Graphs of Linear Systems and Inequalities

Graphs of Linear Systems and Inequalities

Graphing a Line

From Slope-Intercept Form ()

  1. Plot the y-intercept
  2. From there, use slope to find a second point
  3. Draw the line through both points

From Standard Form ()

  1. Find x-intercept: set → point
  2. Find y-intercept: set → point
  3. Draw the line through both intercepts

Example:

xy¡4¡3¡2¡11234¡4¡3¡2¡11234(0;¡1)(3;1)y=23x¡1

Reading Slope from a Graph

Given two visible points and :

Visual clues:

  • Steeper line = larger
  • Flatter line = smaller
  • Horizontal = , Vertical = undefined

Graphing Systems of Equations

The solution of a system is the intersection point of the two lines.

Example: and

xy¡2¡112345¡2¡112345(1;2)y=x+1y=¡x+3

Solution: — where the lines cross.

Visual Interpretation of Number of Solutions

Graph Solutions
Lines cross at one point One solution
Lines are parallel (never cross) No solution
Lines overlap completely Infinitely many

Graphing Linear Inequalities

Steps

  1. Graph the boundary line ( version of the inequality)
  2. Choose line style: dashed (, ) or solid (, )
  3. Shade the correct side: test a point (usually )

Example:

xy¡3¡2¡112345¡3¡2¡112345y∙¡x+2shadedregion
  • Solid line because (boundary is included)
  • Shaded below because
  • Test : ✓ → is in the shaded region

Quick Rule for Isolated

Inequality Shade
or Above the line
or Below the line

Systems of Linear Inequalities

The solution is the overlapping shaded region (feasible region) of all inequalities.

Example: AND

xy¡3¡2¡112345¡3¡2¡112345y>x¡1y∙¡2x+4¡53;23¢

To find the intersection point of the boundary lines:

SAT Strategies for Graph Questions

  1. "Which point is in the solution set?" — plug each answer choice into ALL inequalities; it must satisfy every one
  2. "Which graph represents the system?" — check line styles (solid/dashed), slopes, and shading direction
  3. "How many solutions does the system have?" — count intersection points visually, or compare slopes
  4. Reading solutions from graphs — the coordinates of the intersection point ARE the solution
  5. Estimating from graphs — the SAT sometimes shows a graph and asks for approximate values; read carefully from the axes
  6. Boundary cases — a point ON a dashed line is NOT in the solution set; on a solid line, it IS

Key Formulas Reference

Task Formula
Slope between two points
x-intercept Set , solve for
y-intercept Set , solve for
Midpoint
Distance
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