Linear Equations

The idea

A linear equation in one variable has the general form:

Here, a and b are fixed numbers (coefficients), and x is the unknown. Because the highest power of x is 1, its graph is always a straight line — no curve, no bend. A non-horizontal straight line crosses the x-axis at exactly one point, which is why a linear equation has exactly one root.

Solving it is a matter of isolating x on one side:

Switch to Interactive Grapher to see how changing a and b slides and tilts the line, and moves the root.

Worked example

Method 1 — Factorization

Pull out the common factor of 2:

A product is zero only if one of its factors is zero. Since 2 ≠ 0, it must be that:

Method 2 — Using the formula

Here a = 2 and b = −6. Apply x = −b / a directly:

Both methods agree: x = 3. Check by substituting back: 2(3) − 6 = 0 ✓