In this tutorial, I'll give you an introduction to the concept of equations. We'll be looking at the meaning and importance of equations and the different types of equations.

So, let's dive in.

What Is an Equation?

An equation is a mathematical statement saying two expressions are equal. The statement consists of two expressions separated by an “equals” symbol. For example, 2+3=5\hspace{0.2em} 2 + 3 = 5 \hspace{0.2em}.

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Here are a few more examples —

  • 3x1=15\hspace{0.2em} 3x - 1 = 15 \hspace{0.2em}
  • 4x+5=3x\hspace{0.2em} 4x + 5 = 3 - x \hspace{0.2em}
  • x2+5x=9\hspace{0.2em} x^2 + 5x = 9 \hspace{0.2em}
  • x+y+z=12\hspace{0.2em} x + y + z = 12 \hspace{0.2em}

Now, we use equations all the time while solving problems. Mostly, to indicate that the expressions carry the same value as we restructure or simplify them.

For example, when adding two fractions, we might write something like this.

15+35=45\frac{1}{5} + \frac{3}{5} \, \, = \, \, \frac{4}{5}

Here we are essentially saying — the first expression is equal to the second expression.

But there’s something else that makes equations such an important topic of study.

What Makes Equations So Useful

The real power of equations lies in how we can use them to find unknowns.

As a simple example, say, we want to find a number that will become 10\hspace{0.2em} 10 \hspace{0.2em} if 6\hspace{0.2em} 6 \hspace{0.2em} is added to it. We can represent that unknown number using x and write an equation like this.

x+6=10x + 6 \, \, = \, \, 10

Then using certain techniques, we can solve for x\hspace{0.2em} x \hspace{0.2em}, and hence find the number we were looking for. This is a trivial example, but I hope you get the idea of how equations can be super useful.

Different Types of Equations

Here are the three most common types of equations you will be studying in algebra.

Each of the types mentioned below belong to the larger category of polynomial equations — equations involving only polynomials.

Linear Equations in One Variable

A first-degree equation containing only one variable is called a linear equation in one variable. For example, 2x+1=8\hspace{0.2em} 2x + 1 = 8 \hspace{0.2em}.

The value of the variable for which the equation becomes true is known as the solution of the equation. The solution of the equation x+3=7\hspace{0.2em} x + 3 = 7 \hspace{0.2em} is 4\hspace{0.2em} 4 \hspace{0.2em} (that is, x=4\hspace{0.2em} x = 4 \hspace{0.2em}).

Of the different types of equations, these are generally the simplest to solve.

Linear Equations in Two/Three Variables

Here's an example of a first-degree equation involving two variables (x\hspace{0.2em} x \hspace{0.2em} and y\hspace{0.2em} y \hspace{0.2em}). In other words a linear equation in two variables.

xy=3x - y = 3

A solution for an equation with multiple variables refers to a set of values for the variables for which equations becomes true.

For example, one solution for the equation above is x=3,y=0\hspace{0.2em} x = 3, \, y = 0 \hspace{0.2em}. Another solution would be x=1,y=2\hspace{0.2em} x = 1, \, y = -2 \hspace{0.2em}.

If we have one linear equation with two or more variables, there are infinitely many solutions.

However, generally when we are working with equations with two or more variables/unknowns, we have a system of equations instead of a single equation. Here's an example.

x+y=6xy=4\begin{align*} x + y \, &= \, 6 \\[1em] x - y \, &= \, -4 \end{align*}

Solving such a system of equations is about finding values for all variables for which each of the equations would be true. Such systems may have one unique solution, infinitely many solutions, or no solution at all.

For example, the system of equations above has a unique solution (x=1,y=5)\hspace{0.2em} (x = 1, \, y = 5) \hspace{0.2em}. So, if we plug these values in the given equations, both give us true equalities.

In general, for there to be a unique solution, a system of equations with n\hspace{0.2em} n \hspace{0.2em} unknowns needs to have n\hspace{0.2em} n \hspace{0.2em} independent equations.

So if there are two unknowns, the system of equations must have two independent equations.

Quadratic Equations

Quadratic equations are second-degree equations in one variable. For example, x2+x2=0\hspace{0.2em} x^2 + x - 2 = 0 \hspace{0.2em}.

Again, solving a quadratic equation means finding the value(s) of the variable for which the equation would be true. The solutions for the example above are 1\hspace{0.2em} 1 \hspace{0.2em} and 2\hspace{0.2em} -2 \hspace{0.2em}.

And with that, we come to the end of this tutorial on equations. Until next time.