Lesson 67 Solving and classifying special systems of

Lesson 67 Solving and classifying special systems of linear equations

Inconsistent equations • If no common solution exists, the system consists of inconsistent equations. • The graphs of inconsistent equations never intersect, so they are parallel. • Example: -3 x + y = -4 • y =3 x • Solve by substitution: -3 x + 3 x = -4 • 0 = -4 • THIS STATEMENT IS FALSE, SO THERE IS NO SOLUTION- AN INCONSISTENT SYSTEM

solve • • y = 5 x + 6 y = 5 x

Dependent systems • A dependent system can have an infinite number of solutions. • The equations of a dependent system are called dependent equations, and they have identical solution sets. • Since they have identical solution sets , the equations are the same. • Example: x + 3 y = 6 • 1/3 x + y = 2 • Solve by substitution: x = -3 y+6 • 1/3(-3 y+6) + y = 2 • -y +2 + y = 2 • 2=2 this is a true equation, this means the original equations are true for all values of the variables- there are infinitely many solutions

Method 2 • You can write both equations in slope intercept form. If the equations are exactly the same, then the system is dependent • x + 3 y = 6 3 y = -x + 6 y = -1/3 x + 2 • 1/3 x + y = 2 y = - 1/3 x + 2 • equations in slope-intercept form are identical, so the system is dependent

practice • Solve: • x - 4 y = 8 • 1/4 x - y = 2

Consistent system • A consistent system will have at least 1 common solution. • An independent system will have exactly 1 solution. • An independent system, is also a consistent system. • Graphs of independent systems will intersect in 1 point.

Systems of linear equations • Consistent • Independent consistent dependent • Exactly 1 solution infinitely many solutions no solution • Graphs intersect • in 1 point graphs are same line inconsistent lines are parallel

Classifying systems of equations • Determine if each system is consistent & independent, consistent & dependent, or inconsistent • x - 1/4 y = 3/4 • 2 x+y = 1 y = -2 x + 1 • Solve the system by substitution: • x - 1/4(-2 x+1) = 3/4 • x+ 1/2 x - 1/4 = 3/4 • 3/2 x - 1/4 = 3/4 • + 1/4 • 3/2 x = 4/4 • (2/3)(3/2 x) = 1(2/3) • x = 2/3 y= -2(2/3) + 1 y = -1/3 • Solution is ( 2/3, -1/3) there is exactly 1 solution , so system id consistent and independent

Classify system • • • 3 x + y = 2 y = -3 x -4 Write in slope intercept form y = -3 x +2 y = -3 x-4 Slopes are the same, so the lines are parallel. • There is no solution, the systems is inconsistent

practice • Determine whether each system is consistent & independent, consistent 7 dependent, or inconsistent: • x- 7 y = 3 • 2 x-5 y = 15 • 8 x-5 y= 16 • y = 4 x -5