A linear equation in one variable is an equation where the variable appears only to the first power.
The goal is to isolate the variable by using balanced operations.
1. What is a linear equation?
A one-variable linear equation can usually be simplified into the form:
\[
\begin{aligned}
ax+b=c.
\end{aligned}
\]
Here, \(a\), \(b\), and \(c\) are numbers, and \(x\) is the variable. The equation is linear because
the variable is not squared, cubed, multiplied by itself, or placed in a denominator.
2. The balance rule
The most important rule is:
\[
\begin{aligned}
\text{Whatever you do to one side, you must do to the other side.}
\end{aligned}
\]
This keeps the equation true while you simplify it.
3. Basic isolation method
To solve:
\[
\begin{aligned}
ax+b=c,
\end{aligned}
\]
subtract \(b\) from both sides:
\[
\begin{aligned}
ax=c-b.
\end{aligned}
\]
Then divide by \(a\):
\[
\begin{aligned}
x=\frac{c-b}{a}.
\end{aligned}
\]
4. Multi-step equations with parentheses
Many linear equations need expansion before isolation. For example:
\[
\begin{aligned}
5(2x-3)+4=3x+17.
\end{aligned}
\]
Expand the left side:
\[
\begin{aligned}
10x-15+4=3x+17.
\end{aligned}
\]
Combine constants:
\[
\begin{aligned}
10x-11=3x+17.
\end{aligned}
\]
Move variable terms to the left and constants to the right:
\[
\begin{aligned}
10x-3x=17+11.
\end{aligned}
\]
Simplify:
\[
\begin{aligned}
7x=28.
\end{aligned}
\]
Divide by 7:
\[
\begin{aligned}
x=4.
\end{aligned}
\]
5. Fractions and decimals
Linear equations can include fractions and decimals:
\[
\begin{aligned}
\frac{1}{2}x+3=7.
\end{aligned}
\]
Subtract 3:
\[
\begin{aligned}
\frac{1}{2}x=4.
\end{aligned}
\]
Divide by \(\frac{1}{2}\), or multiply by 2:
\[
\begin{aligned}
x=8.
\end{aligned}
\]
Decimals can also be treated as fractions. For example, \(0.25=\frac{1}{4}\).
6. Variables on both sides
If the variable appears on both sides, collect the variable terms on one side:
\[
\begin{aligned}
4x+5=2x+13.
\end{aligned}
\]
Subtract \(2x\) from both sides:
\[
\begin{aligned}
2x+5=13.
\end{aligned}
\]
Subtract 5:
\[
\begin{aligned}
2x=8.
\end{aligned}
\]
Divide by 2:
\[
\begin{aligned}
x=4.
\end{aligned}
\]
7. One solution, no solution, or infinitely many solutions
After simplifying, a linear equation may have one solution, no solution, or infinitely many solutions.
8. Why some equations are not linear
These expressions are not linear:
\[
\begin{aligned}
x^2,\quad x\cdot x,\quad \frac{1}{x}.
\end{aligned}
\]
They require different methods because the variable is squared, multiplied by itself, or placed in a denominator.
9. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
10. Common mistakes
- Forgetting to distribute a number across every term in parentheses.
- Changing only one side of the equation.
- Moving a term across the equals sign without changing its sign.
- Combining unlike terms, such as \(x\) and constants.
- Dividing by the coefficient before constants are moved away from the variable.
- Assuming every linear equation has one solution, even when variables cancel.
Key idea: expand, collect, move variable terms to one side, move constants to the other side, then divide.
