# Can single variable equations always be solved using basic math principles?

• I
• Fascheue
In summary, not all equations with a single variable can be solved. Some equations, like polynomials of degree five or higher, cannot be solved in terms of elementary functions. However, they can be solved numerically using algorithms like the Newton-Raphson method. The concept of "solving" also depends on the definition and whether complex numbers are allowed.
Fascheue
Can all equations with a single variable be solved?

5 = (x)/(1+x^7)

How can something like this be solved for example. Sometimes I have equations with one variable and I can’t see any way to solve them with the knowledge of math that I currently have. Is it just that a higher level of math is required to solves problems like the one above or am I missing something?

Fascheue said:
Can all equations with a single variable be solved?
No. ##y=xe^x## for example can't.
5 = (x)/(1+x^7)
Not sure about this one. Polynomials up to degree four can be solved in a closed form. Equations of higher degrees like this one can only sometimes be solved, depending on the given case. I haven't checked this one though.
How can something like this be solved for example.
Numerically by algorithms like the Newton-Raphson method.
Sometimes I have equations with one variable and I can’t see any way to solve them with the knowledge of math that I currently have. Is it just that a higher level of math is required to solves problems like the one above or am I missing something?
It is a bit of high level mathematics (Galois theory) to see why from degree five on, polynomials cannot be generally solved anymore, and it is not quite trivial to check a certain polynomial.

Fascheue
It depends what you mean by "solved". If you mean, "write down the answer in terms of elementary functions", then the answer is no, as fresh_42 said. But let's take his example of $y=x \exp(x)$. The solution is $x=W_n(y)$, where Wn is the ProductLog function. This is a tabulated function where you can look up the values, so in this sense it is no different from sin(x). So do you consider the problem "solved"? If by "solved" you mean "find the number(s) for which the equation is satisfied to a sufficient degreee of accuracy", then the solutions for any equation of this type can be solved by numerical methods like Newton-Raphson, if a solution exists.

FactChecker and Fascheue
Are you allowing complex numbers? Complex numbers allow solutions that are not real. Any polynomial can be factored to factors of order 1 and 2 with real numbers. The order 1 factors give a solution and order 2 factors give 2 conjugate complex solutions. Your example can be manipulated to an equivalent polynomial problem, 5 (1 + x⁷) - x = 0, with x=-1 (and some complex numbers) disallowed.

## 1. What is a single variable equation?

A single variable equation is an algebraic expression that contains only one variable. It is used to represent a relationship between two quantities, with the goal of solving for the value of the variable.

## 2. How do you solve a single variable equation?

To solve a single variable equation, you must isolate the variable on one side of the equation and simplify the other side. This can be done by using inverse operations, such as addition, subtraction, multiplication, and division.

## 3. What is the order of operations when solving a single variable equation?

The order of operations when solving a single variable equation is the same as in regular algebra. You must perform operations in the following order: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

## 4. Can you have a single variable equation with fractions?

Yes, a single variable equation can contain fractions. To solve it, you must first get rid of the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This will result in an equation without fractions that can be solved using the regular order of operations.

## 5. What is the difference between an equation and an expression?

An equation is a mathematical statement that shows the equality between two quantities, while an expression is a mathematical phrase that may contain numbers, variables, and operations. An equation must be solved to find the value of the variable, while an expression can simply be simplified.

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