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Solving equations is an essential skill that is an integral part of Algebra. In this lesson, we will learn about some patterns of equations and formulae for solving them, so that instead of deriving the solution from first principles, we can apply these simple formulae to get the answer right away.
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Why is the ability to identify patterns of equations, and solve them by applying formulae important? After all, algebra is about deriving the answer by isolating the unknown variable and equating it to the solution. Consider the case of a generic quadratic equations of the form ax^2 + bx + c = 0. Students are initially taught how to solve such equations by separating b into two parts such that the parts b1 and b2 add up to b, and the ratios a/b1 and b2/c are equal.
That procedure will then lead to the formation of an equation of the form below:
dx(ex + f) + g(ex + f) = 0
which in turn is then simplified to the form:
(dx + g)(ex + f) = 0.
This then gives us the solutions -g/d and -f/e.
Afterwards, we find that this process only works when it is easy to split b up according to the rules. When the required split of b involves fractions and decimals, it is not easy to derive b1 and b2 by trial and error without a lot of effort.
At that point, one may have been taught the derivation of the standard quadratic formula which involves the process of “completing the square”. The end result of this derivation is a formula that is etched deep in most high-schoolers’ brains.
Going forward, when encountering a quadratic equation that needs to be solved, students are expected to use the formula directly rather than going through the process of justifying the use of it by deriving the formula from first principles. In fact, the formula takes the place of going through the process of trying to isolate the unknown variable in order to solve the equation. The formula is a time-saving device that tells us that if and when we go through the process of isolating x (if we actually want to do it that way), we will end up with solutions that are predicted by this formula. The use of the formula is now so widespread that most people who are perfectly capable of solving quadratic equations can not derive the quadratic formula from scratch!
In this lesson, we will deal with four simple types of equations for which one can derive formulae similar to the quadratic formula. Going forward, one can then use these formulae directly instead of either deriving them or going through the steps involved in isolating the unknown variable following first principles. This will not only save us lots of time and effort, but also reduce the probability of errors during the process of isolation itself. Just as the quadratic formula is used widely to solve quadratic equations, the formulae in this lesson should be taught and used widely to solve equations instead of expecting students and others to go through the process of deriving them from first principles.
Note that all the simple forms of equations are provided in their most general form. Many equations may be missing some elements, which makes their identification and classification into one of these general forms a little tricky, but, with practice, it should become easier. As always, the application of the formulae itself will become much easier (and can easily be done mentally) with practice.
All these formulae are derived primarily by using a Vedic Sutra that reads “Paravartya Yojayet”. The literal translation of this sutra is “transpose and adjust”. This is a fairly general phrase that covers pretty much all that we do to solve simple equations.
If your interest is piqued, you can read the full post here. Below is a summary of the equation types and their solution formulae for your reference:
|ax + b = cx + d||x = (d – b)/(a – c)|
|(ax + b)/(cx + d) = p/q||x = (pd – bq)/(aq – cp)|
|(x + a)(x + b) = (x + c)(x + d)||x = (cd – ab)/(a + b – c – d)|
|(ax + b)(cx + d) = (ex + f)(gx + h) Valid only if a*c = e*g||x = (fh – bd)/(ad + bc – eh – fg)|
|p/(x + a) + q/(x + b) = 0||x = (–pb – qa)/(p + q)|
|p/(ax + b) + q/(cx + d) = 0||x = (–pd – qb)/(pc + aq)|
Obviously, identifying the equations by type, and learning to apply the appropriate formula mentally and quickly, comes only with practice. Hope you will spend some time practicing with these different types of equations. As the examples illustrate, one sometimes has to manipulate the given equation in some simple fashion (cross-multiply, move a term from the left hand side to the right hand side or vice versa, etc.) to get it into a form covered by the types above, before the appropriate formula can be applied to the equation. It takes some practice to identify these manipulations also. Good luck, and happy computing!