Solutions of Equations

Overview

Solving equations is one of the most essential aspects in mathematics. Equations act as a tool through which mathematical statements can be verified for different values.
It’s vitour to familiarise with identities and inequalities in the context of equations, as they form a significant part of equation-solving.

Equations can be categorised into several kinds, including linear, quadratic, cubic, and more complex forms.
For each type of equation, various methods may be used to deduce their solutions. Recognising the form of an equation paves the path for its solving process.

An identity is an equation that is true for all values for which both sides are defined.
In mathematical proofs, identities are often leveraged to demonstrate the equality between two mathematical expressions.
Typical examples of identities include trigonometric identities, quadratic identities and equivalent expressions.

An inequality is a statement that one quantity is less than, greater than, less than or equal to, or greater than or equal to another quantity.
It can be utilised to define a range of solutions rather than a specific solution, as is the case with equations.
Basic principles in solving inequalities involve manipulation akin to equation solving, however, the direction of the inequality must be cautiously handled during these operations.

The solution of an equation is a set of values which, when substituted into the equation, make it a true statement.
Different techniques exist for solving different equations, such as factorising, completing the square, and using the quadratic formula for quadratic equations.
For more complex equations, these techniques can be combined, or other methods such as iteration or algebraic division may be needed.

A frequent mistake is the disregard of possible multiple solutions. An equation may have more than one solution, especially in cases of quadratic and cubic equations.
Another error is violation of the rules of inequalities during the solving process. Remember, the inequality sign flips when both sides are multiplied or divided by a negative value.
Precise transcription of mathematical statements into equations or inequalities is crucial. Misinterpretation of the relationship between quantities can lead to incorrect equations or inequalities.

Challenge yourself with a variety of practice problems involving different forms of equations, identities and inequalities. Experiment with various solution methods and ensure a solid understanding of the principles behind them. Continuous practice will enhance your abilities in handling complex problems.