Notes on Graphing Linear Equations

Graphing Linear Equations

Introduction

• This section focuses on analyzing whether given equations are linear or not and how to graph them.

Determining Linear Equations

• An equation is considered linear if it can be expressed in the standard form:
  $Ax + By = C$
  where ( A, B, ) and ( C ) are constants, and both ( x ) and ( y ) are to the first power (no exponents other than 1).

Given Equations and Classification

1. Equation:
   $y = 4 - 3x$

   • Standard Form:
     Rewrite to:
     $3x + y = 4$
   • Classification: Linear
   • Graphing: Yes, this equation can be graphed.

2. Equation:
   $-3y = -1 + 13y$

   • Rearranging:
     $-3y - 13y = -1$
     $-16y = -1$
     $y = \frac{1}{16}$
   • Standard Form:
     $16y = 1$
   • Classification: Linear
   • Graphing: Yes, this equation can be graphed.

3. Equation:
   $6x - x^4 = 4$

   • Classification: Not linear
   • Reason: The presence of the term ( x^4 ) indicates a polynomial of degree 4, thus it cannot be graphed as a linear function.

4. Equation:
   $y = x^2 - 4$

   • Classification: Not linear
   • Reason: The presence of the term ( x^2 ) indicates a quadratic function, which cannot be graphed as a linear function.

5. Equation:
   $3 = y + 8 - 4x$

   • Rearranging:
     $4x + y = 3 - 8$
     $4x + y = -5$
   • Classification: Linear
   • Graphing: Yes, this equation can be graphed.

Conclusion

• The complexity of linear equations varies, and it is crucial to identify the presence of terms with exponents greater than 1, as they denote non-linear functions.
• Each of the linear equations can be graphed, allowing for visual analysis of their relationships.