What does an equation look like with a variable?

Answer 1

A variable is either a symbol for unknown value(s) to be found, or a series/range of values that can be used as input values.

In applied mathematics, the name is normally given by the situation, e.g, v for velocity, m for mass, I for current, μ for friction, A for area etc.

In theoretical maths the variable is conventionally chosen in the series x,y,z etc for the variables being determined and a,b,c etc for input parametres, often acting as constants.

Also n,m are used for discrete (countable) variables and z for complex variables. All can be used with index symbols, e.g. x1, zn.

Reserved symbols like π, e, Σ and ∞ are normally avoided as variables. In the function A=πr2 (circle's area), r and A are variables but π is a constant (approx. 3,14159), and it can be solved as an equation if one of the values is given.

An equation has an equality sign, showing that the formula in the left "balance pan" shall always have the same value as the right one, e.g. x2-2x=sin ex

That is the meaning of "equation", that left equals right.

If there are more than one variable present, it can either be an equation with multiple variables, which normally (the linear case) requires a system with the same amount of equations as unknown variables, e.g. { 2x+y=4 & 3x+1=2y } which has the solution { x=1 & y=2 }

Or it can be a function, i.e. that one variable depends on a formula including another variable, e.g. y=4x-12, p(x)=sin2x. These can be drawn as a graph for the range of x values.

Such a relation can be more complex too, perhaps not separable, e.g. sin(x+y)=x+ey. Another example is x2+y2=1, which is the function of a circle.

Linear equations have exactly the same amount of solutions (roots) as its degree (highest power), including multiple roots, e.g. x3-2x+1=0 has 3 roots by rule. p4=0 has 4 roots at p=0, z2=-1 has 2 complex roots z=±i

Non-linear equations may have any number of roots, zero to infinity, e.g. √x=4 has one solution (x=16), ex=1 has one solution (x=0), sin x=0 has infinite solutions (x=nπ where n={0,±1,±2 etc}) but sin x=2 has no solution.

Thus a valid linear one-variable equation of the first degree has exactly one solution and might look like this: 4x-5(2x+1)=3-7x+2(1-2x) which has the root x=2.

Examples of invalid "equations" could be 1+x=2+x or 1/x=0 , of which neither has a solution.