Math · College algebra · Concept
Systems of linear equations in two variables
A system of linear equations asks for the values that satisfy every equation at once. With two variables each equation is a line, and the solutions are where the lines meet: one point, no point (parallel lines) or infinitely many points (the same line). Substitution and elimination find them exactly, and a check in both original equations confirms them.
A solution satisfies every equation
An ordered pair (x, y) solves the system only if it makes both equations true. Checking a candidate in both original equations is the last step of every method.
Graphing: where the lines cross
Each linear equation in x and y is a line. Two lines meet once, never (they are parallel) or everywhere (they are the same line). A graph shows which case you are in and roughly where, but exact coordinates come from algebra.
Substitution
Solve one equation for one variable, substitute that expression into the other equation, and solve the one-variable equation that results. Substitution is quickest when a variable already has coefficient 1 or −1.
Elimination
Multiply the equations by constants so that one variable has opposite coefficients, then add the equations to eliminate it. Multiplying an equation by a nonzero number and adding two equations do not change the solutions.
One solution, none or infinitely many
Elimination tells the three cases apart.
| Result | The lines | Solutions |
|---|---|---|
| A value for each variable | Cross once | Exactly one |
| A false statement, such as 0 = 3 | Parallel | None: the system is inconsistent |
| A true statement, such as 0 = 0 | The same line | Infinitely many: the system is dependent |
More variables
The same ideas work with three or more equations and unknowns, but the bookkeeping grows. An augmented matrix and row reduction organize it; Matrices & linear systems carries out the row operations exactly.
Common mistakes
- Stopping after one variable: a solution of a two-variable system needs both x and y.
- Multiplying only some terms of an equation, or only one side, before adding.
- Checking the answer in only one equation, or in a rearranged equation that may already contain a slip.
- Reading 0 = 0 as "no solution": it means the equations describe the same line, so there are infinitely many solutions.
Key terms
- Linear system
- Two or more linear equations in the same unknowns, such as 2x + y = 5 and x − y = 1. A solution has to satisfy every equation at once.
- Substitution method
- A way to solve a system of equations: solve one equation for one variable, substitute that expression into the other equation, and solve the one-variable equation that results.
- Elimination method
- A way to solve a system of equations: multiply the equations by constants so that one variable has opposite coefficients, then add them to eliminate that variable.
- Inconsistent system
- A system with no solution because its equations contradict each other; with two variables, the lines are parallel. In row reduction, a row [0 … 0 | c] with c ≠ 0 says 0 = c.
- Dependent system
- A consistent system whose equations do not all carry independent information, so it has infinitely many solutions. With two variables, both equations describe the same line.
- Augmented matrix
- A matrix holding a system’s coefficients, with the constants from the right-hand sides added as a last column. That column holds the equations’ values, not another unknown.
Work through an example
Solve the system 2x + 3y = 12 and x − y = 1.
Solve a system of equations by elimination →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
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