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PREPITUS
AlgebraFoundational

Linear Equations in Two Variables

Read any line's slope and intercepts from any form of its equation — and build equations from points, tables, or stories.

⏱ 7 min📍 Learn first: Linear Equations in One Variable
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Learn Linear Equations in Two Variables in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:37

Linear Equation Forms and Solving Systems

A short spoken refresher on this concept.

00:00 / 20:21

Linear Equations as a Coding Language

The full lesson, worked through in depth.

Concept Snapshot

Definition
An equation like ax + by = c whose solutions (x, y) form a straight line. Every point on the line satisfies the equation.
What the SAT tests
Converting between standard and slope-intercept form, finding intercepts, and writing equations from two points or a context.
Key rule
Slope m = (y₂ − y₁)/(x₂ − x₁). From ax + by = c: slope = −a/b, x-intercept = c/a, y-intercept = c/b.
Most common mistake
Computing slope as Δx/Δy — run over rise — and getting the reciprocal.
Prerequisite
Linear Equations in One Variable

Visual Explanation

What to picture: Slope m = (y₂ − y₁)/(x₂ − x₁). From ax + by = c: slope = −a/b, x-intercept = c/a, y-intercept = c/b.

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

DefinitionAn equation like ax + by = c whose solutions (x, y) form a straight line. Every point on the line satisfies the equation.
Key ruleSlope m = (y₂ − y₁)/(x₂ − x₁). From ax + by = c: slope = −a/b, x-intercept = c/a, y-intercept = c/b.

Linear Equations in Two Variables

Algebra

Two Points → EquationCompute slope, then plug one point into y = mx + b.
Intercept ReadingFind where the line crosses an axis — set the other variable to 0.
Parallel / PerpendicularSame slope = parallel; negative reciprocal = perpendicular.

How the SAT Tests It

Pattern

Two Points → Equation

Compute slope, then plug one point into y = mx + b.

Clue: Two ordered pairs given
Pattern

Intercept Reading

Find where the line crosses an axis — set the other variable to 0.

Clue: "x-intercept", "crosses the y-axis"
Pattern

Parallel / Perpendicular

Same slope = parallel; negative reciprocal = perpendicular.

Clue: "parallel to", "perpendicular to"

Worked Example

A line passes through (2, 7) and (5, 16). What is its equation in slope-intercept form?

Two points determine the line — we need m and b.
m = (16 − 7)/(5 − 2) = 9/3 = 3. Note the order: y-difference on top.
y = 3x + b through (2, 7): 7 = 6 + b, so b = 1.
y = 3x + 1 at x = 5: y = 16. ✓ Equation: y = 3x + 1.

Try It Yourself

The line 3x + 4y = 24 crosses the y-axis at which point?

Common Traps

What students do

Swapping the intercept recipes.

Why it fails

Both intercepts involve setting a variable to zero — it's easy to zero the wrong one under time pressure.

Do this instead

Say it in words: crossing the y-axis means x is 0. The axis you're ON is the variable that survives.

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

Linear Equations as a Coding Language

Slope formula fluency and intercept reading in every form.

00:00 / 20:21

20:21 · the full lesson

What to notice

Slope m = (y₂ − y₁)/(x₂ − x₁). From ax + by = c: slope = −a/b, x-intercept = c/a, y-intercept = c/b.

What the lesson covers

  • 0:00 Slope formula
  • 1:30 Standard form shortcuts
  • 3:00 Building equations
  • 4:30 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 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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