39 Systems of Linear Equations

Systems of Linear Equations is the study of solving two or more linear equations simultaneously, finding the values of the shared variables that satisfy every equation in the set at once, using graphical reasoning and algebraic methods including substitution and elimination.

The Scope of Linear Systems

A system of linear equations is a collection of two or more linear equations considered together, sharing the same set of variables. A solution to a system is an ordered pair (or larger tuple, for systems with more variables) that satisfies every equation in the system simultaneously — substituting the solution into any one of the equations must produce a true statement, and this must hold for all equations in the system at once, not just one of them.

Solving a System Graphically

Because each linear equation in two variables graphs as a straight line, a system of two such equations can be solved graphically by graphing both lines on the same coordinate plane and identifying their point of intersection, which represents the single ordered pair satisfying both equations simultaneously. This method is intuitive and directly illustrates why a system's solution is called simultaneous, but it depends on the intersection point falling on visible, easily readable gridline coordinates to produce an exact answer rather than an estimate.

Solving a System by Substitution

The substitution method solves a system algebraically by isolating one variable in one equation, then substituting that expression into the other equation to produce a single equation in one variable, which is solved using ordinary linear equation techniques.

Substituting the first equation's expression for y into the second gives 3x + (2x + 1) = 11, which simplifies to 5x + 1 = 11 and solves to x = 2; substituting x = 2 back into either original equation gives y = 5, so the system's solution is (2, 5). Substitution is most efficient when one equation already isolates a variable, or can be isolated with minimal effort.

Solving a System by Direct Elimination

The elimination method solves a system algebraically by adding or subtracting the two equations to cancel one variable entirely, made possible when that variable's coefficients in the two equations are already identical or exact opposites.

Subtracting the second equation from the first eliminates x directly, since both equations share a coefficient of 2 on that variable, leaving 4y = 12, which solves to y = 3; substituting y = 3 back into either original equation gives x = 3.5, so the system's solution is (3.5, 3).

Solving a System by Scaled Elimination

When neither variable's coefficients match or are exact opposites in the two given equations, the scaled elimination method multiplies one or both equations by a chosen constant to force a matching pair of coefficients before adding or subtracting.

Multiplying the second equation entirely by -3 produces -3x - 12y = -27, which, when added to the first equation, eliminates x and leaves -10y = -14, solving to y = 1.4; substituting back gives x = 3.4. Choosing the multiplier that produces matching or opposite coefficients on the variable intended for elimination is the key strategic decision in this method.

Classifying Linear System Solutions

A system of two linear equations falls into one of three classification categories, mirroring the identity and contradiction cases seen in single-variable equations. A system with exactly one solution corresponds to two lines that intersect at a single point (differing slopes). A system with no solution corresponds to two parallel, non-identical lines (identical slopes, different y-intercepts), and attempting to solve such a system algebraically produces a false constant statement, such as 0 = 5. A system with infinitely many solutions corresponds to two equations describing the identical line (identical slopes and identical y-intercepts), and attempting to solve such a system algebraically produces a true constant statement, such as 0 = 0.

Applications of Elementary Linear Systems

Systems of two linear equations model situations involving two unknown quantities related by two independent conditions, such as mixture problems, cost-and-revenue break-even problems, and problems comparing two rates or two group compositions. Constructing the system requires translating each of the two stated conditions into its own linear equation using the same verbal translation skills applied to single-equation applications, after which the system is solved using whichever algebraic method is most efficient for its particular coefficients.

Selecting a Method and Verifying a Solution

Choosing between substitution and elimination is typically a matter of efficiency: substitution is favored when a variable is already isolated or nearly so, while elimination is favored when the coefficients of one variable already match or are simple multiples of each other. Regardless of the method used, a system's solution is verified by substituting the found ordered pair into both original equations independently and confirming that each one is satisfied.

Diagnosing Errors in Linear Systems

Common errors in this area include substituting an isolated expression into the same equation it came from rather than into the other equation, adding two equations when subtraction was required for elimination (or the reverse), forgetting to multiply every term of an equation, including its constant term, when scaling for elimination, and misclassifying a system's solution set by stopping the solving process at a false or true constant statement without recognizing what that outcome specifically implies about the number of solutions.