A system of equations is a set of two or more equations that must be satisfied simultaneously.
Solving a system means finding the values of the unknowns that make every equation in the system true at the same time.
Systems of equations are one of the fundamental tools of algebra: they allow us to model simultaneous conditions and arise in an enormous variety of mathematical, geometric, and applied contexts.
From a geometric standpoint, a linear system in two unknowns generally describes the intersection of two lines in the Cartesian plane.
Table of Contents
- What is a system of equations
- General form of a linear system
- Consistent, inconsistent, and dependent systems
- Substitution method
- Elimination method
- Gaussian elimination algorithm
- Linear systems in three unknowns
- Geometric interpretation
- System determinant and Cramer's Rule
- Systems with a parameter
- Nonlinear systems
- Common mistakes
What is a system of equations
Consider two equations in the same unknowns:
\[ \begin{cases} x+y=5\\ x-y=1 \end{cases} \]
Neither equation is to be studied in isolation: we want to find the values of \(x\) and \(y\) that satisfy both equations at the same time.
In this case:
\[ x=3,\qquad y=2 \]
because:
\[ 3+2=5 \]
and simultaneously:
\[ 3-2=1. \]
The ordered pair:
\[ (3,2) \]
is therefore the solution of the system.
In general, a solution of a system is an ordered pair, an ordered triple, or a set of values that makes every equation in the system true.
General form of a linear system
A linear system in two unknowns generally takes the form:
\[ \begin{cases} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2 \end{cases} \]
where:
- \(x\) and \(y\) are the unknowns;
- \(a_1,a_2,b_1,b_2\) are the coefficients;
- \(c_1,c_2\) are the constants (right-hand side terms).
The system is called linear because the unknowns appear only to the first power.
For example:
\[ \begin{cases} 2x+3y=7\\ x-y=4 \end{cases} \]
is a linear system.
On the other hand:
\[ \begin{cases} x+y=5\\ x^2+y^2=13 \end{cases} \]
is not linear, because the term:
\[ x^2 \]
appears.
Consistent, inconsistent, and dependent systems
A system may have:
- exactly one solution;
- no solution;
- infinitely many solutions.
Consistent system
A system is consistent (and independent) when it has exactly one solution.
For example:
\[ \begin{cases} x+y=5\\ x-y=1 \end{cases} \]
has the unique solution:
\[ x=3,\qquad y=2. \]
Inconsistent system
A system is inconsistent when it has no solution.
For example:
\[ \begin{cases} x+y=2\\ x+y=5 \end{cases} \]
is inconsistent, because the same sum cannot equal both \(2\) and \(5\) simultaneously.
Dependent system
A system is dependent when it has infinitely many solutions.
For example:
\[ \begin{cases} x+y=4\\ 2x+2y=8 \end{cases} \]
actually contains two equivalent equations.
Every ordered pair satisfying:
\[ x+y=4 \]
is therefore a solution of the system.
Substitution method
The substitution method consists of isolating one variable in one of the equations and substituting the resulting expression into the other.
Example
Let us solve:
\[ \begin{cases} x-2y=0\\ x+y=6 \end{cases} \]
From the first equation:
\[ x=2y. \]
Substituting into the second:
\[ 2y+y=6. \]
This gives:
\[ 3y=6, \]
so:
\[ y=2. \]
Finally:
\[ x=2\cdot 2=4. \]
The solution of the system is:
\[ (x,y)=(4,2). \]
When to use this method
The substitution method is especially convenient when one variable has a coefficient of \(1\) or \(-1\), since it can then be isolated with minimal effort.
Elimination method
The elimination method (also called the reduction method) works by making the coefficients of one variable equal in magnitude—but opposite in sign—in both equations, and then adding the equations side by side to cancel that variable.
Example
Let us solve:
\[ \begin{cases} 3x+2y=12\\ 5x-2y=4 \end{cases} \]
The coefficients of \(y\) are already opposites:
\[ 2y \qquad \text{and} \qquad -2y. \]
Adding the two equations side by side:
\[ (3x+2y)+(5x-2y)=12+4, \]
we obtain:
\[ 8x=16, \]
so:
\[ x=2. \]
Substituting back into the first equation:
\[ 3\cdot 2+2y=12, \]
that is:
\[ 6+2y=12. \]
This gives:
\[ y=3. \]
Multiplying both equations
When the coefficients are not already opposite, each equation can be multiplied by a suitable nonzero constant.
For example:
\[ \begin{cases} 4x+3y=17\\ 5x-2y=4 \end{cases} \]
We can multiply:
- the first equation by \(2\);
- the second equation by \(3\).
This yields:
\[ \begin{cases} 8x+6y=34\\ 15x-6y=12 \end{cases} \]
Adding:
\[ 23x=46, \]
so:
\[ x=2. \]
Gaussian elimination algorithm
When a system involves three or more unknowns, applying the elimination method in an ad hoc manner can quickly become unwieldy. In such cases, one turns to Gaussian elimination: a systematic algorithm that progressively reduces the system to an equivalent one in row echelon form (a staircase-like triangular structure) by eliminating variables one at a time in an orderly fashion.
Linear systems in three unknowns
A linear system in three unknowns generally takes the form:
\[ \begin{cases} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{cases} \]
In general, the algorithm proceeds by combining pairs of equations to eliminate the same variable, thereby reducing the problem to a smaller system. In cases with particular symmetry, however, the arithmetic can simplify considerably, allowing several unknowns to be eliminated at once.
Example
Consider the following symmetric system:
\[ \begin{cases} x+y+z=6\\ x-y+z=2\\ x+y-z=0 \end{cases} \]
Subtracting the second equation from the first, the unknowns \(x\) and \(z\) cancel out:
\[ (x+y+z) - (x-y+z) = 6 - 2 \implies 2y=4. \]
Therefore:
\[ y=2. \]
Likewise, subtracting the third equation from the first:
\[ (x+y+z) - (x+y-z) = 6 - 0 \implies 2z=6. \]
Hence:
\[ z=3. \]
It now suffices to substitute the values of \(y\) and \(z\) into the first original equation to find the remaining unknown:
\[ x+2+3=6. \]
This gives:
\[ x=1. \]
The solution of the system is the ordered triple:
\[ (x,y,z)=(1,2,3). \]
Geometric interpretation
In a linear system with two unknowns, each first-degree equation represents a line in the Cartesian plane. Solving the system amounts to finding the geometric intersection of these lines.
Intersecting lines
If the lines have different slopes, they meet at exactly one point. The coordinates of that point represent the unique common solution: the system is therefore consistent (and independent).
Parallel and distinct lines
If the lines have the same slope but different \(y\)-intercepts, they are parallel and share no common point. The system has no solution and is therefore inconsistent.
Coincident lines
If the two equations are equivalent, the corresponding lines coincide, sharing infinitely many points. The system has infinitely many solutions and is therefore dependent.
System determinant and Cramer's Rule
Consider a linear system written in standard form:
\[ \begin{cases} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2 \end{cases} \]
The determinant of the system (or principal determinant \(\Delta\)) is defined as the determinant of the coefficient matrix:
\[ \Delta= \begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix} =a_1b_2-a_2b_1. \]
To analyze the system fully without resorting to trial and error, we also introduce the auxiliary determinants \(\Delta_x\) and \(\Delta_y\), obtained by replacing the column of coefficients of the corresponding variable with the column of constants:
\[ \Delta_x= \begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix} =c_1b_2-c_2b_1 \qquad \text{and} \qquad \Delta_y= \begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix} =a_1c_2-a_2c_1. \]
Cramer's Rule allows us to classify the system based on the values of these determinants:
- If \(\Delta \neq 0\): the system is always consistent (unique solution), and the solution is given by: \[ x = \frac{\Delta_x}{\Delta}, \qquad y = \frac{\Delta_y}{\Delta}. \]
- If \(\Delta = 0\) and at least one of \(\Delta_x\) or \(\Delta_y\) is nonzero: the system is inconsistent (parallel and distinct lines).
- If \(\Delta = 0\), \(\Delta_x = 0\), and \(\Delta_y = 0\): the system is dependent (coincident lines).
Systems with a parameter
Some systems contain a parameter: a letter representing an arbitrary, unspecified real number. Analyzing such a system means determining how its nature changes as the parameter varies.
For example:
\[ \begin{cases} x+y=6\\ 2x+ky=12 \end{cases} \]
Isolating \(x\) from the first equation:
\[ x=6-y. \]
Substituting into the second:
\[ 2(6-y)+ky=12. \]
Expanding:
\[ 12-2y+ky=12. \]
Collecting terms in \(y\):
\[ (k-2)y=0. \]
We now carry out the case analysis:
- if \(k\neq 2\), the coefficient of \(y\) is nonzero, so we can divide both sides to obtain a unique solution. The system is consistent;
- if \(k=2\), the equation becomes \(0y=0\), which is always true (\(0=0\)). The system has infinitely many solutions and is therefore dependent.
Nonlinear systems
A system is called nonlinear when at least one of its equations is not of the first degree—that is, when the unknowns appear multiplied together or raised to powers greater than one.
For example:
\[ \begin{cases} x+y=5\\ x^2+y^2=13 \end{cases} \]
In such cases, substitution is almost always the most straightforward and reliable approach.
From the first equation we obtain:
\[ y=5-x. \]
Substituting into the second (quadratic) equation:
\[ x^2+(5-x)^2=13. \]
Expanding the square:
\[ x^2+25-10x+x^2=13. \]
Collecting like terms and writing the resulting quadratic in standard form:
\[ 2x^2-10x+12=0. \]
Dividing every term by the common factor \(2\):
\[ x^2-5x+6=0. \]
Factoring the quadratic (looking for two numbers whose sum is \(-5\) and whose product is \(6\)):
\[ (x-2)(x-3)=0. \]
By the Zero Product Property, we obtain two distinct values of \(x\):
\[ x=2 \qquad \text{or} \qquad x=3. \]
Substituting each value into \(y=5-x\) gives the corresponding \(y\)-values. The two distinct solutions of the system are the ordered pairs:
\[ (2,3) \qquad \text{and} \qquad (3,2). \]
Common mistakes
Sign errors
The most frequent mistakes occur when moving terms from one side of an equation to the other, or when handling the minus sign in front of parentheses during substitution.
Incomplete elimination (multiplication property of equality)
When multiplying an entire equation by a constant to apply the elimination method, students often forget to multiply the constant on the right-hand side as well.
Substituting without parentheses
When substituting a compound expression (a binomial or trinomial) in place of a variable, it is essential to enclose it in parentheses to avoid errors with external coefficients.
For example, if:
\[ x=2y+1, \]
the entire expression must be substituted with its parentheses intact:
\[ (2y+1). \]
Skipping the final check
Once the calculations are complete, it is always good practice to verify the solution by substituting the values found back into both original equations to confirm that the identities hold.
Failing to recognize inconsistent or dependent systems
If, during the algebraic process, all variables cancel and one arrives at a plainly false statement such as:
\[ 0=5, \]
the system has no solution and is inconsistent.
If instead all variables cancel and one arrives at an identity such as:
\[ 0=0, \]
the system has infinitely many solutions and is dependent.
In summary, systems of equations are one of the fundamental tools of algebra: they allow us to express simultaneous conditions and to interpret geometrically the intersections of lines, planes, and curves. A solid command of the substitution method, the elimination method, and the systematic approaches of Gaussian elimination and Cramer's Rule is essential both in secondary-school mathematics and in subsequent studies in linear algebra and mathematical analysis.