Linear Equations in Two Variables
Graphing and Solving Linear Equations in Two Variables
Learning Standards
Content Standard
The learner demonstrates understanding of key concepts of algebraic expressions and linear equations
Performance Standard
The learner is able to model situations using linear equations and solve these algebraically and graphically
Learning Competency
Graph linear equations in two variables and find solutions
Code: M8AL-IIa-b-1
Complete Lesson Plan
Learning Objectives
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Identify linear equations in two variables
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Find solutions (ordered pairs) of linear equations
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Graph linear equations on the Cartesian plane
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Determine if a point is a solution to a linear equation
Lesson Procedures
motivation
Show a real scenario: 'A cellphone plan costs ₱50 plus ₱2 per minute. The equation is: C = 2m + 50, where C = total cost, m = minutes. How can we show all possible costs?'
presentation
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What is a Linear Equation in Two Variables?
General form: Ax + By = C
Examples:
- 2x + y = 5
- x - 3y = 6
- y = 2x + 1
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Finding Solutions (Ordered Pairs)
Example: 2x + y = 6
Method: Choose values for x, solve for y
| x | 2x + y = 6 | y | (x, y) | |---|------------|---|--------| | 0 | 2(0) + y = 6 | 6 | (0, 6) | | 1 | 2(1) + y = 6 | 4 | (1, 4) | | 2 | 2(2) + y = 6 | 2 | (2, 2) | | 3 | 2(3) + y = 6 | 0 | (3, 0) |
All these points are solutions!
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Graphing Linear Equations
Steps:
- Find at least 3 solutions (ordered pairs)
- Plot the points on the Cartesian plane
- Connect the points with a straight line
- Extend the line in both directions (use arrows)
Example: Graph y = 2x + 1
| x | y = 2x + 1 | (x, y) | |---|------------|--------| | -1 | 2(-1) + 1 = -1 | (-1, -1) | | 0 | 2(0) + 1 = 1 | (0, 1) | | 1 | 2(1) + 1 = 3 | (1, 3) | | 2 | 2(2) + 1 = 5 | (2, 5) |
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Slope-Intercept Form: y = mx + b
- m = slope (steepness of line)
- b = y-intercept (where line crosses y-axis)
Example: y = 3x - 2
- Slope (m) = 3
- Y-intercept (b) = -2
- Graph starts at (0, -2)
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Testing if a Point is a Solution
Is (2, 5) a solution to y = 2x + 1?
- Substitute: 5 = 2(2) + 1
- 5 = 4 + 1
- 5 = 5 ✓ YES, it's a solution
Is (3, 6) a solution?
- 6 = 2(3) + 1
- 6 = 7 ✗ NO, not a solution
generalization
Questions:
- What is a linear equation in two variables?
- How do we find solutions?
- What does the graph of a linear equation look like?
- How can we check if a point is a solution?
- What is slope-intercept form?
guided practice
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Find solutions together:
Equation: x + y = 4
Complete the table: | x | y | (x, y) | |---|---|--------| | 0 | ? | (0, ?) | | 1 | ? | (1, ?) | | 2 | ? | (2, ?) |
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Graph the equation together on the board
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Test points:
- Is (5, -1) a solution to x + y = 4?
- Is (2, 2) a solution?
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Practice with: y = -x + 3
- Find 3 solutions
- Plot and graph
independent practice
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Activity 1: Find 4 solutions for each equation:
- x + y = 7
- 2x - y = 4
- y = 3x - 1
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Activity 2: Graph these equations on separate coordinate planes:
- y = x + 2
- y = -2x + 3
- x + y = 5
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Activity 3: Determine if the ordered pair is a solution:
- (3, 4) for x + y = 7: ____
- (2, -1) for y = 2x - 5: ____
- (0, 5) for 2x + y = 5: ____
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Activity 4: Write the equation in slope-intercept form and identify m and b:
- 2x + y = 6
- 3x - y = 9
preliminary activities
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Prayer and greetings
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Review: Cartesian coordinate system (x-axis, y-axis, quadrants)
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Quick drill: Plot points on coordinate plane
Assessment
answers
- (0,8), (1,6), (2,4) or any valid solutions
- Points: (0,4), (1,3), (2,2), etc. with correct line
- Yes (2 = 3(5) - 13 → 2 = 2)
- y = -2x + 5
- C = 15k + 40 (with correct graph)
evaluation
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Answer:
- Find 3 solutions for: 2x + y = 8
- Graph: y = -x + 4 (show at least 3 points)
- Is (5, 2) a solution to y = 3x - 13?
- Write in slope-intercept form: 4x + 2y = 10
- Word Problem: A taxi charges ₱40 base fare plus ₱15 per kilometer. Write a linear equation and graph it.
Materials & Resources
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Graphing paper/Cartesian plane worksheets
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Rulers
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Colored markers or pencils
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Calculator
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Manila paper with coordinate grids
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Graph board for demonstration
assignment
Homework:
- Find 5 solutions and graph: y = 2x - 3
- Solve textbook exercises on page ___
- Create your own linear equation word problem
Remarks:
- Emphasize accuracy in plotting points
- Encourage use of ruler for straight lines
subject matter
Topic: Linear Equations in Two Variables
Key Concepts:
- Linear equation in two variables: Ax + By = C (where A, B, C are constants)
- Variables: Usually x and y
- Solution: An ordered pair (x, y) that makes the equation true
- Graph: A straight line on the Cartesian coordinate plane
- Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
Materials:
- Cartesian plane/graphing paper
- Ruler
- Colored markers
- Calculator
- Manila paper with coordinate grids