Study guide: Systems of Two Linear Equations
A system requires the same ordered pair to satisfy every equation. Substitution and elimination find that shared pair, while parallel or coincident line structure explains whether a system has none or many solutions.
Ideas to explain in your own words
- Substitution uses one equation to express a variable, then replaces that variable in the other equation. The resulting one-variable equation preserves the need to satisfy both original relationships. Use parentheses around a substituted expression, particularly when it is multiplied or subtracted.
- Elimination combines equations so one variable’s coefficients sum to zero. The surviving equation finds one coordinate, and substitution into an original equation finds the other. Add entire equations, including their right sides.
- Lines with equal slopes and different intercepts never meet in the coordinate plane. Their equations demand incompatible outputs for each input, so the corresponding linear system has no solution. Equal slopes alone do not settle the solution count.
- A total count and a weighted total can become two equations. Comparing a low-price baseline with the actual total reveals how many items must use the higher price. The counted groups must be disjoint and cover the whole supplied total.
Useful vocabulary
- system
- Several equations that must be satisfied together.
- substitution
- Replacing a quantity with an equal expression.
- elimination
- Combining equations to remove a variable term.
Watch your reasoning
- Use parentheses around a substituted expression, particularly when it is multiplied or subtracted. Verify both coordinates in both original equations; satisfying only one equation is insufficient for a system.
- Add entire equations, including their right sides. A zero coefficient is a purposeful cancellation here; preserve the sign of a subtracted variable and verify the final pair in both relationships.
- Equal slopes alone do not settle the solution count. Equal intercepts would make the same line with infinitely many solutions; different intercepts establish distinct parallel lines and therefore no shared point.
A worked review example
Complete supplied guide examples: Replace One Variable with an Equal Expression — Solve y=2x+1 and x+y=7. Candidate answers: (A) (2,5) (B) (2,6) (C) (3,5) (D) (5,2). Substitute 2x+1 for y: x+2x+1=7. Then 3x=6, x=2; y=2(2)+1=5. The ordered pair satisfies both equations. Add Equations to Cancel a Variable — Solve x+y=5 and x−y=-1. Candidate answers: (A) (2,3) (B) (3,2) (C) (4,2) (D) (5,-1). Add the equations so y cancels: 2x=4, x=2. Substitute in x+y=5 to get y=3. Recognize Parallel Distinct Lines — Classify y=2x+1 and y=2x+3. Candidate answers: (A) Infinitely many intersections (B) No intersection: parallel distinct lines (C) One intersection at x=0 (D) One intersection at x=2. The lines have equal slope 2 but different y-intercepts 1 and 3. Their vertical separation stays 2, so the system is inconsistent. Use Two Constraints to Recover Two Counts — A fictional stand sells $2 tickets and $3 tickets. It sells 5 tickets for $13. How many $3 tickets were sold? Candidate answers: (A) 2 tickets (B) 3 tickets (C) 4 tickets (D) 6 tickets. If all 5 tickets were $2, revenue would be $10. Each $3 ticket adds $1. The extra $3 therefore means 3 of the $3 tickets.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.