Equation solving

In mathematics, solving an equation means finding the values that make the equation true. Equations have two parts connected by an equals sign, and often include one or more variables that we don't know yet. These unknown values are called solutions. When we find a solution, we replace the unknown variables with numbers or expressions that make both sides of the equation equal.

For example, in the equation x + y = 2, if we know y is 1, we can solve for x by saying x = 1. This works because 1 + 1 equals 2. Sometimes, equations can have many solutions. In the equation x + y = 2, any pair of numbers that add up to 2, like x = 0 and y = 2 or x = 1.5 and y = 0.5, are solutions.

Solving equations can be done in different ways. We can solve them numerically, meaning we find actual numbers that work. Or we can solve them symbolically, meaning we find expressions that show the relationship between unknowns. Both methods help us understand how different values are connected and are important tools in many areas of math and science.

Overview

In math, solving an equation means finding the values that make it true. An equation usually has two parts connected by an equals sign, and we look for numbers that fit the mystery spots, called unknowns.

For example, in the equation 3x + 2y = 21z, x, y, and z are unknowns. There aren’t just a few answers — there are many possible sets of numbers that work. One simple answer is x = 0, y = 0, z = 0. Other answers include x = 3, y = 6, z = 1 and x = 8, y = 9, z = 2. All these answers lie on a flat surface, or plane, in three-dimensional space.

Solution sets

Main article: Solution set

The solution set of an equation or inequality is all the values that make the equation true. For example, in the equation x² = 2, there are no whole number solutions, but there are two real number solutions: √2 and –√2.

When equations have more unknowns than numbers, there can be infinitely many solutions. These solutions can sometimes be shown as shapes like lines or planes.

Methods of solution

The methods for solving equations depend on the type of equation and the values that the unknowns can take. There are many different kinds of equations, so there are many different ways to solve them. Sometimes, there is no known way to solve an equation, and it may take a long time to find a solution.

For some types of equations, we can use computer programs to help find solutions. But sometimes, we can solve equations using just a pencil and paper. In other cases, we can use methods that are not always guaranteed to work but often do, like trying different values to see what works.

One common way to solve simple equations is by using basic algebra. For example, equations like 8x + 7 = 4x + 35 or (4x + 9)/(3x + 4) = 2 can be solved with algebraic methods. Larger systems of equations can also be solved using algebra, or with special methods for dealing with many equations at once.

For equations with higher powers, like x^4 - 5x^3 + 6 = 0, we can sometimes find exact solutions using special algebraic tricks. But for even higher powers, we often need to use numerical methods, which give us very close answers instead of exact ones.

We can also solve equations by using inverse functions. For example, if we have an equation like h(x) = c, we can sometimes find the solution by using the inverse of the function h. This works for functions like square roots, logarithms, and others.

Another method is to rewrite the equation so that it can be broken into simpler parts. For example, an equation like tan(x) + cot(x) = 2 can be rewritten and solved by finding the values of x that make the equation true.

For more complicated equations, we can use numerical methods, which are like step-by-step guesses that get closer and closer to the right answer. These methods are useful when simple algebra does not work.

Main article: Solving polynomial equations

See also: System of polynomial equations