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21.5 Chapter summary

  • When working with integers, pay a special attention to operations with negative numbers. For example, when you need to substitute \(x = - 10\) into the equation \(y = x^{2}\), make sure you use brackets around the input value: \(y = ( - 10)^{2} = 100\).
  • If the formula has fractions and integers, look out for any sign change after substituting a negative number in the numerator or denominator. For example, if \(x = - 2\) and \(y = - \frac{2x}{6}\), then

    \[\begin{align} y &= - \frac{2( - 2)}{6} \\ &= - \frac{- 4}{6} \\ &= - \left( - \frac{2}{3} \right) \\ &= + \frac{2}{3} \\ &= \frac{2}{3} \end{align}\]
  • The table of values can help you to sketch graphs and look for patterns in a relationship.
  • All the pairs \((x;y)\) of corresponding \(x\) and \(y\) values are called the solutions of the given equation. This is because when one value is substituted into the equation, it holds true for the second value in the same pair.
  • Some formulas have more than two variables. When there are three variables, we need to find a “triplet” \((x;y;z)\) that satisfies the relationship.