Systems of linear equations

Substitution is efficient when one equation already isolates a variable, or when a variable has coefficient 1 or -1. Rearrange one equation, replace that variable in the other equation, then back-substitute.

Use parentheses around the substituted expression. This protects signs and ensures that every term is multiplied correctly.

If substitution produces a true identity such as $0=0$, the equations are dependent and further interpretation is needed. If it produces a contradiction such as $0=5$, the system is inconsistent.

Elimination combines equations so that one variable cancels. Multiply one or both equations by nonzero constants to create equal or opposite coefficients, then add or subtract the complete equations.

Write equations in aligned standard form before combining them. This reduces sign errors and makes omitted terms visible.

After finding one variable, substitute into an original equation. A final check is especially important when equations were multiplied or subtracted.

For a 3 by 3 system, eliminate the same variable from two different pairs of equations. This creates a 2 by 2 system in the remaining variables.

Label the equations and record each operation. A clear elimination chain earns method marks and makes the work auditable.

Once two variables are known, back-substitute to find the third, then verify the ordered triple in all three original equations.

A linear system can be written compactly as $A\mathbf{x}=\mathbf{b}$, where $A$ is the coefficient matrix, $\mathbf{x}$ is the column of unknowns and $\mathbf{b}$ is the constants column.

The augmented matrix places the constants column beside the coefficients. The vertical divider is organizational; row operations act across the complete row, including the augmented entry.

Fix the variable order before building the matrix. A missing variable must be represented by a zero coefficient.

Three reversible row operations preserve the solution set: interchange two rows, multiply a row by a nonzero constant, or add a multiple of one row to another row.

Row replacement is written as a single instruction, for example $R_{2}$ to $R_{2}$-$2R_{1}$. Calculate every new entry from the old rows before moving to the next operation.

Multiplying by zero is forbidden because it destroys information and is not reversible.

Gaussian elimination uses row operations to reach row-echelon form: pivots move to the right as rows descend, and entries below each pivot are zero.

Once the system is triangular, solve the last equation and back-substitute upward. Gauss-Jordan elimination continues until each pivot is the only nonzero entry in its column, producing reduced row-echelon form.

Exact fractions expose structure. Premature decimal rounding can turn a dependent row into an apparently nonzero row or distort a parameter condition.

A unique solution occurs when every variable column contains a pivot. There are no free variables, so one ordered tuple satisfies the system.

No solution occurs when row reduction creates a contradiction row such as $0=1$. The augmented column introduces a pivot that is absent from the coefficient matrix.

Infinitely many solutions occur when the system is consistent but at least one variable is free. A zero row $0=0$ indicates redundant information, not a contradiction.

Rank is the number of pivots. The system $A\mathbf{x}=\mathbf{b}$ is consistent exactly when $\operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf{b}])$. If that common rank is smaller than the number of unknowns, free variables produce infinitely many solutions.

In two dimensions, each linear equation represents a line. Two nonparallel lines meet once, distinct parallel lines never meet, and coincident lines share every point.

In three dimensions, a linear equation represents a plane. Three planes may meet at one point, share a line or a plane, or have no common point. Pairwise intersections alone do not guarantee one point common to all three planes.

The algebraic and geometric classifications agree: unique means one common point, inconsistent means no common point, and dependent means a common family of points.

Intersecting, parallel and coincident lines correspond to one solution, no solution and infinitely many solutions.

A consistent system has at least one solution. An inconsistent system has no solution.

Independent equations contribute distinct constraints. Dependent equations contain redundant information because one equation is a linear combination of others.

Dependency does not automatically mean infinitely many solutions. A larger system may contain one redundant equation while the remaining independent equations still determine a unique solution.

For a square 2 by 2 system, a nonzero determinant guarantees a unique solution. A zero determinant tells you only that the system is nonunique; compare constants or row-reduce to distinguish no solution from infinitely many solutions.

A parameter changes the coefficients or constants, so the solution type can change at special values. Identify the expression that controls a pivot or determinant, then split into cases.

Never divide by an expression involving a parameter until the zero case has been handled separately. Dividing by k-1 silently discards the case $k=1$.

For a square system, determinant not equal to zero gives a unique solution. At determinant zero, inspect the augmented equations to decide whether the system is dependent or inconsistent.

A modeling problem begins by defining unknowns with units. Translate each independent condition into a linear equation, solve, then interpret the tuple in the original context.

Check feasibility as well as algebra: counts may need to be nonnegative integers, masses and times should be nonnegative, and a monetary total should match the stated revenue.

If the equations are inconsistent, the data or assumptions cannot all hold simultaneously. If they are dependent, the information is insufficient for one unique answer.

A GDC may provide a simultaneous-equation solver, matrix inverse command, row-reduced echelon form command, or numerical solve feature. Enter the coefficients in a fixed variable order and include zeros for missing variables.

For a nonsingular square matrix, $x=A^{-1}b$ is valid. If A is singular, the inverse does not exist; use row reduction to distinguish no solution from infinitely many solutions.

GDC output must still be interpreted. A decimal tuple should be checked in the original equations, and parameter-based classification requires algebraic reasoning rather than sampling a few values.

On examinations, record the equations or augmented matrix and state the solution. Technology is a tool, not a replacement for mathematical communication.

Sign errors often enter when subtracting equations. Put brackets around the full equation being subtracted or add a clearly written negative multiple instead.

A row operation must change every entry in the row, including the augmented constant. Applying it only to selected coefficients changes the system.

Do not confuse a zero row with a contradiction row, dependent equations with inconsistent equations, or a singular matrix with proof of no solution.

In parameter questions, dividing by an expression before treating its zero case loses a possible classification. In contextual problems, failing to interpret units or feasibility leaves the solution incomplete.

Premature decimal rounding can hide exact dependency. Keep exact values during elimination and check the final tuple in every original equation.

Start by deciding whether substitution, elimination, matrices or technology is most efficient. For small exact systems, algebra usually communicates the structure best; for larger numerical systems, a GDC can reduce arithmetic.

Show enough structure for method marks: define variables, align equations, label row operations, identify parameter cases and state the final solution type.

If the system has infinitely many solutions, introduce free parameters and specify their domain. If it has no solution, show the contradiction. If it has one solution, verify the tuple.

For a contextual response, finish with a sentence containing units and a reasonableness check. For approximate GDC results, use the requested accuracy and avoid rounding intermediate values.

Attempt each problem without the solution first. Use exact arithmetic unless an approximation is requested, and verify numerical solutions in the original equations.

The set progresses from two-variable techniques through row reduction, classification, parameters, modeling and three-variable reasoning.

A solution satisfies every equation simultaneously. Use substitution for an isolated variable, elimination for convenient matching coefficients, and Gaussian elimination for systematic matrix work.

One pivot per variable gives a unique solution. A contradiction row gives no solution. A consistent system with free variables has infinitely many solutions.

In two dimensions these cases correspond to intersecting, parallel and coincident lines. In three dimensions, interpret equations as planes and focus on their common intersection.

For parameters, split the zero and nonzero pivot or determinant cases before dividing. With a GDC, record the setup, interpret singular output and verify the result.