Systems Sleuths Packet
Systems Sleuths Investigation
Case File: The Mystery of the Intersection
Name:
Date:
Do Now: Detect the Solution
Circle the point where the lines cross and write the (x, y) coordinates.
Solution: (____ , ____)
Solution: (____ , ____)
Key Investigative Fact:
A system of equations is when we have two lines at once. The solution is the exact point (x, y) where those two lines cross paths.
Method 1: Graphing
Guided Practice #1
y = 1x + 2
y = -1x + 4
Line 1: y = 1x + 2
Slope (m): _________
Intercept (b): _________
Line 2: y = -1x + 4
Slope (m): _________
Intercept (b): _________
Final Intersection
( ____ , ____ )
Start at b on the y-axis. Use slope to move to the next point.
Method 2: Substitution
Guided Practice #2
y = 2x
x + y = 9
Step 1: Replace y with (2x)
x + ( ________ ) = 9
Step 2: Solve for x
x = ________
Step 3: Solve for y
y = ________
Solution: ( ____ , ____ )
Method 3: Elimination
Guided Practice #3
3x + y = 10
2x - y = 5
Step 1: Add the Columns
3x + y = 10
2x - y = 5
5x + 0 = 15
Step 2: Solve for x
5x = 15
x = ________
Step 3: Solve for y
y = ________
Solution: ( ____ , ____ )
Independent Practice
1. Graph the system: y = x + 1 and y = 2x - 1
Solution: ( ____ , ____ )
2. Graph the system: y = -x + 3 and y = x - 1
Solution: ( ____ , ____ )
3. Substitution: y = 4 and x + y = 10
Solution: ( ____ , ____ )
4. Substitution: x = 2y and x + y = 6
Solution: ( ____ , ____ )
5. Elimination: x + y = 10 and x - y = 4
Solution: ( ____ , ____ )
6. Elimination: 3x + y = 13 and x - y = -1
Solution: ( ____ , ____ )
Final Case: Exit Ticket
y = x + 4
x + y = 10
Which method did you use? Why?
Graphing
Substitution
Elimination
Investigation Result
x = ____ , y = ____
Solution Key Teacher Guide
L1: Graphing intersections
L2: Solution as a point
5 mins + 5 mins Review
Systems Sleuths Investigation
Case File: The Mystery of the Intersection
Name: TEACHER KEY
Date: 05/08/2026
T
AT
FD
Do Now: Detect the Solution
TT Spotting Solutions, CC S/O
Circle the point where the lines cross and write the (x, y) coordinates.
Solution: (2 , 2)
Solution: (1 , -2)
BID: Intersection = Solution
Key Investigative Fact:
A system of equations is when we have two lines at once. The solution is the exact point (x, y) where those two lines cross paths.
L1: Slope-Intercept Review
L2: Graphing systems
10 mins
Method 1: Graphing
T
AT
FD
Guided Practice #1
y = 1x + 2
y = -1x + 4
Line 1: y = 1x + 2
Slope (m): 1
Intercept (b): 2
Line 2: y = -1x + 4
Slope (m): -1
Intercept (b): 4
Final Intersection
( 1 , 3 )
Start at b on the y-axis. Use slope to move to the next point.
BID: Check b first!
L1: Identify isolated var
L2: Substitute expression
10 mins
Method 2: Substitution
T
AT
FD
Guided Practice #2
y = 2x
x + y = 9
Step 1: Replace y with (2x)
x + ( 2x ) = 9
Combine: 3x=9
Step 2: Solve for x
3x = 9
x = 3
x = 3
Plug x=3 into eq 1
Step 3: Solve for y
y = 2(3)
y = 6
y = 6
Solution: ( 3 , 6 )
L1: Line up columns
L2: Identify opposites
10 mins
Method 3: Elimination
T
AT
FD
Guided Practice #3
3x + y = 10
2x - y = 5
BID: +y/-y cancel!
Step 1: Add the Columns
3x + y = 10
2x - y = 5
5x + 0 = 15
Step 2: Solve for x
