1. Linear Equations & Slope

Linear equations are the backbone of SAT® Algebra, the largest math domain on the test. A linear equation graphs as a straight line and is usually written in slope-intercept form:
\[ y = mx + b, \]
where \(m\) is the slope (steepness) and \(b\) is the y-intercept (where the line crosses the y-axis).

To solve a linear equation, undo the operations to isolate the variable. For \(3x + 7 = 22\): subtract 7 to get \(3x = 15\), then divide by 3 to get \(x = 5\).

Key facts the SAT® tests:

• Slope between two points: \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) (rise over run).
• Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals (like \(2\) and \(-\tfrac{1}{2}\)).
• A positive slope rises left to right; a negative slope falls.

⚠️ Common trap: in \(y = mx + b\), the slope is the number multiplied by \(x\), not the constant. In \(y = -2x + 5\), the slope is \(-2\) and the y-intercept is \(5\).

💡 Tip: whatever you do to one side of an equation, do to the other — that keeps it balanced.