Solving Systems of Equations with Graphing

Solving Systems with Graphing

Warm Up

• Introduction to graphing the line given by the function
  • Function: $f(x) = 3x + 2$

Solving Systems of Equations with Graphing

• Objective: Find values for $x$ and $y$ that satisfy both equations simultaneously.
• Given equations:
  1. $y = 2x + 1$
  2. $y = 4x - 7$
• Note: We will use the graphing method as we learned how to graph equations in the form of $y = mx + b$.

Finding the Solution to a System of Equations

• The solution to the system of equations is identified at the intersection point of two lines on the graph.
• General form of a system of equations:
  • $A x + B y = C$
  • $D x + 2y = F$

Steps to Solve a System of Equations by Graphing

1. Rearrange each equation to the form $y = mx + b$.
2. Graph both lines on the same coordinate plane.
3. Identify the point where the two lines intersect. This point represents the solution to the system.

Example: Solve the System of Equations

Given Equations

• $y = 2x + 1$
• $y = 4x - 7$
• The intersection point (solution) after graphing is (4, 9).

Example: Solve Another System by Graphing

Given Equations

• $3x + y = -1$
• Rearranged:
  • From $3x + y = -1$, subtract $3x$:
  • $y = -3x - 1$
  • From $-3x + y = 5$, add $3x$:
  • $y = 3x + 5$
• Solution (intersection point):
  • $(-1, 2)$

Special Cases in Systems of Equations

1. Identical Lines
   • When two lines are the same, the solution is "all real numbers." This means there are infinitely many solutions since every point on the line satisfies both equations.
2. Parallel Lines
   • When the lines never touch, it indicates that there is no solution. This occurs when two lines have the same slope but different y-intercepts.

Additional Examples

Example: Solve the System of Equations

Given:

• $4x + 2y = 2$
• Rearrangement:
  • $2y = -4x + 2$
  • Thus, $y = -2x + 1$

Example: Solve the System of Equations

Given:

• $6x + 2y = 2$
• Rearrangement:
  • From $6x + 2y = 2$, subtract $6x$:
  • $2y = -6x + 2$
  • Again, rearranging for the second equation $3x + y = 3$ gives:
  • $y = -3x + 3$
• Result is that there is no solution, indicating parallel lines as identified in special case 2 above.