Systems of Linear Equations

Systems of Linear EquationsAS OcampoSystem of equations or simultaneous equationsSimultaneous solution or root of equation or solution setKinds of Systems of Equationa. Consistent Equations- have a unique solutionSolution setNo point of intersectionInfinite points of intersectionb. Inconsistent Equations- have no solutionc. Dependent Equations- have an infinite number of solutionsIntersecting linesparallel linescoinciding linesMethods commonly used in finding the solution of a system of linear equationsGRAPHICAL METHOD OF SOLVING A SYSTEM OF LINEAR EQUATIONS

Example 1. Find the solution set of the following system of linear equations using the graphical method.

2x + 2y = 12 and 4x 2y = 0Solution: Use the intercept method(0,6)(2,4)(6,0)(1,2)(0,0)4x 2y = 02x +2y = 12xy06602x +2y = 12xy00124x 2y = 0YXCheck by substituting this values in the equation and see if both equations are satisfied. Solution set: (2,4)

Check:2x + 2y= 122 (2) + 2 (4)= 12 12= 12

4x 2y= 04 (2) 2 (4)= 0 0= 0

Solution: Use the intercept method(0,4)(4,0)(8,0)(0,2)0x + 2y = 4x +2y = 8xy0480x +2y = 8xy0240x + 2y = 4YXSolution: Use the intercept method(4,0)(0,2)0x + 2y = 4xy0240x +2y = 4xy02403x + 6y = 12YX3x + 6y = 12ANALYTICAL SOLUTION OF SYSTEMS OF LINEAR EQUATIONSExample 1. Solve analytically2x + 2y = 12 and 4x 2y = 0Solution: To solve for x: Use addition 2x + 2y = 12(+) 4x 2y = 0 6x = 12 x = 2To solve for y: Use subtraction2x + 2y = 12 (Multiply by 2) 4x + 4y = 24( - )4x 2y = 0 ( - ) 4x 2y = 06y = 24Solution set: (2,4) y = 4