What is a system of equations?

A system of equations is a set of two or more equations containing the same variables, solved simultaneously to find values that satisfy all of them.

What are the methods for solving a system of equations?

The primary algebraic methods are substitution (replacing a variable with an equivalent expression), elimination (adding or subtracting equations to eliminate a variable), and graphing (finding the intersection point on a plot).

What is an inconsistent system?

An inconsistent system of equations has no solution, which occurs when the equations represent parallel lines that never intersect.

What is a dependent system?

A dependent system has infinitely many solutions because the equations represent the exact same line, meaning every point on the line is a solution.

How many solutions can a linear system have?

A system of linear equations in two variables can have zero solutions (parallel lines), one unique solution (intersecting lines), or infinitely many solutions (coincident lines).

How many solutions can a nonlinear system have?

The number of solutions depends on the degrees of the polynomials. For example, a line and a circle can intersect at 0, 1, or 2 points.

What is Cramer's Rule?

Cramer's Rule is an algebraic method that uses determinants to solve systems of linear equations. It is highly efficient for small systems.

What is the determinant of a system?

The determinant $D$ of a linear system is calculated from the coefficients of the variables. If $D = 0$, the system does not have a unique solution.

Can this calculator solve quadratic systems?

Yes, it can solve systems containing quadratic terms, such as a line intersecting a parabola.

What is the graphical representation of a system's solution?

The solution is represented graphically by the coordinates $(x, y)$ of the point or points where the graphs of the equations intersect.

How do you check if a solution is correct?

Substitute the calculated values of the variables back into all the original equations. The values must satisfy every equation in the system.

What is a homogeneous system?

A homogeneous system of linear equations is one in which all the constant terms are zero ($c_1 = 0, c_2 = 0$). It always has at least the trivial solution ($x=0, y=0$).

math calculator

System of Linear & Nonlinear Equations Solver

Solve systems of 2 linear or nonlinear equations using substitution, elimination, and graphing methods.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Equation System Configuration

Line 1: a₁x + b₁y = c₁

Coefficient a₁

Coefficient b₁

Constant c₁

Line 2: a₂x + b₂y = c₂

Coefficient a₂

Coefficient b₂

Constant c₂

Results

Solution Status

UNIQUE

Intersection Point 1(x = 3.0000, y = 2.0000)

Equations Intersection Graph

Blue: Eq 1 | Green: Eq 2 | Orange dot: Intersections

Step-by-Step Derivation

Given the linear system:

\[(1)\quad 2x +3y = 12\]

\[(2)\quad 1x -1y = 1\]

Apply Cramer\'s Rule:

Calculate the determinant of the coefficient matrix D:

\[D = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (1)(3) = -5\]

Calculate Dx and Dy:

\[D_x = \begin{vmatrix} 12 & 3 \\ 1 & -1 \end{vmatrix} = (12)(-1) - (1)(3) = -15\]

\[D_y = \begin{vmatrix} 2 & 12 \\ 1 & 1 \end{vmatrix} = (2)(1) - (1)(12) = -10\]

Solve for x and y:

\[x = \frac{D_x}{D} = \frac{-15}{-5} = 3.0000\]

\[y = \frac{D_y}{D} = \frac{-10}{-5} = 2.0000\]

Formula

Equations System Solver formulas

x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}

Linear systems use determinants. Nonlinear systems substitute linear bounds into quadratic terms to compute real intersection points.

Step by step

How the calculation works

1Select the equation model (Linear vs. Nonlinear).
2For Linear systems, enter coefficients for equations 1 & 2 in standard form: ax + by = c.
3For Nonlinear systems, define the straight line slope/offset and the parabola polynomial.

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Learn more about this calculation

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Matrix Solver

Solve larger systems using determinants and matrices.

Quadratic Solver

Solve polynomial roots.

math calculator

System of Linear & Nonlinear Equations Solver

Solve systems of 2 linear or nonlinear equations using substitution, elimination, and graphing methods.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Equation System Configuration

Line 1: a₁x + b₁y = c₁

Coefficient a₁

Coefficient b₁

Constant c₁

Line 2: a₂x + b₂y = c₂

Coefficient a₂

Coefficient b₂

Constant c₂

Results

Solution Status

UNIQUE

Intersection Point 1(x = 3.0000, y = 2.0000)

Equations Intersection Graph

Blue: Eq 1 | Green: Eq 2 | Orange dot: Intersections

Step-by-Step Derivation

Given the linear system:

\[(1)\quad 2x +3y = 12\]

\[(2)\quad 1x -1y = 1\]

Apply Cramer\'s Rule:

Calculate the determinant of the coefficient matrix D:

\[D = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (1)(3) = -5\]

Calculate Dx and Dy:

\[D_x = \begin{vmatrix} 12 & 3 \\ 1 & -1 \end{vmatrix} = (12)(-1) - (1)(3) = -15\]

\[D_y = \begin{vmatrix} 2 & 12 \\ 1 & 1 \end{vmatrix} = (2)(1) - (1)(12) = -10\]

Solve for x and y:

\[x = \frac{D_x}{D} = \frac{-15}{-5} = 3.0000\]

\[y = \frac{D_y}{D} = \frac{-10}{-5} = 2.0000\]

Formula

Equations System Solver formulas

x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}

Linear systems use determinants. Nonlinear systems substitute linear bounds into quadratic terms to compute real intersection points.

Step by step

How the calculation works

1Select the equation model (Linear vs. Nonlinear).
2For Linear systems, enter coefficients for equations 1 & 2 in standard form: ax + by = c.
3For Nonlinear systems, define the straight line slope/offset and the parabola polynomial.

Related guides

Learn more about this calculation

Guide

Ratio and Proportion Guide: Simplify, Solve, Split Totals

Learn how to simplify ratios, solve proportions, and split totals with worked examples tied to the live DTC ratio calculator.

Site Guide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Learn the fastest reliable workflow for using Do The Calculation calculators, reading results, checking formulas, and using save, print, share, and export actions correctly.

Matrix Solver

Solve larger systems using determinants and matrices.

Quadratic Solver

Solve polynomial roots.