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Systems of Two Linear Equations in Two Variables

Solve two equations at once — and know from the slopes alone whether a system has one solution, none, or infinitely many.

⏱ 8 min📍 Learn first: Linear Functions
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13 free resources
00:00 / 01:34

Linear Systems Shortcuts and Setup

A short spoken refresher on this concept.

00:00 / 23:57

Cheat Codes for DSAT Linear Systems

The full lesson, worked through in depth.

Concept Snapshot

Definition
Two linear equations sharing x and y. The solution is the point (or points) satisfying both — graphically, where the lines meet.
What the SAT tests
Solving by substitution or elimination, counting solutions from structure, and building systems from word problems.
Key rule
Different slopes → one solution. Same slope, different intercepts → none (parallel). Same line → infinitely many.
Most common mistake
Adding equations to eliminate a variable but forgetting to multiply EVERY term first.
Prerequisite
Linear Functions

Visual Explanation

What to picture: Different slopes → one solution. Same slope, different intercepts → none (parallel). Same line → infinitely many.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionTwo linear equations sharing x and y. The solution is the point (or points) satisfying both — graphically, where the lines meet.
Key ruleDifferent slopes → one solution. Same slope, different intercepts → none (parallel). Same line → infinitely many.

Systems of Two Linear Equations in Two Variables

Algebra

Solve the SystemFind the (x, y) pair satisfying both equations.
Count the SolutionsChoose k so the system has no solution — make slopes match, intercepts differ.
Word Problem SetupTwo totals (e.g., tickets and revenue) become two equations.

How the SAT Tests It

Pattern

Solve the System

Find the (x, y) pair satisfying both equations.

Clue: "the solution to the system"
Pattern

Count the Solutions

Choose k so the system has no solution — make slopes match, intercepts differ.

Clue: "no solution", "infinitely many"
Pattern

Word Problem Setup

Two totals (e.g., tickets and revenue) become two equations.

Clue: Two quantities, two constraints

Worked Example

Solve the system: 2x + y = 11 and 3x − y = 4.

Find the one (x, y) that satisfies both equations.
The y-coefficients are +1 and −1 — elimination by adding is fastest.
Add: 5x = 15 → x = 3. Substitute: 2(3) + y = 11 → y = 5.
3(3) − 5 = 4 ✓ and 2(3) + 5 = 11 ✓. Solution: (3, 5).

Try It Yourself

For what value of a does this system have NO solution? ax + 2y = 8 and 6x + 4y = 5

Common Traps

What students do

Solving the whole system when the question only counts solutions.

Why it fails

Structure questions are answerable in ten seconds from slope comparison — full solving wastes a minute and invites errors.

Do this instead

Compare coefficient ratios first: a₁/a₂ vs b₁/b₂ vs c₁/c₂. Their pattern tells you one / none / infinite immediately.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Cheat Codes for DSAT Linear Systems

Elimination, substitution, and the ratio shortcut for counting solutions.

00:00 / 23:57

23:57 · the full lesson

What to notice

Different slopes → one solution. Same slope, different intercepts → none (parallel). Same line → infinitely many.

What the lesson covers

  • 0:00 Elimination
  • 1:40 Substitution
  • 3:00 The ratio test
  • 4:40 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Systems of Two Linear Equations in Two Variables notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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