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Linear System (3 × 3)

Algebra

Cramer's rule with full determinant working.

Augmented matrix

Coefficients for aᵢx + bᵢy + cᵢz = dᵢ.

x
y
z
=

Solution

|A|-1
|Aₓ|-2
|Aᵧ|-3
|A_z|1
ClassificationUnique solution
x = |Aₓ| / |A|2
y = |Aᵧ| / |A|3
z = |A_z| / |A|-1

Cofactor expansions

|A| cofactor expansion
2 · (-4) − 1 · (-2) + -1 · (-5) = -1
|Aₓ| cofactor expansion
8 · (-4) − 1 · (-16) + -1 · (-14) = -2
|Aᵧ| cofactor expansion
2 · (-16) − 8 · (-2) + -1 · (-13) = -3
|A_z| cofactor expansion
2 · (14) − 1 · (-13) + 8 · (-5) = 1

Frequently asked questions

▶What does the Linear System (3 × 3) do?

The Linear System (3 × 3) is an interactive algebra lab where you change inputs and the answers update live, helping you build intuition without doing the arithmetic by hand.

▶Who should use the Linear System (3 × 3)?

Students from primary to senior secondary classes, parents helping with homework, and anyone revising algebra can use this lab to check answers and explore "what-if" scenarios.

▶Is the Linear System (3 × 3) free to use?

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▶Does the Linear System (3 × 3) work offline?

Once the page is loaded, all calculations run locally in JavaScript, so you can keep using it even on a flaky connection.

▶How accurate are the results?

Results use standard JavaScript double-precision arithmetic and the textbook formula for algebra. We never display scientific notation : numbers are shown as ordinary decimals you can read at a glance.