Take Note Concept Summary: Graphical Solutions of Linear Systems

Problem 4 Classifying a System Without Graphing

Without graphing, is the system independent, dependent, or inconsistent?

{ 4 y − 2 x = 6 8 y = 4 x − 12 left brace . table with 2 rows and 1 column , row1 column 1 , 4 y minus 2 x equals 6 , row2 column 1 , 8 y equals 4 x minus 12 , end table

Rewrite each equation in slope-intercept form. Compare slopes and y-intercepts.

4 y − 2 x = 6 8 y = 4 x − 12 y = 1 2 x + 3 2 y = 1 2 x − 3 2 m = 1 2 ; y -intercept is  3 2 m = 1 2 ; y -intercept is  − 3 2 table with 3 rows and 4 columns , row1 column 1 , 4 y minus 2 x , column 2 equals 6 , column 3 8 y , column 4 equals 4 x minus 12 , row2 column 1 , y , column 2 equals , 1 half , x plus , 3 halves , column 3 y , column 4 equals , 1 half , x minus , 3 halves , row3 column 1 , m , column 2 equals , 1 half , semicolon y . minusintercept is . 3 halves , column 3 m , column 4 equals , 1 half , semicolon y . minusintercept is . minus , 3 halves , end table

The slopes are equal and the y-intercepts are different. The lines are different but parallel. The system is inconsistent.

✓ Got It?

4. Without graphing, is each system independent, dependent, or inconsistent?
   1. { − 3 x + y = 4 x − 1 3 y = 1 left brace . table with 2 rows and 1 column , row1 column 1 , negative 3 x plus y equals 4 , row2 column 1 , x minus , 1 third , y equals 1 , end table
   2. { 2 x + 3 y = 1 4 x + y = − 3 left brace . table with 2 rows and 1 column , row1 column 1 , 2 x plus 3 y equals 1 , row2 column 1 , 4 x plus y equals negative 3 , end table
   3. { y = 2 x − 3 6 x − 3 y = 9 left brace . table with 2 rows and 1 column , row1 column 1 , y equals 2 x minus 3 , row2 column 1 , 6 x minus 3 y equals 9 , end table