Grid Lock Graphing Worksheet
Algebra I Module 4 • Systems of Equations
Grid Lock: Systems by Graphing
Name: ___________________
Date: _____________
Period: _________________
Score: _____ / 20
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One Solution \((x, y)\)
Lines have different slopes. They intersect at exactly one coordinate point.
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No Solution \((\emptyset)\)
Lines have the same slope but different y-intercepts. Lines are parallel.
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Infinitely Many Solutions
Lines have the same slope and same y-intercept. Lines coincide.
Guided Model Solve by Graphing: \(\begin{cases} y = -\frac{1}{2}x + 3 \\ y = x - 3 \end{cases}\)
Step-by-step verification
Line 1:
\(y = -\frac{1}{2}x + 3\) → Start at y-intercept \((0, 3)\). Use slope \(m = -\frac{1}{2}\) (down 1, right 2).
Line 2:
\(y = x - 3\) → Start at y-intercept \((0, -3)\). Use slope \(m = \frac{1}{1}\) (up 1, right 1).
Intersection: \((4, 1)\)
Check: \(1 = -\frac{1}{2}(4)+3 = 1\) ✓ | \(1 = (4)-3 = 1\) ✓
x y (4, 1)
Level 1 Slope-Intercept Form (\(y = mx + b\)) • Identify slope and y-intercept, graph both lines, label intersection
Problem 1 \(\begin{cases} y = 2x - 4 \\ y = -x + 2 \end{cases}\)
Line 1: \(y = 2x - 4\)
Slope \(m\) = _______
y-int \(b\) = (0, _____)
Line 2: \(y = -x + 2\)
Slope \(m\) = _______
y-int \(b\) = (0, _____)
x y 4 -4 4 -4
Solution: ( _____ , _____ ) Check:
Problem 2 \(\begin{cases} y = \frac{1}{2}x + 1 \\ y = -x + 4 \end{cases}\)
Line 1: \(y = \frac{1}{2}x + 1\)
Slope \(m\) = _______
y-int \(b\) = (0, _____)
Line 2: \(y = -x + 4\)
Slope \(m\) = _______
y-int \(b\) = (0, _____)
x y 4 -4 4 -4
Solution: ( _____ , _____ ) Check:
Unit 4: Systems of Linear Equations Page 1 of 2 • Practice Continued on Reverse
Level 2 & 3
Converting Forms, Special Cases & Real-World Modeling
Page 2 of 2
Problem 3: Standard Form \(\begin{cases} x + y = 3 \\ y = 2x - 3 \end{cases}\)
Step 1: Rewrite \(x + y = 3\) in slope-intercept form:
\(y =\) ______________________
Line 1
\(m\) = _______
\(b\) = (0, _____)
Line 2: \(y = 2x - 3\)
\(m\) = _______
\(b\) = (0, _____)
x y 4 -4 4 -4
Solution: ( _____ , _____ )
Problem 4: Special Case \(\begin{cases} y = -\frac{3}{2}x + 4 \\ 3x + 2y = -2 \end{cases}\)
Step 1: Rewrite \(3x + 2y = -2\) in \(y = mx + b\):
\(y =\) ______________________
Line 1
\(m\) = _______
\(b\) = (0, _____)
Line 2
\(m\) = _______
\(b\) = (0, _____)
x y 4 -4 4 -4
Solution Type: One / None / Infinitely Many
Reason: _________________________________________
Level 3 • Real-World Context The Fitness Club Comparison
Break-Even Analysis
FitZone charges a registration fee of $10 plus $2 per gym visit. Peak Fitness charges $0 registration fee but $4 per gym visit. Let \(x\) = number of visits and \(y\) = total cost in dollars.
a) Write equations:
FitZone: \(y =\) ______________
Peak: \(y =\) ______________
b) Intersection point:
( _____ , _____ )
c) Interpret the solution:
What does the intersection coordinate \((x, y)\) mean in terms of gym visits and total cost?
d) Make a recommendation:
If a student plans to visit the gym 8 times this month, which gym is the better deal? Explain why.
Error Analysis: Spot the Flaw
Marcus graphed a system and saw two lines that never intersect. He wrote down: "The answer is infinitely many solutions." What error did Marcus make?
Self-Confidence Check
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Algebra I • Linear Systems Unit Page 2 of 2
