Heart of Algebra

Lines & Graphs

The heart of the SAT Math section. Master slope, standard form, and interpreting linear models.

Foundation: The 4 Essential Forms

Before you learn the hacks, you must fluent in the four languages of linear equations.

Slope-Intercept Form

Slope-Intercept Form

y = mx + b

The most common form. It tells you exactly where to start (b) and where to go (m).

  • 👉 m = slope (rise/run)
  • 👉 b = y-intercept (starting value)
  • Best for: Graphing & Word Problems
Point-Slope Form

Point-Slope Form

\( y - y_1 = m(x - x_1) \)
where
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \tan(\theta) \)

The most flexible form. Use this when you have a random point (not the intercept) and the slope.

  • 👉 (x₁, y₁) = any point on the line
  • 👉 m = slope = \( \frac{y_2 - y_1}{x_2 - x_1} = \tan(\theta) \)
  • Best for: Writing equations from 2 points
Intercept Form

Intercept Form

\( \frac{x}{a} + \frac{y}{b} = 1 \)

The "Magic" form. If you know where the line crosses the axes, you write this instantly.

  • 👉 a = x-intercept (where y=0)
  • 👉 b = y-intercept (where x=0)
  • Best for: Speed-running graph problems
Standard Form Intercepts

Standard Form

Ax + By = C

The "pretty" form where A, B, and C are integers. It hides the slope, but reveals intercepts easily.

  • 👉 A, B, C = Integers (No Fractions)
  • 👉 Slope = -A/B (The Hack!)
  • Best for: Systems of Equations & Intercepts

The "Standard Form" Secret

Most students waste time converting \(Ax + By = C\) into \(y = mx + b\). The SAT knows this and times you out. Here is the Practix method.

Feature School Way (Slow) Practix Way (Fast)
Find Slope (m) Subtract Ax, divide by B, simplify... \[ m = -\frac{A}{B} \]
Find y-intercept Set x=0, solve for y... \[ y = \frac{C}{B} \]
Find x-intercept Set y=0, solve for x... \[ x = \frac{C}{A} \]
⚡ Example:

"What is the slope of \(3x + 4y = 12\)?"

Practix Way: \(-A/B\) means \(-3/4\). Done in 1 second.

💡
Desmos Pro-Tip: The "Double-X" Trap

Did you get a vertical line when you expected a diagonal one? Check your variables! Typing 3x + 4x = 12 simplifies to 7x = 12. Desmos only sees one variable, so it graphs a constant $x$ value. Always ensure you have both an x and a y for diagonal lines!

Interpreting Word Problems

In SAT word problems, "Slope" and "Y-Intercept" are code words.

Slope (m) means:

  • 👉 Rate of change
  • 👉 "Per", "Each", "Every"
  • 👉 Speed, Cost per unit

Y-Intercept (b) means:

  • 👉 Starting amount
  • 👉 Flat fee / Initial cost
  • 👉 At time = 0
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The Cheat Sheet

Slope formula, Point-Slope form, and the "Standard Form Hack".

View Formulas →
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Desmos Hacks

How to graph standard form instantly and find intersections without algebra.

Open Calculator Guide →
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The Traps

"Infinite Solutions" vs "No Solution" logic puzzles that trick 80% of students.

Review Hard Questions →