Graphs of Linear Systems and Inequalities: Foundations
[Video Placeholder - Introduction to graphing systems]
Duration: 7:10
Graphing Systems of Equations
A system of linear equations can be solved graphically by finding the intersection point(s) of the lines.
Example 1: One Solution
Graph the system and find the solution:
y = 2x + 1
y = -x + 4
Solution: The lines intersect at (1, 3)
This means x = 1, y = 3 satisfies both equations.
Example 2: No Solution
Graph the system:
y = 3x + 2
y = 3x - 1
Solution: Parallel lines never intersect
This system has no solution (inconsistent).
Try it!
How many solutions does this system have?
y = ½x - 3
2y = x - 6
Graphing Linear Inequalities
To graph linear inequalities:
Graph the corresponding equation (solid line for ≤/≥, dashed for <>)
Test a point not on the line to determine which side to shade
Shade the appropriate region
Example 3: Single Inequality
Graph: y > 2x - 1
1. Graph y = 2x - 1 as a dashed line (since it's > not ≥)
2. Test (0,0): 0 > 2(0)-1 → 0 > -1 (true)
3. Shade the region containing (0,0)
Example 4: System of Inequalities
Graph the system:
y ≤ x + 2
y ≥ -x
x < 3
1. Graph y = x + 2 (solid), shade below
2. Graph y = -x (solid), shade above
3. Graph x = 3 (dashed), shade left
Solution is the overlapping (purple) region
Practice: Graph Matching
Which system matches this graph?
SAT-Style Application
Example 5: Word Problem
A company makes chairs (x) and tables (y). Each chair needs 2 hours of labor, each table needs 4 hours. They have at most 40 labor hours available. They must make at least 5 chairs. Graph the feasible region.
Inequality 1: 2x + 4y ≤ 40 (labor constraint)
Inequality 2: x ≥ 5 (minimum chairs)
Inequality 3: y ≥ 0 (can't make negative tables)
Feasible region is the purple area showing possible production combinations