System Gridlock Worksheet
Algebra 2 & Honors Unit 2: Linear Systems
Solving Linear Systems by Graphing
Graph each system, determine the exact point of intersection, and classify system consistency.
Name:
Date: Pd:
Score:
One Solution \((x,y)\)
Intersecting lines (\(m_1 \neq m_2\)).
Consistent & Independent
No Solution \((\emptyset)\)
Parallel lines (\(m_1 = m_2, b_1 \neq b_2\)).
Inconsistent
Infinitely Many Sol.
Coincident lines (\(m_1 = m_2, b_1 = b_2\)).
Consistent & Dependent
Part I: Standard & Slope-Intercept Systems
Rewrite in slope-intercept form if needed, graph, and state the solution.
Problem 1 Slope-Intercept
\(\begin{cases} y = 2x - 3 \\ y = -\frac{1}{2}x + 2 \end{cases}\)
L1: \(m = \_\_\_\), \(b = \_\_\_\)
L2: \(m = \_\_\_\), \(b = \_\_\_\)
-4 -2 2 4 6 4 2 -2 -4 -6 x y
Solution: ( , )
□ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 2 Standard Form
\(\begin{cases} 3x + 2y = 6 \\ x - 2y = -6 \end{cases}\)
L1: \(y = \_\_\_\_\_\_\_\_\)
L2: \(y = \_\_\_\_\_\_\_\_\)
-4 -2 2 4 6 4 2 -2 -4 -6 x y
Solution: ( , )
□ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 3 Standard Form
\(\begin{cases} x + y = 1 \\ 2x - y = 5 \end{cases}\)
L1: \(y = \_\_\_\_\_\_\_\_\)
L2: \(y = \_\_\_\_\_\_\_\_\)
-4 -2 2 4 6 4 2 -2 -4 -6 x y
Solution: ( , )
□ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 4 Special Case
\(\begin{cases} y = -\frac{2}{3}x - 1 \\ 2x + 3y = 9 \end{cases}\)
L1: \(m = \_\_\_\), \(b = \_\_\_\)
L2: \(y = \_\_\_\_\_\_\_\_\)
-4 -2 2 4 6 4 2 -2 -4 -6 x y
Solution: ( , )
□ Cons. Indep. □ Cons. Dep. □ Inconsistent
System Gridlock • Algebra 2 / Honors Algebra 2 Document Page 1 of 2
Part II & Honors Extensions
Special Cases & Bounded Geometry
Student Name:
Problem 5 Transform & Compare
\(\begin{cases} 2x - 4y = 8 \\ -x + 2y = -4 \end{cases}\)
L1: \(y = \_\_\_\_\_\_\_\_\)
L2: \(y = \_\_\_\_\_\_\_\_\)
-44 4-4 x y
Solution: _________________
□ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 6 Vertical Line Mix
\(\begin{cases} x = -3 \\ 2x - 3y = 6 \end{cases}\)
L1: \(x = -3\) (vertical)
L2: \(y = \_\_\_\_\_\_\_\_\)
-44 4-4 x y
Solution: ( , )
□ Cons. Indep. □ Cons. Dep. □ Inconsistent
Honors Level
Multi-Line Bounded Systems & Algebraic Parameters
Problem 7: Enclosed Triangular Region Coordinate Geometry
Graph the three boundary lines on the grid to enclose a triangle:
\(L_1: y = x + 3\) • \(L_2: y = -x + 5\) • \(L_3: y = -1\)
a) Determine all 3 vertices of the triangle:
\(A:\) ( ___, ___ )
\(B:\) ( ___, ___ )
\(C:\) ( ___, ___ )
b) Calculate the exact area of the bounded triangle:
Base \(b =\) _____ Height \(h =\) _____ Area = _________ sq units
-4-2246 42-2-4-6 xy
Problem 8: Parameter Analysis (Honors Extension) Algebraic Reasoning
Consider the linear system containing unknown constant \(k\): \(\begin{cases} 4x - ky = 16 \\ 2x + 3y = 6 \end{cases}\)
a) Find \(k\) such that the system is inconsistent:
Value of \(k\): \(k =\)
b) Can \(k\) be chosen so the system is dependent? Explain:
Possible? Circle: YES / NO
System Gridlock • Algebra 2 / Honors Algebra 2 Document Page 2 of 2
System Gridlock Answer Key
Teacher Resource Answer Key & Solutions
Solving Linear Systems by Graphing
Complete graphical plots, algebraic verifications, and classification criteria.
Unit 2: Systems of Equations
Course: Algebra 2 & Honors
Total Points: 100% / Key
One Solution \((x,y)\)
Lines intersect at exactly 1 point.
✓ Consistent & Independent
No Solution \((\emptyset)\)
Same slope, different intercepts.
✓ Inconsistent
Infinitely Many Sol.
Equations represent the same line.
✓ Consistent & Dependent
Part I: Standard & Slope-Intercept Solutions
Line 1 Line 2 Intersection
Problem 1 Solution One Solution
\(\begin{cases} \color{#2563eb}{y = 2x - 3} \\ \color{#dc2626}{y = -\frac{1}{2}x + 2} \end{cases}\)
L1: \(m = 2\), \(b = -3\)
L2: \(m = -\frac{1}{2}\), \(b = 2\)
-44 4-4 (2, 1)
Solution: ( 2 , 1 )
■ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 2 Solution One Solution
\(\begin{cases} 3x + 2y = 6 \\ x - 2y = -6 \end{cases}\)
L1: \(y = -\frac{3}{2}x + 3\)
L2: \(y = \frac{1}{2}x + 3\)
-44 4-4 (0, 3)
Solution: ( 0 , 3 )
■ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 3 Solution One Solution
\(\begin{cases} x + y = 1 \\ 2x - y = 5 \end{cases}\)
L1: \(y = -x + 1\)
L2: \(y = 2x - 5\)
-44 4-4 (2, -1)
Solution: ( 2 , -1 )
■ Cons. Indep. □ Cons. Dep. □ Inconsistent
Problem 4 Solution No Solution
\(\begin{cases} y = -\frac{2}{3}x - 1 \\ 2x + 3y = 9 \end{cases}\)
L1: \(m = -\frac{2}{3}\), \(b = -1\)
L2: \(y = -\frac{2}{3}x + 3\)
-44 4-4 Parallel Lines
Solution: No Solution (\(\emptyset\))
□ Cons. Indep. □ Cons. Dep. ■ Inconsistent
System Gridlock Answer Key • Algebra 2 / Honors Key Page 1 of 2
Teacher Resource • Solutions Guide
Special Cases & Honors Extensions Solutions
Module 2 • Page 2 Answers
Problem 5 Solution Infinitely Many
\(\begin{cases} 2x - 4y = 8 \\ -x + 2y = -4 \end{cases}\)
L1: \(y = \frac{1}{2}x - 2\)
L2: \(y = \frac{1}{2}x - 2\)
-44 4-4 Coincident Lines
Solution: Infinitely Many Solutions
□ Cons. Indep. ■ Cons. Dep. □ Inconsistent
