BIZOMAT MAKEOS · Mathematics
Simultaneous Equations and 3 × 3 Matrix
What solves these two or three simultaneous linear equations, and what are the determinant and inverse?
Solution of two or three simultaneous linear equations, with the determinant, inverse and transpose of the coefficient matrix.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Size | choice | Two or three equations (the third row and column are ignored for two). |
| Equation 1 | ||
| Coefficient of x1 | Coefficient of x1 in equation 1. | |
| Coefficient of x2 | Coefficient of x2 in equation 1. | |
| Coefficient of x3 | Coefficient of x3 in equation 1. | |
| Right-hand side | Constant on the right-hand side of equation 1. | |
| Equation 2 | ||
| Coefficient of x1 | Coefficient of x1 in equation 2. | |
| Coefficient of x2 | Coefficient of x2 in equation 2. | |
| Coefficient of x3 | Coefficient of x3 in equation 2. | |
| Right-hand side | Constant on the right-hand side of equation 2. | |
| Equation 3 | ||
| Coefficient of x1 | Coefficient of x1 in equation 3. | |
| Coefficient of x2 | Coefficient of x2 in equation 3. | |
| Coefficient of x3 | Coefficient of x3 in equation 3. | |
| Right-hand side | Constant on the right-hand side of equation 3. | |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Size
- What it is
- How many equations and unknowns the system has: two equations in x1 and x2, or three in x1, x2 and x3.
- Units and format
- Choose one of 2 options from the list: 2 equations, 2 unknowns; 3 equations, 3 unknowns.
- Example
- The form opens with “3 equations, 3 unknowns” selected as a worked example. Choose the option that matches your case.
- Get it right
- With two equations, the third column and Equation 3 are ignored.
Equation 1
What this means: Coefficient of x1
- What it is
- The number multiplying x1 in the first equation, once the equation is written with all unknowns on the left and the constant on the right.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type 0 if x1 does not appear in this equation.
What this means: Coefficient of x2
- What it is
- The number multiplying x2 in the first equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Include the sign; a term written as −x2 has a coefficient of −1.
What this means: Coefficient of x3
- What it is
- The number multiplying x3 in the first equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when the size is two equations.
What this means: Right-hand side
- What it is
- The constant on the right-hand side of the first equation, after every term containing an unknown has been moved to the left.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A constant moved across the equals sign changes sign.
Equation 2
What this means: Coefficient of x1
- What it is
- The number multiplying x1 in the second equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type 0 if x1 does not appear in this equation.
What this means: Coefficient of x2
- What it is
- The number multiplying x2 in the second equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Include the sign; a term written as −x2 has a coefficient of −1.
What this means: Coefficient of x3
- What it is
- The number multiplying x3 in the second equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when the size is two equations.
What this means: Right-hand side
- What it is
- The constant on the right-hand side of the second equation, after every term containing an unknown has been moved to the left.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -11 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A constant moved across the equals sign changes sign.
Equation 3
What this means: Coefficient of x1
- What it is
- The number multiplying x1 in the third equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Equation 3 is ignored when the size is two equations.
What this means: Coefficient of x2
- What it is
- The number multiplying x2 in the third equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Equation 3 is ignored when the size is two equations.
What this means: Coefficient of x3
- What it is
- The number multiplying x3 in the third equation, with all unknowns on the left-hand side.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Equation 3 is ignored when the size is two equations.
What this means: Right-hand side
- What it is
- The constant on the right-hand side of the third equation, after every term containing an unknown has been moved to the left.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Equation 3 is ignored when the size is two equations.
Method and formulas
- Determinant by cofactor expansion; the system has a unique solution when it is not zero.
- Inverse = adjugate ÷ determinant; solution x = A⁻¹ b (Cramer’s rule).
- Transpose swaps rows and columns.
- Example: the default system gives x₁ = 2, x₂ = 3, x₃ = -1.
Limitations
- Nearly singular matrices (determinant close to zero) give unreliable solutions.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Solve force balance and reaction equations
- Invert small matrices for transformations
- Check hand matrix calculations
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Solution x₁, x₂ (, x₃)
- Determinant
- Inverse and transpose of the matrix
Interpreting the result
- Substitute the solution back into the original equations as a check.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 122–127. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · Library collection
Simultaneous equations and matrices in practice: blending, product mix, structures and circuits
In Practical Mathematics for Engineering and Manufacturing
What it covers: How pairs and systems of equations solve blending, product-mix, structural and circuit problems, how matrices and spreadsheets handle larger systems, and how to spot unreliable solutions.
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