exploring systems of linear equations ( math)
Introduction to Systems of Linear Equations (math)
Explore systems of linear equations.
Lesson Objective: By the end, be able to determine if a point is a solution of a system of two linear equations.
Specific Goals:
Determine if a point is a solution of a system of two linear equations.
Identify the solution of a system of two linear equations from a graph and verify the solution algebraically.
Vocabulary Words
Substitute (green): A general academic term used across subjects.
Intersect (purple): The point where two lines meet in a graph.
Ordered pair (purple): A pair of numbers (x, y) that represent a point on a Cartesian plane.
Solution to a system (purple): The ordered pair that satisfies all equations in the system.
System of linear equations (purple): A set of two or more linear equations with the same variables (e.g., x and y).
Key Question
How can you determine if a point is a solution to a system of linear equations?
Real-World Example
Scenario: Buying fruit with a specific amount of money. This represents many real-world situations where systems of equations apply.
Review of Linear Equations
Solution of a Linear Equation (Example): Solve the equation 3x + y = 6.
For a single variable, solutions are expressed as $x = k$.
For two variables: Solutions are ordered pairs ($x, y$) that satisfy the equation.
Example Solution: Graph of the equation shows a line of solutions, each satisfying the equation.
Testing a Solution for a Linear Equation
Testing Point (1, 3) in Equation 3x + y = 6:
Substitute into the equation: 3(1) + 3 = 6.
Calculation: 3 + 3 = 6.
True statement confirmed: (1, 3) is a solution for the equation.
Systems of Linear Equations
Definition: A system of linear equations consists of two or more linear equations with shared variables (e.g., x, y).
A solution of a system will also be presented as an ordered pair $(x, y)$, which must satisfy both equations.
Example: Testing a Solution to a System of Equations
Given equations:
3x + y = -1
y = -x + 1
Tested solution point: (-1, 2).
Testing in the First Equation:
Substitute values: 3(-1) + 2 = -1.
Calculations: -3 + 2 = -1
True statement: Thus, it satisfies the first equation.
Testing in the Second Equation:
Substitute values: 2 = -(-1) + 1.
Calculations: 2 = 1 + 1
Confirmed true: therefore, it satisfies the second equation.
Conclusion: The point (-1, 2) is a solution to the system of equations.
Solutions vs. Non-Solutions
Example of a solution: (1, 3)
Example of a non-solution: (-2, -3)
Rationale for (1, 3):
Substitute into both equations:
1st equation: x = 1 gives true statement.
2nd equation: x = 1 verifies as well.
Rationale for (-2, -3):
Substitute into both equations:
1st equation: x = -2 leads to a true statement.
2nd equation: x = -2 yields a false stmt (i.e., not satisfying both equations).
Graphical Representation
Example of intersection of linear equations:
Graphing shows where two lines intersect, representing the solution to the system.
Each point on a line is a solution for that equation; hence intersection point is the common solution for both equations.
Example: Intersection point (-1, 2) confirmed as the solution.
Verifying Solution from Graph
Regardless of graph accuracy, represented solutions can be approximated but should be checked algebraically.
Real-World Application Example
Scenario with Helene organizing a party with fruit:
Goal: 10 pieces of fruit representing apples and oranges, with $8 total budget.
Equations derived:
x + y = 10 (total pieces of fruit)
x + 0.5y = 8 (total cost)
Variables: x = number of apples, y = number of oranges.
Find the solution by graphing the equations to identify where they intersect.
Found intersection point: (6, 4).
Understanding Solution Points
Result: Helene buys 6 apples and 4 oranges.
Verification:
For first equation: 6 + 4 = 10 (true statement).
For second equation: costs yield 6 + 0.5(4) = 8 (true statement).
Confirmed: Solution meets both logical and budget constraints.