Table of Contents
When solving a system of linear equations, the graphical method means drawing the graphs of the equations and using the picture to find their common solution(s). This chapter focuses on how to do that and what the different pictures mean.
Linear equations as lines in the plane
In two variables $x$ and $y$, a linear equation like
$$
y = 2x + 1
$$
or
$$
2x - 3y = 6
$$
represents a straight line on the coordinate plane.
In earlier chapters, you learn about slope, intercepts, and how to graph a line. Here we will assume you can already graph each equation and focus on using those graphs to solve systems.
A system of two linear equations in two variables looks like:
$$
\begin{cases}
\text{Equation 1} \\
\text{Equation 2}
\end{cases}
$$
Each equation has its own line. The graphical method is about how these two lines relate to each other.
The idea of the graphical method
The solution to a system of linear equations in two variables is an ordered pair $(x, y)$ that makes both equations true at the same time.
On a graph, this means:
- Each equation is a line made of all points that satisfy that equation.
- Points that satisfy both equations lie on both lines.
- So the solution(s) to the system are the point(s) where the lines intersect.
In summary:
- Graph each equation on the same coordinate plane.
- Look for intersection(s).
- Read the coordinates of the intersection point(s).
- These coordinates are the solution(s) to the system.
Step-by-step: solving by graphing
Here is a practical outline of how to solve a system by graphing.
Consider:
$$
\begin{cases}
y = x + 1 \\
y = -x + 5
\end{cases}
$$
- Write each equation in a graphable form
The slope–intercept form $y = mx + b$ is especially convenient, because it clearly shows the slope $m$ and the $y$-intercept $b$.
In our example, both are already in slope–intercept form:
- First line: $y = x + 1$ (slope $1$, $y$-intercept $(0, 1)$)
- Second line: $y = -x + 5$ (slope $-1$, $y$-intercept $(0, 5)$)
- Plot each line
For each equation:
- Plot the $y$-intercept.
- Use the slope to find at least one more point.
- Draw a straight line through those points.
For $y = x + 1$:
- Intercept: $(0, 1)$
- Slope $1$ means “up 1, right 1,” so another point is $(1, 2)$.
For $y = -x + 5$:
- Intercept: $(0, 5)$
- Slope $-1$ means “down 1, right 1,” so another point is $(1, 4)$.
- Find the intersection point
On the graph, locate the point where the two lines cross. Suppose your drawing shows them crossing at $(2, 3)$.
- Read the solution from the graph
The intersection is $(2, 3)$, so the solution to the system is:
$$
x = 2,\quad y = 3.
$$
- (Recommended) Check the solution
Substitute $x = 2$, $y = 3$ back into each equation:
- First equation: $y = x + 1$ becomes $3 = 2 + 1$ (true).
- Second equation: $y = -x + 5$ becomes $3 = -2 + 5$ (true).
Since $(2,3)$ satisfies both, it is indeed the solution.
Types of solutions you can see on the graph
Depending on how the lines are positioned, a system of two linear equations can have:
- One solution (lines intersect once)
- No solution (lines are parallel and distinct)
- Infinitely many solutions (lines are the same line)
These three cases show up very clearly when you graph.
1. One solution: intersecting lines
If the two lines cross at a single point, the system has exactly one solution.
Example:
$$
\begin{cases}
y = 2x - 1 \\
y = -x + 5
\end{cases}
$$
Graphing:
- $y = 2x - 1$: intercept $(0, -1)$, slope $2$.
- $y = -x + 5$: intercept $(0, 5)$, slope $-1$.
On the plane, these meet at a point, say $(2, 3)$ (the exact intersection can also be found algebraically, but here we are focusing on the picture).
Interpretation:
- The two equations describe two different lines.
- There is exactly one point that lies on both lines.
- So the system has one unique solution.
2. No solution: parallel lines
If the two lines never meet (they are parallel), there is no point that lies on both lines. The system has no solution.
Example:
$$
\begin{cases}
y = 3x + 1 \\
y = 3x - 4
\end{cases}
$$
Graphing:
- Both lines have slope $3$.
- First line: intercept $(0, 1)$.
- Second line: intercept $(0, -4)$.
Two lines with the same slope and different intercepts are parallel. On the graph, they run side by side and never intersect.
Interpretation:
- The equations are inconsistent: nothing makes both true at once.
- The system has no solution.
In symbols, sometimes you will see this written as “no solution” or using the empty set symbol $\varnothing$.
3. Infinitely many solutions: the same line
If the two equations actually describe the same line, every point on that line is a solution to both equations, so there are infinitely many solutions.
Example:
$$
\begin{cases}
y = -\dfrac{1}{2}x + 2 \\
x + 2y = 4
\end{cases}
$$
Rewrite $x + 2y = 4$ into $y$-form:
$$
x + 2y = 4 \\
2y = -x + 4 \\
y = -\dfrac{1}{2}x + 2
$$
Both equations are the same when simplified. Graphing them will give just one line; you won’t see two separate lines.
Interpretation:
- The two equations are really the same condition written differently.
- Every point on the line is a solution.
- The system has infinitely many solutions.
Recognizing the case from the equations
Although this chapter focuses on graphs, it is useful to connect the picture with the algebraic form.
Putting both equations in slope–intercept form $y = mx + b$:
- Different slopes ($m_1 \neq m_2$):
- The lines intersect at exactly one point.
- One solution.
- Same slope, different intercepts ($m_1 = m_2$ but $b_1 \neq b_2$):
- The lines are parallel and do not meet.
- No solution.
- Same slope and same intercept ($m_1 = m_2$ and $b_1 = b_2$):
- The equations describe the same line.
- Infinitely many solutions.
On a graph, these three cases appear as: intersecting lines, parallel lines, or overlapping lines.
Accuracy and limitations of the graphical method
The graphical method is visual and helps build intuition, but it has some limitations:
- Accuracy depends on your drawing
If your graph is hand-drawn on paper, it is easy to be slightly off, especially if:
- The intersection is at non-integer coordinates.
- The scales on the axes are large or small.
- You do not draw the lines carefully.
You may only get an approximate solution, like “the intersection looks close to $(1.3, 2.7)$,” rather than an exact pair.
- Harder for large or awkward numbers
Systems with very large or very small coefficients or solutions may force you to:
- Use a very large or very small scale,
- Or zoom in or out a lot,
which can make the intersection difficult to identify precisely.
- Not practical beyond two variables
Graphing works clearly with two variables (a plane). With three variables, you would need to visualize planes in 3D space. Beyond three variables, you cannot visualize directly. For larger systems, algebraic methods (like substitution or elimination) are more practical.
Because of these issues, the graphical method is often used:
- To get a rough idea of the solution,
- To check if two lines are likely to intersect, be parallel, or be the same,
- To build intuition about what solutions to systems “look like”.
For precise answers, especially with non-integer solutions, the substitution and elimination methods are usually preferred.
Using technology for graphing
Graphing calculators, computer software, and online graphing tools can:
- Plot each equation accurately.
- Show the intersection point numerically.
- Let you zoom in to see the intersection clearly.
When using technology:
- Enter each equation in a graphable form, often $y = \dots$.
- Plot both on the same coordinate plane.
- Use built-in features (like “intersect”) to calculate the exact coordinates of the intersection point.
- Interpret these coordinates as the solution of the system.
This combines the visual intuition of the graphical method with the accuracy of algebraic computation.
Interpreting graphs in word problems
In word problems, each equation often represents a real-world relationship, like cost, distance, or quantity. When you graph these:
- Each line shows how one quantity depends on another.
- The intersection point often answers questions like:
- When are two quantities equal?
- At what time do two things meet?
- At what production level do two plans cost the same?
For example, if:
- One line shows the cost of Plan A as a function of the number of hours used,
- Another line shows the cost of Plan B as a function of the same number of hours,
then the intersection point’s $x$-coordinate might tell you the number of hours at which the two plans cost the same, and the $y$-coordinate gives that common cost.
The steps are the same:
- Write equations from the problem.
- Graph them.
- Find and interpret the intersection point based on the context.
Summary
- In two variables, each linear equation graphs as a straight line.
- Solving a system by the graphical method means:
- Graph both lines on the same coordinate plane,
- Find their intersection point(s),
- Read the coordinates as the solution(s).
- The relationship between the lines and the type of solution:
- Intersecting lines → one solution.
- Parallel, distinct lines → no solution.
- Same line → infinitely many solutions.
- Graphs are powerful for building intuition and understanding real-world situations, but hand-drawn graphs may only give approximate answers. Algebraic methods are usually used for exact solutions, especially when the intersection does not have simple coordinates.