Common Points - Algebra-1

1. Fundamental Concepts

Graph Definition: A graph in the context of solving systems of equations is a visual representation of the relationship between variables (typically x and y) in an equation. For linear equations, the graph is a straight line, where each point on the line represents a solution to that individual equation.
System of Equations: A set of two or more equations with the same variables. When solving such a system graphically, we focus on finding the relationship between the graphs of these equations.

Solution as Common Points: The solution to a system of equations is the set of all points that lie on all the graphs of the equations in the system. In other words, it is the intersection point(s) where the graphs cross each other.
- For linear systems, there are three possible scenarios:
  - One solution: The lines intersect at exactly one point (consistent and independent system).
  - No solution: The lines are parallel and never intersect (inconsistent system).
  - Infinitely many solutions: The lines are identical, so every point on the line is a solution (consistent and dependent system).
Verification: To confirm a point is a solution, substitute the coordinates of the point into all equations in the system. If the point satisfies all equations, it is a valid solution.

Solve the system graphically:

Solution:

Graph \(y = 2x + 1\) (a line with slope 2 and y-intercept 1) and \(y = 3\) (a horizontal line).
They intersect at \((1, 3)\).
Verification: Substitute \(x = 1\), \(y = 3\) into \(y = 2x + 1\): \(3 = 2(1) + 1 = 3\), which is true.

Solve the system graphically:

Solution:

Graph \(y = x + 2\) (slope 1, y-intercept 2) and \(y = 2x + 1\) (slope 2, y-intercept 1).
They intersect at \((1, 3)\).
Verification: For \(y = x + 2\): \(3 = 1 + 2 = 3\); for \(y = 2x + 1\): \(3 = 2(1) + 1 = 3\), both are true.

Solve the system graphically:

Solution:

Graph \(y = 2x + 1\) (slope 2, y-intercept 1) and \(y = -x - 2\) (slope -1, y-intercept -2).
They intersect at \((-1, -1)\).
Verification: For \(y = 2x + 1\): \(-1 = 2(-1) + 1 = -1\); for \(y = -x - 2\): \(-1 = -(-1) - 2 = -1\), both are true.

Graph Each Equation: For linear equations, determine the slope and y-intercept to plot the lines accurately. For other types of equations, identify key points (e.g., intercepts) to sketch the graph.
Identify Intersection Points: Locate the point(s) where the graphs cross each other. These are the potential solutions.
Verify the Solution: Substitute the coordinates of the intersection point into all equations in the system to ensure they satisfy each equation.
Classify the System: Based on the number of intersection points, determine if the system has one solution, no solution, or infinitely many solutions.