REVIEW OF      CARTESIAN COORDINATE            SYSTEM    Cartesian Coordinate System consists of:         two coplanar per...
REVIEW OF CARTESIAN COORDINATE       SYSTEMCartesian Coordinate System consists of:    four regions called quadrants      ...
SYSTEMS OF LINEAR      EQUATIONS IN TWO         VARIABLESA system of linear equations in two variables refers totwo or mor...
SYSTEMS OF LINEAR     EQUATIONS IN TWO        VARIABLESJust like in solving the linear equations, the system of linearequa...
SYSTEMS OF LINEAR      EQUATIONS IN TWO         VARIABLESFrom the previous example, check whether the ordered pair (1,2)is...
SYSTEMS OF LINEAR     EQUATIONS IN TWO        VARIABLES For Eq. 2:                         Since both equations hold,   2x...
SYSTEMS OF LINEAR EQUATIONS IN TWO    VARIABLES   2x + y = 4                        x–y=–1                (2,1)
Determine whether the given point is a solution of thegiven system of linear equations. a. (3,-1)       x–y=4             ...
y = -2x + 5                       x–y=4              (3,-1)
Determine whether the given point is a solution of thegiven system of linear equations. b. (- 1,- 3)       2x – y = 1     ...
(-1,-3)
DIFFERENTSYSTEMS OF   LINEAREQUATIONS
Geometrically, solutions of systems of linear equations are  the points of intersection of the graph of the equations.    ...
CONSISTENT - INDEPENDENT         SYSTEMintersecting                exactly one    lines                    (unique)       ...
CONSISTENT - DEPENDENT         SYSTEMcoinciding                infinitely   lines                    many                 ...
INCONSISTENT   SYSTEMparallel lines               no solution                a1 b1 c1                  = ≠                ...
Without graphing, identify the kind of system, and state   whether the system of linear equations has exactly one   soluti...
ASSIGNMENT: • Look for the methods on how to solve the solutions of the systems of linear equations.                      ...
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  1. 1. REVIEW OF CARTESIAN COORDINATE SYSTEM Cartesian Coordinate System consists of: two coplanar perpendicular number lines y-axis or the vertical line x-axis or thevertical line . origin
  2. 2. REVIEW OF CARTESIAN COORDINATE SYSTEMCartesian Coordinate System consists of: four regions called quadrants Quadrant II Quadrant I (–,+) (+,+) . Quadrant III Quadrant IV (–,–) (+,–)
  3. 3. SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLESA system of linear equations in two variables refers totwo or more linear equations involving two unknowns,for which, values are sought that are common solutionsof the equations involved.Example: x–y=–1 (Eq. 1) 2x + y = 4 (Eq. 2)
  4. 4. SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLESJust like in solving the linear equations, the system of linearequations also have their solutions, wherein this time, thesolution is an ordered pair that makes both equations true.To check whether the given ordered pair is the solution for thesystem, simply substitute the values of x and y to theequations then see whether both equations hold. (If the leftside of the equation is equal to its right side)
  5. 5. SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLESFrom the previous example, check whether the ordered pair (1,2)is the solution to the system. For Eq. 1: Remember: x – y = – 1 ; (1,2) It is not enough to check (1)– (2) = – 1 whether the given order – 1 = –1  pair is true in one of the given equations. You still have to check the other Eq. 1 is true in the equation to see if both ordered pair (1,2) equations hold.
  6. 6. SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES For Eq. 2: Since both equations hold, 2x + y = 4 ; (1,2) this implies that the point 2(1) +(2) = 4 (1,2) is a common point of 2 +2 =4 the lines whose equations are x – y = – 1 & 2x + y = 4. 4=4  Eq. 2 is also true inthe ordered pair (1,2) Hence, (1,2) is the point of intersection of the lines.
  7. 7. SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES 2x + y = 4 x–y=–1 (2,1)
  8. 8. Determine whether the given point is a solution of thegiven system of linear equations. a. (3,-1) x–y=4 (Eq.1) y = – 2x + 5 (Eq. 2) For Eq. 1: For Eq. 2: x – y =4 y = - 2x + 5 (3) – (-1) = 4 (-1) = - 2(3) + 5 3+1=4 -1 = -6 + 5 4=4  -1 = -1  Since both of the equations hold, the solution of the given system of linear equations is (3,-1).
  9. 9. y = -2x + 5 x–y=4 (3,-1)
  10. 10. Determine whether the given point is a solution of thegiven system of linear equations. b. (- 1,- 3) 2x – y = 1 (Eq.1) 2x + y = 5 (Eq. 2) For Eq. 1: For Eq. 2: 2x – y = 1 ; (-1,-3) 2x + y = 5 ; (-1,-3) 2(-1) – (-3) = 1 2(-1) + (-3) = 5 -2 + 3 = 1 -2 – 3 = 5 1=1 -5≠-5 Since one of the equations doesn’t hold, the lines of the equations will not meet @ point (-1,-3)
  11. 11. (-1,-3)
  12. 12. DIFFERENTSYSTEMS OF LINEAREQUATIONS
  13. 13. Geometrically, solutions of systems of linear equations are the points of intersection of the graph of the equations. INDEPENDENT CONSISTENTSYSTEMS OF DEPENDENT LINEAREQUATIONS INCONSISTENT
  14. 14. CONSISTENT - INDEPENDENT SYSTEMintersecting exactly one lines (unique) solution a1 b1 c1 a2 ≠ b ≠c 2 2
  15. 15. CONSISTENT - DEPENDENT SYSTEMcoinciding infinitely lines many solutions a1 = b1 = c1 a2 b2 c2
  16. 16. INCONSISTENT SYSTEMparallel lines no solution a1 b1 c1 = ≠ a2 b2 c2
  17. 17. Without graphing, identify the kind of system, and state whether the system of linear equations has exactly one solution, no solution or infinitely many solutions.a. x + 2y = 7 1 2 7 *consistent – independent 2x + y = 4 2 ≠ 1≠ 4 *one unique solutionb. 4x = -y – 9 4 1 -9 *inconsistent 2y = -8x – 5 8 = 2≠ -5 *no solutiona. 3x + 4y = -12 3 4 -3 *consistent – dependent y = - ¾x – 3 ¾ = 1= -3 *one unique solution
  18. 18. ASSIGNMENT: • Look for the methods on how to solve the solutions of the systems of linear equations. END…

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