using graphs to solve systems (math)

Lesson Introduction

• Topic: Using graphs to solve systems of two linear equations.
• Learning Outcome: By the end of the lesson, the student will be able to graph two linear equations and determine their solution from the graph.

Lesson Goals

• Rewrite linear equations in slope-intercept form.
• Graph both linear equations on a coordinate plane.
• Determine the solution of a linear system from a graph.

Vocabulary

• Substitute (General Academic): A term used universally across subjects.
• Graph (Domain Specific): A visual representation of equations on a coordinate plane.
• Slope-Intercept Form (Domain Specific): A common way to write linear equations, represented as $y = mx + b$ where m is the slope and b is the y-intercept.
• Solution to a System of Equations (Domain Specific): The point(s) where the equations intersect.
• System of Equations (Domain Specific): A set of equations that have common variables.

Lesson Question

• How do you use graphs to solve a system of two linear equations?

Conceptual Overview

• The concept is introduced using a metaphor of a lighthouse aiding ships to navigate; similarly, two lines in a graph meet at a point defined by coordinates.
• If two linear equations are graphed, their intersection represents the solution of the system if they intersect at a single point.

Steps to Solve Systems of Equations Graphically

1. Graphing Equations in Slope-Intercept Form

• Each equation must be converted into slope-intercept form: $y = mx + b$.
• Ensure both equations are in slope-intercept form to facilitate graphing.

Example 1: Graphing Specific Equations

• Equation 1: $y = x + 2$

  • Slope (m): 1
  • Y-Intercept (b): 2
  • Graphing Steps:
  • Plot y-intercept (0, 2).
  • From (0, 2), use slope 1 (rise over run) to find the next points.
    • Up 1, over 1 to (1, 3).
    • Up 1, over 1 to (2, 4).

• Equation 2: $y = 5x - 6$

  • Slope (m): 5
  • Y-Intercept (b): -6
  • Graphing Steps:
  • Plot y-intercept (0, -6).
  • Use slope 5 (rise over run, i.e., 5/1).
    • Up 5 to (-5, -6) and over 1 to (-4, -5).

• Intersection Point: The lines intersect at (2, 4) indicates the system's solution.

2. Checking the Solution

• After graphing $y = -x$ and $y = 2x + 6$, intersection occurs at point (-2, 2).

Steps to Verify the Solution:

1. Substitute x and y into both equations.
2. For the first equation:
   • Substitute: $y = -x$ yields $2 = -(-2)$ (True, as 2=2).
3. For the second equation:
   • Substitute: $y = 2x + 6$ yields $2 = 2(-2) + 6$.
   • Simplifies to $2 = 2$ (True).

• Confirmed as solution (-2, 2).

System Types

• A system of equations can have:
  • One Solution: Lines intersect at one point.
  • No Solution: Lines are parallel (same slope, different intercepts).
  • Infinitely Many Solutions: Lines overlap completely (same slope and y-intercept).

3. Converting Equations to Slope-Intercept Form

Example: Converting and Graphing Equations

• Equation 1: $3x + y = 1$

  • Rearranged to isolate y:
  • $y = -3x + 1$ (in slope-intercept form).

• Equation 2: $2(x + 2y) = -16$

  • Distribution yields:
  • After rearranging and simplifying gives: $y = -\frac{1}{2}x - 4$ (in slope-intercept form).

• Analysis of Slope and Y-Intercept for both equations:

  • First equation slope = -3; y-intercept = 1.
  • Second equation slope = -rac{1}{2}; y-intercept = -4.

4. Graphing Process

• First Line: Start at (0, 1) and plot 3 down and 1 to the right for further points.
• Second Line: Start at y-intercept (0, -4) and plot down 1 to the right 2.
• Intersection Point: Found to be (2, -5).

Verification of Solution

• Substitute into the equations to check.

1. For $3x + y = 1$:
   • Check $3(2) + (-5)$ should equal 1:
   • Verifies as $6 - 5 = 1$ (True).
2. For $2x + 2y = -16$:
   • Check $2(2) + 2(-5)$ should equal -16:
   • Simplifies to $4 - 10 = -6$ (wrong assumption). Correct phrasing needed here - ensure correct substitution.

5. Example: Finding Solution by Graphing

Problem Statement

• Given equations need to be in slope-intercept form to graph easily:

1. Convert: $6x + 3y = 9$
   • $3y = -6x + 9$
   • $y = -2x + 3$.
2. Convert: $x + \frac{1}{2}y = -2$
   • $\frac{1}{2}y = -x - 2$
   • Multiplying through by 2 gives: $y = -2x - 4$.

Graphing Confirmation

• Parallel lines confirmed:
• No solution because lines never intersect.
• Final conclusion drawn from visual result of graphs.

Conclusion

• The lesson encapsulates how to solve systems of equations graphically using slope-intercept form, checks for solutions and recognizes different types of systems based on their graphical representation.