Table of Contents
Understanding Systems of Linear Equations
In earlier algebra work you dealt with single linear equations in one variable, such as $2x + 3 = 7$. Systems of linear equations extend this idea: you look at several linear equations at the same time and ask whether there is a single set of variable values that makes all of them true together.
This chapter focuses on what systems of linear equations are, the types of solutions they can have, and how to interpret these solutions algebraically and graphically. Later subsections will develop specific solution methods (graphical, substitution, elimination) in more detail.
What Is a System of Linear Equations?
A system of linear equations is a collection of two or more linear equations that all involve the same variables.
For example, in two variables $x$ and $y$:
$$
\begin{cases}
2x + 3y = 7 \\
x - y = 1
\end{cases}
$$
This is a system of two linear equations in two variables.
The goal is to find all ordered pairs $(x, y)$ that satisfy every equation in the system at the same time.
In three variables $x, y, z$:
$$
\begin{cases}
x + y + z = 6 \\
2x - y + 3z = 4
\end{cases}
$$
This is a system of two linear equations in three variables. An ordered triple $(x, y, z)$ is a solution if it makes both equations true simultaneously.
In this course level, most of the time you will work with two equations and two variables.
Solutions of a System
For a system in two variables, a solution is an ordered pair $(x, y)$ that satisfies each equation in the system.
- If you plug the $x$ and $y$ values into every equation, all of them should come out true (left side equals right side).
For the system
$$
\begin{cases}
2x + 3y = 7 \\
x - y = 1
\end{cases}
$$
if you try $(x, y) = (2, 1)$:
- First equation: $2(2) + 3(1) = 4 + 3 = 7$ ✓
- Second equation: $2 - 1 = 1$ ✓
Since both equations are true, $(2, 1)$ is a solution of the system.
If an ordered pair makes even one of the equations false, it is not a solution of the system.
Types of Systems and Numbers of Solutions
A system of two linear equations in two variables can have:
- Exactly one solution
- No solution
- Infinitely many solutions
These three possibilities are closely tied to the graphs of the equations and the relationships between the two lines.
1. Exactly One Solution (Consistent, Independent System)
This occurs when the two lines intersect at a single point. The coordinates of the intersection point give the unique solution of the system.
Example:
$$
\begin{cases}
y = 2x + 1 \\
y = -x + 4
\end{cases}
$$
- Graphically, these are two lines with different slopes ($2$ and $-1$).
- Lines with different slopes cross exactly once.
- The intersection point $(x, y)$ is the only ordered pair that satisfies both equations.
Algebraic signs of “one solution”:
- When you solve the system (by any method), you end up with specific values like $x = 1$, $y = 3$.
- After simplification, you do not get a contradiction like $0 = 5$ and you do not get an identity like $0 = 0$ for every step; instead you isolate values for variables.
Such a system is called:
- Consistent (it has at least one solution), and
- Independent (its two equations represent different lines).
2. No Solution (Inconsistent System)
This happens when the two lines are parallel and distinct; they never meet, so there is no point that lies on both lines.
Example:
$$
\begin{cases}
y = 2x + 1 \\
y = 2x - 3
\end{cases}
$$
- Both lines have slope $2$, but different $y$-intercepts ($1$ and $-3$).
- Same slope, different intercept → parallel, non-overlapping lines.
- Since they never intersect, there is no ordered pair $(x, y)$ that satisfies both equations.
Algebraic signs of “no solution”:
- When you try to solve the system, the variables cancel out and you get a false statement like:
$$
0 = 4 \quad \text{or} \quad 5 = 2
$$
which is impossible.
A system with no solution is called inconsistent.
3. Infinitely Many Solutions (Consistent, Dependent System)
This occurs when the two equations actually describe the same line. Every point on the line satisfies both equations, so there is not just one solution, but infinitely many.
Example:
$$
\begin{cases}
y = 3x + 2 \\
2y = 6x + 4
\end{cases}
$$
If you simplify the second equation:
$$
2y = 6x + 4 \Rightarrow y = 3x + 2
$$
Both equations are equivalent, just written differently. They represent the same line.
Graphically:
- If you draw both equations, you only see one line because they sit exactly on top of each other.
- Every point on this line is a common solution, so there are infinitely many.
Algebraic signs of “infinitely many solutions”:
- When solving, the variables cancel and you end up with a true identity like:
$$
0 = 0 \quad \text{or} \quad 4 = 4
$$ - This shows the second equation doesn’t add any new restriction; it’s just the first equation in disguise.
Such a system is:
- Consistent (there are solutions), and
- Dependent (the equations are not independent; one can be obtained from the other).
Graphical Interpretation in Two Variables
Each linear equation in two variables represents a line on the coordinate plane. A system of two equations in two variables represents two lines drawn on the same plane.
Thinking in terms of graphs:
- The solution of the system is the set of all intersection points of the lines.
- Unique intersection point → exactly one solution.
- No intersection (parallel lines) → no solution.
- Same line (coincident) → infinitely many solutions (all points on the line).
When you learn the graphical method in detail later, you will use this idea: sketch both lines on the same axes and look for where they meet.
Writing Systems in Different Forms
Just as a single linear equation can be written in different equivalent forms (for example, slope–intercept form, standard form), an entire system can be written in corresponding ways.
For two variables $x$ and $y$, a common standard form is:
$$
\begin{cases}
a_1x + b_1y = c_1 \\
a_2x + b_2y = c_2
\end{cases}
$$
where $a_1, b_1, c_1, a_2, b_2, c_2$ are constants.
For example:
$$
\begin{cases}
3x - 2y = 7 \\
x + 5y = -1
\end{cases}
$$
You may also see systems written with one or both equations in slope–intercept form:
$$
\begin{cases}
y = -\dfrac{1}{2}x + 3 \\
y = 2x - 1
\end{cases}
$$
The choice of form can make one method of solution (graphical, substitution, or elimination) more convenient than another. Later sections focus on how to convert and choose forms effectively.
Checking a Solution
No matter which method you use to solve a system, always check your candidate solution.
Suppose you think $(x, y) = (-1, 2)$ solves
$$
\begin{cases}
3x - 2y = 7 \\
x + 5y = -1
\end{cases}
$$
Check each equation:
- First equation:
$$
3(-1) - 2(2) = -3 - 4 = -7 \neq 7
$$
So the first equation is false.
Since one equation is false, $(-1, 2)$ is not a solution of the system, even if it works in the other equation. A correct solution must satisfy all equations simultaneously.
Systems in More Than Two Variables (Preview Idea)
While this chapter focuses on two equations in two variables, the ideas generalize:
- In three variables, each linear equation represents a plane in 3D space.
- A system of three equations in three variables asks for the point(s) common to all three planes.
Possible outcomes remain similar:
- A single solution (planes intersect at one point),
- No solution (they never share a common point), or
- Infinitely many solutions (they share a line or an entire plane).
You will encounter more of this viewpoint when you study systems in linear algebra.
Word Problems and Interpretation (Overview)
Word problems involving systems typically describe two or more unknown quantities and give you several conditions about them. You:
- Choose variables to represent the unknowns (for example, $x$ for number of adults, $y$ for number of children).
- Translate each condition into a linear equation.
- Solve the resulting system.
- Interpret the solution in the context of the problem (and check if it makes sense: values should often be non-negative and whole numbers, depending on the situation).
The “Word problems” subsection for systems will guide you through systematic steps and examples; here it is enough to recognize that systems are powerful tools to model and solve multi-unknown situations.
Summary of Key Ideas
- A system of linear equations is a set of linear equations with the same variables, considered together.
- A solution of a system is a set of variable values (like $(x, y)$) that satisfies every equation in the system simultaneously.
- For two linear equations in two variables, there are three possibilities:
- Exactly one solution: lines intersect at one point (consistent, independent).
- No solution: lines are parallel and distinct (inconsistent).
- Infinitely many solutions: lines coincide; one equation is a multiple or rewrite of the other (consistent, dependent).
- Graphically, solving a system corresponds to finding the intersection of lines.
- Algebraic clues:
- Specific values for variables → one solution.
- A contradiction like $0 = 5$ → no solution.
- An identity like $0 = 0$ (after variables cancel) → infinitely many solutions.
- Always check any proposed solution by substituting into all equations.
In the next subsections you will learn and practice specific strategies to find these solutions: by graphing, by substitution, and by elimination.