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In addition to the revision notes for Variables, Coefficients and Constants. First Order Equations with One Variable on this page, you can also access the following Equations learning resources for Variables, Coefficients and Constants. First Order Equations with One Variable
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
|---|---|---|---|---|---|---|
| 9.1 | Variables, Coefficients and Constants. First Order Equations with One Variable |
In these revision notes for Variables, Coefficients and Constants. First Order Equations with One Variable, we cover the following key points:
By definition, a mathematical sentence, also called a mathematical statement, is a sentence written by using math symbols and which can be identified as either true or false.
Open sentences are mathematical sentences where the value of which is not given. All algebraic expressions are open sentences. On the other hand, if the value of a mathematical sentence is given, it is a close sentence. This value may be true or false. If true, the close sentence involved represents an equation.
An equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions (open sentences) that have the same value. When combined, these two equal open sentences form a single closed sentence.
The difference between closed sentences and equations is that closed sentences may be either true or false while equations are supposed to be always true.
Like in algebraic expressions, every equation contains variables, coefficients and constants.
Variables are unknown numbers expressed by means of letters; for example x, y, z, a, b, c, m, n, etc. The ultimate goal when dealing with an equation is to calculate the value of their variables.
Coefficients are numbers that multiply (or divide) variables and that precede them. It is a norm that if during calculations a coefficient appears after the variable, the equation is rearranged so that all coefficient precede their corresponding variable.
Constants are "free" numbers that are not associated with any variable. They usually appear after the equal sign (giving the value of equation), but we may see constants appearing in the side that contains variables as well, especially in the original form of equations (before making any operation). The correct value of the constant is obtained when there is a single number appearing on the same side with variables. If the constant is on the other side of the equation, it must change direction. This process is associated with a change in the constant sign.
Any part of the equation containing the product of a coefficient and one or more variables (they may also be in a certain power) is called a term. Equation terms are separated from each other by 'plus' or 'minus' symbols. This means addition and subtraction act as separators of any equation terms, which on the other hand bear the sign preceding them. The constant also represents a separate term in an equation.
The regular form of all equations is
where E(x, y, z) is the equation written in terms of variables x, y, z, etc.
The simplest type of equation is the first - order equation with one variable. As the name implies, such equations have a single variable, which is in the first power.
Solving first - order equations with one variable means isolating the variable and expressing it in terms of the other elements of the equation (numbers and non - variable letters).
The letter that represents more frequently the variable in equations with one variable is 'x'. Therefore, the general formula of the first - order equation with one variable is
where a and b are numbers (a is a coefficient and b is a constant).
Solving a first - order equation with one variable means calculating the value of its variable. In math language, this process is called "finding the root of the equation". This has nothing to do with the concept of roots
The solution of the first - order equations is
When solving any first - order equation with one variable, the following transformations can take place:
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