A system of linear equations is a group of equations that must be true at the same time.
In two variables, each equation usually represents a line. The solution is the point where
the lines meet, if such a point exists.
This page focuses on the standard school methods: substitution, elimination, and graphing.
For matrices, determinants, row-reduced echelon form, vector spaces, and larger systems,
use the Linear Algebra section of the website.
1. Standard form
A two-variable linear equation is usually written as:
\[
ax+by=c.
\]
A three-variable linear equation is usually written as:
\[
ax+by+cz=d.
\]
A system means several such equations must be solved together.
\[
\begin{aligned}
a_1x+b_1y&=c_1,\\
a_2x+b_2y&=c_2.
\end{aligned}
\]
2. What a solution means
A solution is a value of every variable that makes every equation true.
For example:
\[
\begin{aligned}
2x+3y&=8,\\
4x-y&=7.6.
\end{aligned}
\]
The solution is:
\[
x=2.2,\qquad y=1.2.
\]
Check:
\[
\begin{aligned}
2(2.2)+3(1.2)&=8,\\
4(2.2)-1.2&=7.6.
\end{aligned}
\]
3. Substitution method
In substitution, solve one equation for one variable and substitute that expression into another equation.
Example:
\[
\begin{aligned}
x+y&=5,\\
x-y&=1.
\end{aligned}
\]
From the second equation:
\[
x=y+1.
\]
Substitute into the first equation:
\[
(y+1)+y=5.
\]
Then:
\[
2y+1=5,\qquad 2y=4,\qquad y=2.
\]
Substitute back:
\[
x=2+1=3.
\]
So the solution is:
\[
\boxed{x=3,\ y=2}.
\]
4. Elimination method
In elimination, multiply or add equations so that one variable cancels.
Example:
\[
\begin{aligned}
2x+3y&=8,\\
4x-y&=7.6.
\end{aligned}
\]
Multiply the first equation by \(2\):
\[
4x+6y=16.
\]
Subtract the second equation:
\[
(4x+6y)-(4x-y)=16-7.6.
\]
This gives:
\[
7y=8.4,\qquad y=1.2.
\]
Substitute \(y=1.2\) into \(4x-y=7.6\):
\[
4x-1.2=7.6,\qquad 4x=8.8,\qquad x=2.2.
\]
5. Graph method
The graph method is mainly used for two-variable systems. Each equation is written as a line.
The solution is the point where the lines intersect.
If the two lines cross once, the system has one solution.
If the two lines are parallel and distinct, the system has no solution.
If the two equations describe the same line, the system has infinitely many solutions.
6. One solution
A system has one solution when all equations meet at exactly one point.
For two lines, this means the lines have different slopes.
\[
\boxed{\text{one solution}=\text{one intersection point}}.
\]
7. No solution
A system has no solution when the equations contradict each other.
During elimination, this can appear as something impossible:
\[
0=5.
\]
In a two-variable graph, this usually means the lines are parallel and never meet.
8. Infinitely many solutions
A system has infinitely many solutions when one equation repeats the same information as another equation.
During elimination, this can appear as:
\[
0=0.
\]
Example:
\[
\begin{aligned}
x+y&=5,\\
2x+2y&=10.
\end{aligned}
\]
The second equation is just two times the first equation, so both equations describe the same line.
9. Parametric answers
When there are infinitely many solutions, the answer may use a parameter.
For example:
\[
x+y=5.
\]
Let:
\[
y=t.
\]
Then:
\[
x=5-t.
\]
So:
\[
\boxed{x=5-t,\ y=t}.
\]
The parameter \(t\) can be any real number.
10. Three-variable systems
For three variables, substitution and elimination still work, but there are more steps.
A typical system looks like:
\[
\begin{aligned}
ax+by+cz&=d,\\
ex+fy+gz&=h,\\
ix+jy+kz&=\ell.
\end{aligned}
\]
The goal is to eliminate variables until one equation has only one variable, then substitute backward.
11. Verification
Always check the final answer in the original equations:
\[
\begin{aligned}
\text{LHS}_1&\stackrel{?}{=}\text{RHS}_1,\\
\text{LHS}_2&\stackrel{?}{=}\text{RHS}_2.
\end{aligned}
\]
Checking is especially important when decimals or fractions are involved.
12. Simple-method summary
The table uses plain-text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
13. Advanced methods disclaimer
Matrices, determinants, Gaussian elimination, row-reduced echelon form, and vector-space language are powerful
tools for more advanced systems. They are usually taught in Linear Algebra, so this calculator keeps them out
of the main explanation and focuses on simpler classroom methods.
14. Common mistakes
- Changing only one side of an equation during elimination.
- Forgetting to distribute a multiplier across every term.
- Substituting into the wrong equation.
- Rounding too early before checking the answer.
- Calling a dependent system “no solution” just because one equation disappears.
- Trying to use this linear calculator for nonlinear terms such as \(x^2\) or \(xy\).
Key idea: use substitution, elimination, or graphing to simplify the system, then check whether the final result gives one solution, no solution, or infinitely many solutions.
