simultaneous equations are a set of equations containing multiple variables.
  • Simultaneous equations are solved by finding the values of the variables that satisfy all the equations in the set simultaneously.
  • These can be solved using various methods, including graphically, by substitution, or by elimination.
  • When solving graphically, the point where the graphs intersect gives the solution.
  • When using substitution, one equation is rearranged to express one variable in terms of the others, then this expression is substituted into the other equations.
  • In the method of elimination, the equations are manipulated to cancel out one of the variables, allowing the other variable to be solved.

Quadratic Equations

  • Quadratic equations are of the form axe^2 + bx + c = 0, where a, b, and c are constants.
  • A quadratic equation has zero, one or two distinct solutions, known as roots.
  • These roots can be found using the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a.
  • The term under the square root in the formula, b^2 - 4ac, is called the discriminant. It can determine the number of roots:
    • If the discriminant is positive, there are two distinct roots.
    • If it’s zero, there’s one repeated root.
    • If negative, there are no real roots.

Linear Equations

  • Linear equations are of the form y = mx + c, where m is the gradient and c is the y-intercept.
  • To solve a linear equation, isolate y (or x) and use algebra to solve for the unknown.
  • Linear equations can be graphed as a straight line, where the gradient dictates the steepness of the line.
  • You can use this format to find the equation of a line given two points.

Polynomial Equations

  • Polynomial equations are composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and positive integer exponents.
  • To solve polynomial equations, it may be necessary to factorise, use the quadratic formula, or synthetic division.
  • Polynomial division can be used to simplify expressions, factor polynomials, and divide polynomials by other polynomials.

By concentrating on these parts of equations, critical techniques can be learnt that will help in the Algebra section of the Mathematics examination.