December 28, 2020 1 min to read

Sum and Difference Problem

Bài toán tổng hiệu

• Key Tools

  If the sum of two variables is \(S\) and their difference is \(D\), then the larger variable equals \[L={{S+D}\over 2} \] and the smaller variable equals \[Sm={{S-D}\over 2}\]

• General Theory

  An equation with two variables \(x,y\):\[ax+by=c \tag{1} \] has infinitely many real roots. Therefore, in order to reduce the number of possible cases, we often need another equation\[dx+ey=f \tag{2} \] To solve this system of equation, we first need to bring them to one equation with one variable.

  This can be done by multiplying (1) with d and multiplying (2) with a, then we have the following function of equations\[ \begin{cases}adx+bdy=cd \\ adx+aey=af\end{cases}\]

  This apparently yields \(bdy-aey=cd-af\), which is a solvable equation with one variable. <\p>

• Other takeaways

  1. In general, we need \(n\) equations to find \(n\) unknown variables, so always find the equations first.

  2. Sum and Differences are not limited to number, but can include fractions, such as the parts of the road covered by two vehicles.

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