Equation Solver

About Equation Solver

Three Equation Types

Type	Form	Method
Linear	ax + b = c	Isolate x by subtracting and dividing
Quadratic	ax² + bx + c = 0	Quadratic formula: x = (-b ± √(b²-4ac)) / 2a
System (2 equations)	a₁x + b₁y = c₁ and a₂x + b₂y = c₂	Cramer's rule using determinants

Solving Linear Equations

Worked example: Solve 3x + 7 = 22

1. Start: 3x + 7 = 22
2. Subtract 7 from both sides: 3x = 15
3. Divide both sides by 3: x = 5

Edge cases:

Situation	Example	Result
Normal	3x + 7 = 22	x = 5 (one solution)
a = 0, consistent	0x + 5 = 5	Infinitely many solutions (any x works)
a = 0, inconsistent	0x + 5 = 3	No solution

Solving Quadratic Equations

The quadratic formula finds both roots of ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

Worked example: Solve 2x² - 5x - 3 = 0

1. a = 2, b = -5, c = -3
2. Discriminant: (-5)² - 4(2)(-3) = 25 + 24 = 49
3. √49 = 7
4. x = (5 ± 7) / 4
5. x₁ = (5 + 7) / 4 = 12/4 = 3
6. x₂ = (5 - 7) / 4 = -2/4 = -0.5

The discriminant determines the nature of the roots:

Discriminant (b²-4ac)	Value	Roots	Example
Positive	> 0	Two distinct real roots	x² - 5x + 6 = 0 → x = 2, 3
Zero	= 0	One repeated real root	x² - 6x + 9 = 0 → x = 3
Negative	< 0	Two complex conjugate roots	x² + 1 = 0 → x = ±i

Solving Systems of Two Equations

For a system of two equations with two unknowns, the solver uses Cramer's rule:

Worked example:

• Equation 1: 2x + 3y = 12
• Equation 2: 4x - y = 5

1. Main determinant: D = (2)(-1) - (3)(4) = -2 - 12 = -14
2. D ≠ 0, so a unique solution exists
3. Dx = (12)(-1) - (3)(5) = -12 - 15 = -27
4. Dy = (2)(5) - (12)(4) = 10 - 48 = -38
5. x = Dx/D = -27/-14 = 27/14 ≈ 1.929
6. y = Dy/D = -38/-14 = 38/14 ≈ 2.714

Verify: 2(1.929) + 3(2.714) = 3.857 + 8.143 = 12 ✓

System solution types:

Determinant	Geometric Meaning	Solution
D ≠ 0	Lines intersect at one point	Unique solution (x, y)
D = 0, Dx ≠ 0 or Dy ≠ 0	Lines are parallel	No solution
D = 0, Dx = 0, Dy = 0	Lines are the same	Infinitely many solutions

Common Equation-Solving Strategies

Equation Type	Strategy	Example
Linear (simple)	Isolate variable	5x = 20 → x = 4
Linear (multi-step)	Combine like terms, then isolate	3x + 7 = 2x + 12 → x = 5
Quadratic (factorable)	Factor and set each factor to zero	x² - 5x + 6 = (x-2)(x-3) = 0
Quadratic (general)	Quadratic formula	Works for any quadratic
System of 2	Substitution or elimination	Or Cramer's rule (this tool)

For a dedicated quadratic tool with parabola graphing, the quadratic formula calculator includes vertex and axis of symmetry. For fraction arithmetic that often comes up when solving equations, the fraction calculator handles operations step by step.

What Does the Discriminant Tell You?

The discriminant b² - 4ac is the single most informative number in a quadratic equation because it reveals the nature of the roots before you finish solving. A positive discriminant means the parabola crosses the x-axis twice, zero means it touches the axis at a single vertex point, and a negative value means the parabola never meets the axis at all and the two roots are complex conjugates.

The formula traces back to the Babylonians around 2000 BCE, though they used geometric "completing the square" methods rather than symbolic algebra. The modern form was popularised by Persian mathematician al-Khwarizmi in his 820 CE treatise al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, from which the word "algebra" is derived. The unified quadratic formula as we write it today only appeared once negative numbers were accepted into European mathematics in the 16th and 17th centuries.

Worked example with complex roots: For x² + 2x + 5 = 0, the discriminant is 2² - 4(1)(5) = 4 - 20 = -16. Since it is negative, the roots are complex: x = (-2 ± √-16) / 2 = (-2 ± 4i) / 2 = -1 ± 2i.

Cramer's Rule vs Substitution vs Elimination

Systems of linear equations can be solved three common ways, and each has a best use case. Cramer's rule, named after Swiss mathematician Gabriel Cramer and published in 1750, scales poorly for large systems because it requires computing a determinant for every variable. For a 2×2 system it is the fastest method by hand; for a 10×10 system, Gaussian elimination finishes in roughly one-hundredth the time.

Method	Best For	Main Drawback
Substitution	One variable has a coefficient of 1	Gets messy with three or more variables
Elimination (add/subtract)	Equations already aligned nicely	Requires matching coefficients first
Cramer's rule	Small 2×2 or 3×3 systems	Scales as O(n!) - impractical past n = 4
Gaussian elimination	Larger systems, computer implementation	More bookkeeping by hand
Matrix inverse	Repeated solves with the same coefficient matrix	Computing the inverse is itself expensive

This tool uses Cramer's rule because the step-by-step output maps cleanly onto three determinant calculations, which are easy to verify by hand. For systems larger than 2×2 you would reach for row reduction instead.

Common Mistakes When Solving Equations

Most errors when solving equations come from a handful of slips that even experienced students make under time pressure. According to research from the National Council of Teachers of Mathematics (NCTM), sign errors account for roughly 40% of wrong answers on standardised algebra exams, followed by errors in applying the distributive property and forgetting the ± in the quadratic formula.

• Sign errors on -b: The quadratic formula uses -b, so if b is already negative the -b becomes positive. For b = -5, the term -b = 5, not -5.
• Forgetting both roots: A quadratic has two roots (counting multiplicity). Always write x₁ and x₂ even if they happen to be equal.
• Dividing by a variable: If you divide both sides of x² = 5x by x, you lose the root x = 0. Factor instead: x² - 5x = 0, so x(x - 5) = 0.
• Squaring both sides: Introduces extraneous solutions. Always check answers against the original equation.
• Mishandling 0 = 0: This indicates infinitely many solutions, not no solution. The opposite slip is reading 5 = 3 as "answer is 2" rather than recognising a contradiction.

If you find yourself getting fraction arithmetic wrong during substitution, the fraction calculator can handle the intermediate steps without rounding errors.

Where Equation-Solving Shows Up in Real Life

Algebraic equations are not just exam fodder. Quadratics model the motion of any object under gravity, so physicists and engineers solve them constantly. When you drop a ball from height h₀ with initial velocity v₀, its height at time t follows h(t) = -½gt² + v₀t + h₀, where g ≈ 9.81 m/s² on Earth. Setting h(t) = 0 and solving the quadratic tells you exactly when the ball hits the ground.

Linear systems appear in chemistry (balancing reactions), economics (supply-demand equilibrium), and nutrition (mixing foods to hit target calories and protein). A common finance example: allocating £10,000 between two funds with target returns 4% and 7% to hit an overall 5.5% return requires solving a simple 2×2 system. If x is the amount in the 4% fund and y is the amount in the 7% fund:

1. x + y = 10,000 (total invested)
2. 0.04x + 0.07y = 550 (target return = 5.5% of 10,000)

Applying the solver gives x = 5,000 and y = 5,000, so an even split hits the target. The same approach underpins much of mean-variance portfolio construction that is used in the modern investment management industry.

Calculator vs Pencil: When to Use Each

Research published by the American Mathematical Society in 2024 found that students who used calculators alongside hand-working performed 18% better on conceptual follow-up questions than those who used either exclusively. The takeaway is that the solver is most useful as a checker: work a problem by hand, then compare against the tool's step-by-step output to see which step went wrong. Over-reliance on the calculator alone leaves gaps in fluency that show up when you need to manipulate an equation inside a larger proof.

For quadratic equations specifically, factoring should always be your first attempt when the coefficients are small integers. Only if factoring fails should you fall back on the quadratic formula. For larger or non-integer coefficients, the formula wins. The step-by-step output from this tool mirrors the working you would write on paper, so it doubles as a worked example for revision.

Related tools on ToolboxKit include the scientific calculator for the arithmetic inside each step, and percentage calculations which often appear when converting word problems into equations.

All solving happens in your browser. No data is sent to any server.

Sources

• Wolfram MathWorld - Quadratic Formula
• Wolfram MathWorld - Cramer's Rule
• Britannica - al-Khwarizmi and the origin of algebra
• National Council of Teachers of Mathematics (NCTM)
• American Mathematical Society
• Khan Academy - Quadratic Functions and Equations