Algebra · Core concept

Linear equations

A linear equation is one of the simplest mathematical models of an unknown quantity. Learn exactly what it means, why the solution rule works, how to interpret it graphically, and how to solve equations reliably.

\(ax+b=0\)
for \(a\neq0\):
\(x=-\frac{b}{a}\)

1. Definition

A linear equation in one variable is an equation that can be written as

\[ax+b=0\]

where \(a\) and \(b\) are constants and \(x\) is the unknown. The equation is genuinely linear when \(a\neq0\). The word linear means that the variable appears only to the first power: there is no \(x^2\), \(\sqrt{x}\), \(1/x\), or other nonlinear dependence on \(x\).

Equivalent form. You will also often see \(ax=b\). It is the same idea, because any equation \(ax+c=d\) can be rearranged into \(ax+(c-d)=0\).

2. Why the solution formula works

Starting from \(ax+b=0\), use operations that preserve equality.

Subtract \(b\) from both sides.
\[ax=-b\]
Divide both sides by \(a\). This is allowed only because \(a\neq0\).
\[x=-\frac{b}{a}\]

This solution is unique. If both \(x_1\) and \(x_2\) satisfy \(ax+b=0\), then \(ax_1=ax_2\). Since \(a\neq0\), division by \(a\) gives \(x_1=x_2\).

3. Special cases

\(a\neq0\)Exactly one solution: \(x=-b/a\).
\(a=0,\ b\neq0\)No solution, because the equation becomes \(b=0\), which is false.
\(a=0,\ b=0\)Every real number is a solution, because the equation becomes \(0=0\).

4. Graphical interpretation

The equation \(ax+b=0\) asks: where does the line \(y=ax+b\) cross the x-axis? The x-coordinate of that intercept is the solution.

Graph of y equals 2x minus 4 A straight line crosses the x-axis at x equals 2, illustrating the solution of 2x minus 4 equals zero. x = 2 y = 2x − 4 xy

For \(2x-4=0\), the line \(y=2x-4\) intersects the x-axis at \(x=2\). Algebra and geometry describe the same result.

5. Worked examples

Example A — variables on one side

Solve \(5x-15=0\).

\[5x-15=0\quad\Rightarrow\quad5x=15\quad\Rightarrow\quad x=3\]

Example B — variables on both sides

Solve \(7x-3=2x+12\).

\[ \begin{aligned} 7x-3&=2x+12\\ 5x-3&=12\\ 5x&=15\\ x&=3 \end{aligned} \]

Checking the result: left side \(=7\cdot3-3=18\), right side \(=2\cdot3+12=18\). Both sides agree.

Example C — fractions

Solve \(\frac{x}{3}+2=5\).

\[\frac{x}{3}=3\quad\Rightarrow\quad x=9\]

A useful strategy is to remove denominators early by multiplying every term by a common denominator.

6. Interactive linear equation solver

Enter coefficients for \(ax+b=0\). The solver handles the unique-solution and degenerate cases.

7. Common mistakes

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Changing only one side. Any addition, subtraction, multiplication or division used as an equivalence transformation must be applied to both sides.
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Dividing by zero. The step \(x=-b/a\) requires \(a\neq0\). Always treat \(a=0\) separately.
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Sign errors. From \(ax+b=0\), subtracting \(b\) gives \(ax=-b\), not \(ax=b\).
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Expanding unnecessarily. Simplify strategically. The goal is to isolate the variable while keeping the algebra easy to verify.

8. Practice

ProblemSkill
\(9x-4=2x+17\)Collect variable terms
\(3(x-2)=12\)Distribute or divide first
\(\frac{x}{4}-3=2\)Fractions
\(0x+5=0\)Recognize no-solution case
Show answers

1) \(x=3\)   2) \(x=6\)   3) \(x=20\)   4) no solution.

9. Related topics

Equation fundamentals

Equality, transformations and solution sets.

Quadratic equations

What changes when the highest power is two?

Systems of equations

Solve several linear equations simultaneously.