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In algebra, a quadratic equation is usually in this form
ax²+bx+c=0
where x represents an unknown or the value we have to find out, and a, b, and c represent known numbers, with a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no ax² term. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant having fix or defined value.
One variable
Frequently the term linear equation refers implicitly to the case of just one variable. This case, in which the name unknown for the variable is sensibly used, is of particular importance, since it offers a unique value as solution to the equation. According to the above definition such an equation has the form
ax+b=0
and, for a ≠ 0, a unique value as solution
x= -b/a
In the case of a=0, two possibilities emerge:
1.b=0 Every value of x is a solution to the equation 0. and
2.b is not equal zero, there is no solution for the equation 0. x+b=0 the equation is said to be inconsistent.
Two variables
In the case of just two variables the indexed variable names x1 and x2 and the respective coefficients a1 and a2 are often replaced, for the convenience of not having to deal with indices, by x, y, a and b, respectively. As a consequence, the so-called constant term, named coefficient b in the above notation, must also be renamed; c suggests itself. A linear equation in two variables is then denoted as
ax+by=c
Any change to such an equation that does not alter the set of solutions, i.e., the set of pairs(x,y) , that satisfy this equation, generates an equivalent equation. It is immediate that changing the involved names (e.g. capitalizing names or using other letters) and also reordering the equation (for example, by moving terms to the other side), does not change this set of solutions, and thus results in an equivalent equation, like,
Ax+By=c with A=a, B=b, C=c
These equivalent variants are sometimes given generic names, such as general form or standard form, but contribute no new concepts.
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