Systems of Equations
A system of equations is a set of two or more equations considered at the same time. Solving a system means finding the values of the unknown variables, called common solutions, that satisfy all the equations simultaneously.
Here is an example of a system made of two equations with two unknowns:
$$ \begin{cases} 2x + y = 3 \\ \\ 6x - 3y = 3 \end{cases} $$
The equations are written one beneath the other and grouped by a large curly brace on the left. A system may contain any number of equations.
Each equation may include one or more unknowns, and it is not necessary for every equation in the system to contain the same variables.
Note. A system is said to be in standard form if each equation is written in the form ax + by = c, where a and b are real coefficients and c is a constant term. In the system $$ \begin{cases} 2x + y = 3 \\ \\ 6x - 3y = 3 \end{cases} $$ both equations are already in standard form, so the system as a whole is in standard form.
How to solve a system of equations
Solving a system of equations means finding the values of the unknowns that satisfy every equation in the system at the same time.
The solution set of the system is the intersection of the solution sets of each individual equation.
A linear system of equations can be classified as follows:
- consistent
if it has one or a finite number of solutions - inconsistent (impossible)
if it has no common solutions - dependent
if it has infinitely many solutions
Note. Two or more systems of equations are called equivalent systems if they share the same set of solutions.
There are several methods for solving a system of equations:
Every method leads to the same solution, so the choice is largely a matter of preference.
However, depending on the structure of the system, one method may lead to quicker or simpler calculations than another.
A practical example
Consider the following system with two unknowns, x and y:
$$ \begin{cases} 2x + y = 3 \\ \\ 6x - 3y = 3 \end{cases} $$
We will solve it using the substitution method.
First, isolate y in the first equation:
$$ \begin{cases} y = 3 - 2x \\ \\ 6x - 3y = 3 \end{cases} $$
Now substitute y = 3 - 2x into the second equation:
$$ \begin{cases} y = 3 - 2x \\ \\ 6x - 3(3 - 2x) = 3 \end{cases} $$
The second equation now contains only one unknown. We simplify and solve for x:
$$ \begin{cases} y = 3 - 2x \\ \\ 6x - 9 + 6x = 3 \end{cases} $$
$$ \begin{cases} y = 3 - 2x \\ \\ 12x = 3 + 9 \end{cases} $$
$$ \begin{cases} y = 3 - 2x \\ \\ 12x = 12 \end{cases} $$
$$ \begin{cases} y = 3 - 2x \\ \\ x = \frac{12}{12} \end{cases} $$
$$ \begin{cases} y = 3 - 2x \\ \\ x = 1 \end{cases} $$
Now that we have x = 1, we substitute this value back into the first equation:
$$ \begin{cases} y = 3 - 2 \cdot 1 \\ \\ x = 1 \end{cases} $$
This gives us the value of y:
$$ \begin{cases} y = 1 \\ \\ x = 1 \end{cases} $$
The system has exactly one solution, x = 1 and y = 1. Therefore, the system is determined.
Check. To verify the result, substitute x = 1 and y = 1 into the original equations. If both are satisfied, the solution is correct. $$ \begin{cases} 2x + y = 3 \\ \\ 6x - 3y = 3 \end{cases} $$ Testing our values: $$ \begin{cases} 2 \cdot (1) + (1) = 3 \\ \\ 6 \cdot (1) - 3 \cdot (1) = 3 \end{cases} $$ $$ \begin{cases} 2 + 1 = 3 \\ \\ 6 - 3 = 3 \end{cases} $$ $$ \begin{cases} 3 = 3 \\ \\ 3 = 3 \end{cases} $$ Since both equations check out, the solution is correct.
The degree of a system of equations
The degree of a system of equations is defined as the product of the degrees of its individual equations.
A system is called a linear system if all its equations are first-degree equations.
A system is of second degree if it contains at least one second-degree equation, while the others are first-degree.
Example 1
This system consists of three equations:
$$ \begin{cases} 2x + y = 3 \\ \\ 4x + y = 5 \\ \\ 6x^2 - 3y = 2 \end{cases} $$
The first two equations are first degree, and the third is second degree.
Therefore, the system is of second degree, since:
$$ 1 \cdot 1 \cdot 2 = 2 $$
Example 2
This system also consists of three equations:
$$ \begin{cases} 7x + 2y = 1 \\ \\ 3x^2 + 4y = 8 \\ \\ 2x^2 - 4y = -2 \end{cases} $$
The first equation is first degree, and the other two are second degree.
The system is therefore of fourth degree, since:
$$ 1 \cdot 2 \cdot 2 = 4 $$
Example 3
This system consists of two equations:
$$ \begin{cases} 2x^3 + y^2 = 3 \\ \\ 6x^2 - 3y = 2 \end{cases} $$
The first equation is third degree, and the second is second degree.
$$ 2 \cdot 3 = 6 $$
The system is therefore of sixth degree.
And so on.
