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Systems of Equations
Graphing, Substitution, and Elimination
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A system of linear equations is a set of two or more equations that share the same variables. The solution is the point (or points) where all equations are satisfied simultaneously - graphically, the intersection of the lines. Three methods for solving systems exist: graphing (visual but imprecise), substitution (isolate one variable, substitute into the other equation), and elimination (add or subtract equations to cancel a variable).
Choosing the right method speeds up calculation. Substitution works best when one equation already has an isolated variable. Elimination shines when coefficients are easy to match. Graphing is ideal for visualizing the relationship and checking answers. All three give the same solution - pick whichever is most efficient for the problem.
Key Facts
- One solution: lines intersect at exactly one point (consistent and independent).
- No solution: parallel lines - same slope, different y-intercept (inconsistent).
- Infinite solutions: same line - identical slope and intercept (dependent).
- Substitution: solve one equation for a variable, substitute into the other.
- Elimination: multiply equations to match a coefficient, then add/subtract to eliminate that variable.
- Check: substitute your solution back into both original equations to verify.
Vocabulary
- System of equations
- Two or more equations with the same set of variables, solved simultaneously.
- Solution of a system
- An ordered pair (x, y) that satisfies all equations in the system simultaneously.
- Consistent system
- A system with at least one solution.
- Inconsistent system
- A system with no solution; lines are parallel.
- Dependent system
- A system with infinitely many solutions; both equations describe the same line.
Common Mistakes to Avoid
- Stopping after finding x without solving for y. The solution is an ordered pair - you need both values.
- During elimination, multiplying only one term in an equation instead of the entire equation by the scalar.
- Forgetting to check the solution in both original equations. An arithmetic error in one step can pass undetected without checking.
- Misidentifying parallel lines (no solution) as having a negative solution. If elimination gives (), the system is inconsistent with no solution.
Practice Questions
- 1 Solve by substitution: 2x + y = 7 and x - y = 2.
- 2 Solve by elimination: 3x + 2y = 12 and 5x - 2y = 4.
- 3 Identify the type of system: y = 2x + 3 and 4x - 2y = -6. Justify your answer.