How do you solve a linear equation step by step?

A linear equation in the form ax + b = 0 is solved by isolating x. First subtract b from both sides: ax = −b. Then divide both sides by a: x = −b/a. For example, 2x + 6 = 0 → 2x = −6 → x = −3. Always check by substituting back: 2(−3) + 6 = −6 + 6 = 0 ✓. General steps for any linear equation: • Move all x terms to one side and all constants to the other • Combine like terms on each side • Divide both sides by the coefficient of x • Check your answer by substituting back into the original equation • If the coefficient of x is 0, check whether the constants balance (infinite solutions) or conflict (no solution)

What does it mean when a linear equation has no solution?

A linear equation has no solution when it leads to a contradiction — a statement that is always false regardless of x. This happens when the x terms cancel out and you are left with a false numeric equation like 3 = 7. Graphically, this means the two sides of the equation represent parallel lines that never intersect. Example: 4x + 3 = 4x + 7. Subtract 4x from both sides: 3 = 7. This is never true — no value of x can make it true. The solution set is the empty set ∅. How to recognise it: • After simplification, all x terms disappear • What remains is a false equation (e.g. 0 = 5, 3 = −1) • The coefficients of x on both sides are equal but the constants differ • In systems of equations, this corresponds to parallel lines (same slope, different intercepts)

What does it mean when a linear equation has infinite solutions?

A linear equation has infinite solutions (also called an identity) when it is true for every value of x. This happens when both sides simplify to the same expression — the equation is always satisfied regardless of what x is. Graphically, the two sides represent the same line. Example: 2x + 1 = 2x + 1. Subtract 2x from both sides: 1 = 1. This is always true. The solution set is all real numbers, written x ∈ ℝ. How to recognise it: • After simplification, all variables and all constants cancel • What remains is a true equation (0 = 0 or 5 = 5) • The coefficients of x AND the constants are equal on both sides • Any value substituted for x will satisfy the equation

How do I solve ax + b = cx + d for x?

Move all x terms to one side and all constants to the other. Subtract cx from both sides: (a−c)x + b = d. Then subtract b from both sides: (a−c)x = d−b. Finally divide by (a−c): x = (d−b)/(a−c), provided a ≠ c. Example: 3x + 4 = x + 10 • Subtract x from both sides: 2x + 4 = 10 • Subtract 4 from both sides: 2x = 6 • Divide by 2: x = 3 • Check: LHS = 3(3)+4 = 13, RHS = 1(3)+10 = 13 ✓ Special cases: • If a = c and b ≠ d: no solution (parallel lines) • If a = c and b = d: infinite solutions (same line) • If a ≠ c: exactly one solution x = (d−b)/(a−c)

What is the difference between a linear and a quadratic equation?

A linear equation has the highest power of x equal to 1 — it produces a straight line when graphed. A quadratic equation has the highest power of x equal to 2 — it produces a parabola. Linear equations have at most one solution. Quadratic equations can have zero, one, or two real solutions. Key differences: • Linear (ax + b = 0): highest power is 1, always one solution (unless a=0) • Quadratic (ax²+bx+c = 0): highest power is 2, zero/one/two real solutions • Linear graph: straight line — crosses x-axis at most once • Quadratic graph: parabola — can cross x-axis 0, 1, or 2 times • Linear solving: simple algebra (isolate x) • Quadratic solving: quadratic formula, factoring, or completing the square • Real-world: linear equations model constant rates; quadratics model acceleration, projectile motion, optimisation

Linear Equation Solver — Step-by-Step with Number Line

The Linear Equation Solver finds the solution to any linear equation in one variable. Two modes: simple form (ax + b = 0) and two-side form (ax + b = cx + d). The solver handles all three cases — a unique solution, no solution (contradiction), and infinite solutions (identity) — and shows the result...

LINEAR SOLVER

2x + 6 = 0

a (coefficient of x)

b (constant)

UNIQUE SOLUTION

x = -3/1

= -3 (decimal)

STEP-BY-STEP SOLUTION

Start: 2x + 6 = 0

Subtract 6 from both sides: 2x = -6

Divide both sides by 2: x = -6 / 2

x = -3/1 = -3

QUICK EXAMPLES

NUMBER LINE

SOLUTION

-3/1

LHS CHECK

RHS CHECK

Live diagram · updates as you type

SOLUTION PROPERTIES

Solution-3/1

Decimal-3

SignNegative

LHS = RHS0 = 0 ✓

Created with❤️byeaglecalculator.com

HOW TO USE

1
Select your mode: ax + b = 0 for simple equations with all terms on one side, or ax + b = cx + d for equations with x terms on both sides.
2
Enter the coefficients. For ax + b = 0, enter a and b. For ax + b = cx + d, enter all four values a, b, c, and d.
3
The solution appears instantly in the result box. If there is a unique solution, the exact fraction form is shown alongside the decimal.
4
Read the step-by-step solution to see every algebraic manipulation — useful for checking your own working or understanding the method.
5
Check the number line diagram — the solution point is marked in red. The LHS and RHS check values confirm the solution is correct.

WORKED EXAMPLE

Example 1: 2x + 6 = 0. Subtract 6: 2x = −6. Divide by 2: x = −3. Check: 2(−3)+6 = 0 ✓. Example 2: 3x + 4 = x + 10. Subtract x: 2x + 4 = 10. Subtract 4: 2x = 6. Divide by 2: x = 3. Check: LHS = 13, RHS = 13 ✓. Example 3 (no solution): 4x + 3 = 4x + 7 → 3 = 7 — contradiction. Example 4 (infinite): 2x + 1 = 2x + 1 → 1 = 1 — identity.

REFERENCE FORMULAS

FORMULA REFERENCE

NAME	FORMULA	NOTES
Simple form	ax + b = 0 → x = −b/a	Unique solution when a ≠ 0
Two-side form	ax + b = cx + d → x = (d−b)/(a−c)	Unique solution when a ≠ c
No solution	(a−c) = 0, (d−b) ≠ 0	Contradiction — parallel lines, empty set
Infinite solutions	(a−c) = 0, (d−b) = 0	Identity — same line, all reals
Check	Substitute x back: LHS = RHS	Always verify by substituting your answer

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Last updated: April 28, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.